{"id":50,"date":"2023-05-18T11:13:35","date_gmt":"2023-05-18T15:13:35","guid":{"rendered":"https:\/\/pressbooks.library.upei.ca\/bio3310\/?post_type=chapter&p=50"},"modified":"2024-08-05T11:40:47","modified_gmt":"2024-08-05T15:40:47","slug":"part-4-basic-applied-statistics-using-minitab-21","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/bio3310\/chapter\/part-4-basic-applied-statistics-using-minitab-21\/","title":{"raw":"Part 4: Basic Applied Statistics using Minitab 21","rendered":"Part 4: Basic Applied Statistics using Minitab 21"},"content":{"raw":"
\r\n

Part 4: Basic Applied Statistics using Minitab 21<\/h1>\r\n<\/div>\r\n
\r\n\r\nA note about statistical significance:\r\n\r\n\r\n\r\n
\r\n
\r\n\r\nAll statistical tests are designed to test whether a pattern you see in your data is \u201cstatistically significant\u201d<\/strong>. We say something is statistically significant if our test confirms that the pattern is unlikely to have occurred by chance. To test this we look at the \u201cp-value\u201d\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe p-value is related to the hypotheses about the data:<\/strong>\r\n
    \r\n \t
  • H<\/strong>0<\/strong>\u00a0<\/strong>(Null Hypothesis) is that there is no difference between groups, or no relationship between variables. This is a \u201cno effect\u201d hypothesis<\/li>\r\n \t
  • H<\/strong>A<\/strong>\u00a0<\/strong>(Alternate Hypothesis)\u00a0\u00a0 is that there is a difference or a relationship. This is sometimes called the \u201cactive\u201d hypothesis, since it indicates some sort of effect<\/li>\r\n<\/ul>\r\nTherefore, to interpret your data, you need to examine the graph of the data and clearly state an hypothesis, or you won\u2019t know what the p-value means!<\/strong>\r\n\r\n\r\n\r\n
    \r\n
    \r\n\r\nIf p<0.05, reject your Null Hypothesis<\/strong>\r\n\r\nIf p>0.05, accept your Null Hypothesis<\/strong>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor each of the tests detailed in the next pages, note what the<\/strong> Null Hypothesis is, so that you can determine how to interpret<\/strong> the p-value from the test.<\/strong>\r\n\r\n<\/div>\r\n\"\"\r\n
    \r\n

    Getting started\u00a0<\/strong><\/h2>\r\nDATA<\/strong> ENTRY<\/strong>\r\n\r\nWhen working in a spreadsheet, the common method of entering your data is in adjoining columns. For example, if you have data such as we looked at in class on the crab temperatures, you would put your crab data for time 1 in one column, and your crab data for time 2 in the next column. However, for advanced stats, you must enter your data so that all the responses for a single variable (in this case, temperature) are in a single column, with another column giving the key for the variable.\r\n\r\nNote that data in the \u201cadvanced stats\u201d format are set up so that the responses are given in the columns, and the variables (e.g. whether they ran or not, what their sex was\/is given as a category number in another column. This method of data entry is necessary for most statistical packages.\r\n\r\n\r\n\r\n
    \r\n
    \r\n\r\nNotes<\/strong> about the data examples<\/strong>\r\n\r\nFor this manual, most of the examples will be from a dataset on an imaginary group of students that were asked to take a bunch of measurements on themselves before and after running in place for one minute.\r\n\r\nOne group of students was asked to run in place for a minute, and another group (the control) did not run.\r\n\r\nStudents (in both groups) were asked to take their pulses (in heartbeats per minute) before and after the running exercise, then were asked to indicate whether they were male or female, whether they smoked or not, whether they thought of themselves as active or not, and so on. This data set allows us to illustrate a wide variety of statistical analyses. The dataset is at the back of this manual, and will be placed on your Moodle site, so that you can practice the exercises in this manual, and see what the answers should look like.\r\n
      \r\n \t
    • Ran = 1 means they ran,<\/li>\r\n \t
    • Ran = 2 are the nonrunners (control)<\/li>\r\n \t
    • Smokes = 1 means they smoke<\/li>\r\n \t
    • Sex = 1 means male<\/li>\r\n \t
    • Sex = 2 means female<\/li>\r\n \t
    • Activity levels: 1 = slight, 2 = moderate, 3 = high<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
      \r\n\r\nTo start in Minitab:\r\n
        \r\n \t
      • Open Minitab 21\r\n
          \r\n \t
        • You will see a \u201csession\u201d window on top where you\u2019ll find the record of what you\u2019ve done, as well as the text results of statistical tests.<\/li>\r\n \t
        • The worksheet on the bottom will contain your data.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nNavigate in Minitab through the menus on the upper property toolbar.<\/strong>\r\n\r\n\"\"\r\n\r\nEntering data:<\/strong>\r\n\r\nData can be entered (typed in) directly, or copied from a spreadsheet\r\n
            \r\n \t
          • From spreadsheet: open your spreadsheet to the desired database and select and copy the data. O<\/strong>nly copy the numbers<\/strong>... do not copy column headers (that will designate your columns as text columns, and cause problems when doing the statistical analyses)<\/li>\r\n \t
          • Reenter Minitab, paste the data into the worksheet.<\/li>\r\n \t
          • Name your columns by clicking on the blank space below the column number, and typing in the name<\/li>\r\n \t
          • Save your worksheet onto your data disk or personal drive. It will save as a .mpx file<\/li>\r\n<\/ul>\r\n<\/div>\r\n
            \r\n\r\nSimple Column Statistics\u00a0<\/strong>\r\n\r\nDescriptive statistics:<\/strong>\r\n
              \r\n \t
            • Select Stat <\/strong>from the property bar, and click on \u201cbasic statistics\u201d.<\/li>\r\n \t
            • Select \u201cdisplay descriptive statistics<\/strong>\u201d to see the Display window<\/li>\r\n \t
            • Click on the box labelled \u201cStatistics\u201d to see the range of statistics available.<\/li>\r\n \t
            • The list of variables is on the left and an empty box for the variable(s) to test is on the right.<\/li>\r\n<\/ul>\r\nHighlight the variable you want to test, and click select<\/strong>\r\n\r\nThere is a long list of potential statistics you can have the computer calculate, all in one operation. Check the boxes of all you would like.\r\n
                \r\n \t
              • Click ok <\/strong>to return to the display descriptive stats window<\/li>\r\n \t
              • Click ok <\/strong>again to obtain the data.<\/li>\r\n<\/ul>\r\nYour results will appear in the upper Minitab Session Window. You can copy and paste the results into your word processor spreadsheet if you want to.\r\n\r\n<\/div>\r\n
                \r\n\r\nVariables<\/strong> to<\/strong> choose:<\/strong> Variables must be ordinal (such as Pulse 1, Pulse 2, Weight, Height in this example; Note that your categorical variables won\u2019t work here.\r\n\r\nThis section also allows you to see some basic graphs for your variables, by clicking on \u201cgraphs\u201d rather than\u201cstatistics\u201d\r\n
                  \r\n \t
                • Click on all of these options to see how these graphs can help you get a feel for what your data looks like.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                  \r\n

                  Descriptive statistics for sub-groups within each column of data (e.g. males and females in your group; age groups, etc.)<\/h3>\r\nOptions:\r\n\r\na)\u00a0 cut and paste in your spreadsheet, then copy into Minitab and run analysis twice\r\n\r\nb)\u00a0 Split the data <\/strong>in Minitab (see page 46 for method) and run analysis twice\r\n\r\nc)\u00a0 use the \u201cBy variables<\/strong>\u201d<\/strong> option and run the analyses simultaneously.\r\n\r\ne.g. In the pulses dataset, each of the two pulse columns (Pulse 1, Pulse 2) include groups (runners & non-runners, males and females, smokers and non-smokers). If you want to compare one subgroup to the other within a single column of data, you will need descriptive statistics and normality testing on each subgroup.\r\n\r\nExample: <\/strong>calculate the descriptive statistics for the runners and the non-runners separately, in the pulses2 column (the second pulse rate, measured after running).\r\n
                    \r\n \t
                  • Go into stat <\/strong>on the property bar, and click on \u201cbasic statistics<\/strong>\u201d, then \u201cdisplay descriptive statistics<\/strong>\u201d. Select the Pulse2 column, and click select. Then place your cursor in the By variables <\/strong>window, and select the group variable, Ran<\/li>\r\n<\/ul>\r\nYour output (in the Session box) will have information for both subgroups in your variable (i.e. Ran 1 & 2), and your graphs will also show both subgroups.\r\n\r\nYou can see how this lets you look at multiple subgroups separately without having to do a lot of cutting and pasting. you can choose this option in your \u201cstore descriptive statistics\u201d section as well.\r\n\r\n<\/div>\r\n\"\"\r\n
                    \r\n

                    Normality Testing<\/h3>\r\nMany statistical tests depend on data being parametric (one part of which is normality). Normality testing is a first step for most statistical testing.\r\n\r\n\r\n\r\n
                    \r\n
                    \r\n\r\nThe normal distribution is a type of frequency distribution which has a characteristic bell curve shape with a particular height and width for its mean (average) and Standard Deviation. We can assess normality (by eye) by plotting the frequency distribution of the data, and comparing it to the normal curve that is calculated for a data set with this mean and standard deviation. However, it can be hard to see from the frequency plot so there are other methods to assess normality.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSeveral methods to test normality: Generally use at least two, since not all work well for all data<\/strong>\r\n\r\na.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 frequency histogram<\/em><\/strong>: <\/strong>important to see what data look like, but not very accurate for assessing whether they fit the normal distribution\r\n\r\nb.\u00a0\u00a0\u00a0\u00a0\u00a0 normal probability plot<\/em><\/strong>:<\/strong> This modifies the frequency scale, so that if data are normal, they fall on a straight line. **This is usually the best method to assess normality<\/em><\/strong>\r\n\r\nc.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 determining whether the shape of the curve fits a mathematical range <\/em><\/strong>(based on \u201cskew\u201d (tails) and \u201ckurtosis\u201d (height of curve)<\/strong>). This is a great method to do as a check on the other methods, in case you just aren\u2019t sure of the interpretation.\r\n\r\nd.\u00a0\u00a0\u00a0\u00a0\u00a0 Statistical methods: <\/em><\/strong>These give some comfort because they seem quantitative, but in fact, they are not accurate in many cases so should be used with caution. Several do not work well with small sample sizes, and several don\u2019t work well if there are many \u201ctied values\u201d (the same number repeated frequently in the dataset)\r\n

                    Frequency Distributions<\/h3>\r\nFirst, assess the frequency distribution as a first step to see what the data look like.\r\n
                      \r\n \t
                    • Plot the histogram through the Column Statistics <\/strong>menu as described on p. 83, or through Graph, Histogram, <\/strong>as described on pp. 73-75.<\/li>\r\n \t
                    • In the graphing menu, choose \u201cwith fit\u201d <\/strong>for the histogram with the normal curve.<\/li>\r\n \t
                    • Select your variables as before, then click on \u201cdataview<\/strong>\u201d. Choose the \u201cdistribution tab, and make sure your distribution says \u201cnormal\u201d and click ok<\/li>\r\n<\/ul>\r\n\"\"\r\n\r\n<\/div>\r\n
                      \r\n\r\nDoes your graph match the bell?<\/strong>\r\n

                      The normal probability plot<\/h3>\r\n
                        \r\n \t
                      • Select Graph <\/strong>from the property bar, and then choose Probability plot from the drop-down menu. When prompted, choose the \u201csingle\u201d graph, and click okay<\/li>\r\n \t
                      • Choose your variable as before. The normal probability plot is the default, but if you want to test another distribution (e.g. random), you can click on \u201cdistribution\u201d.<\/li>\r\n<\/ul>\r\nThis is produces a plot of your frequency histogram on a special probability scale, so that if the data are normal, your points should fall on a straight line. (Remember that this is for the entire column of data; if you need a subset of the data, such as males vs females, you\u2019ll need to separate them)\r\n\r\nYou can check this by eye, or you can do a statistical normality test on the data.\r\n\r\n\"\"\r\n\r\n<\/div>\r\nThis plot includes the 95% confidence limit for the line. If the points generally fall along the line and are within the confidence limits lines, then you can assume normality\r\n
                        <\/div>\r\nThe conclusion from this <\/strong>graph is that the data are normal. <\/strong>The dots deviate from the line very slightly, but not by much, and most fall within the confidence limits. We can run through other methods to see if they confirm our impression from the graph.\r\n
                        \r\n

                        Using skew and kurtosis calculations<\/h3>\r\n\"\"Data are normally distributed if the standard error of the skew (SE<\/strong>skew<\/strong>)<\/strong> and the standard error of the kurtosis (SE<\/strong>kurtosis<\/strong>)<\/strong> fall between -1.96 and +1.96.<\/strong>\r\n\r\nMethod: Determine the Standard Error (SE) of the kurtosis and skew from the skew and kurtosis values in the Descriptive stats analysis (see p. 84) <\/strong>(note that some statistical packages do this calculation for you). The equations below provide an approximation of the SE values.\r\n\r\n\"\"\r\n
                          \r\n \t
                        • SEskew\u00a0= Skew \u00f7 \u221a (6\/n)<\/li>\r\n \t
                        • SEkurtosis\u00a0= kurtosis \u00f7 \u221a (24\/n)<\/li>\r\n \t
                        • n=no. of obs.<\/li>\r\n<\/ul>\r\nFor the Pulse 1 column:\r\n
                            \r\n \t
                          • SEskew\u00a0= skew \u00f7 \u221a(6\/n)\u00a0\u00a0 = .43 \u00f7 \u221a (6\/91) = 1.67<\/li>\r\n \t
                          • SEkurt = kurtosis \u00f7 \u221a (24\/n\u00a0 = -.58 \u00f7 \u221a (24\/91) = -1.13<\/li>\r\n<\/ul>\r\nThese both fall between 1.96 and -1.96, so data are statistically normal<\/strong>\r\n\r\nNote: the SE skew value is close to non-normal, as you can see by the bars in the figure looking a bit crowded towards the left side, but it still falls in the statistical range.<\/strong>\r\n\r\n\u00a0<\/strong>This test confirms that the Pulse 1 data are normally distributed<\/strong>\r\n
                              \r\n \t
                            • \u00a0Note: \u201cData is\u201d ??\u00a0\u00a0 \u201cData are\u201d ?? The word \u201cdata\u201d is plural (the singular version is \u201cdatum\u201d) so should always be given with the plural form of the verb when written.\r\n<\/strong><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n
                              \r\n
                              \r\n\r\nUsing Statistical Normality tests:<\/strong>\r\n\r\nOne big problem with statistical normality tests is that they are adversely affected by both small sample sizes and large numbers of \u201ctied values\u201d, i.e. when we have a number of duplicate values in our list of numbers. Tied values often occur if we have a large data set. Therefore the normality test must always be treated with caution, and results should ALWAYS be checked against the normal probability curve.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                              \u00a0<\/strong>Using statistical normality tests<\/h3>\r\n<\/div>\r\n
                              \r\n
                                \r\n \t
                              • Minitab provides 3 statistical normality tests. All three will plot the \u201cnormal probability curve\u201d as part of the analysis so you can compare the test result to the graph.<\/li>\r\n \t
                              • The purpose of the tests is to see whether the dots are significantly different from the line.<\/li>\r\n<\/ul>\r\nMethod:<\/strong>\r\n
                                  \r\n \t
                                • Choose Stat<\/strong>, Basic Statistics, then Normality Test <\/strong>(near the bottom of the drop down menu).<\/li>\r\n \t
                                • In the Normality Test window, select your variable, choose your test, and click ok.<\/li>\r\n<\/ul>\r\n\"\"<\/strong>\r\n\r\nWhich test to choose?<\/strong>\r\n
                                    \r\n \t
                                  • Anderson Darling<\/strong>: quite strongly affected by tied values, which are often encountered in large sample sizes. If you pick the AD test, then make sure you view the probability plot to see if there are tied values.<\/li>\r\n \t
                                  • Ryan<\/strong> Joiner (Shapiro Wilk<\/strong>): this test is useful for large sample sizes (>) as it doesn\u2019t react as strongly to tied values<\/li>\r\n \t
                                  • The Kolmogorov-Smirnov test<\/strong>: Avoid this test unless it is \/illefors corrected: It is not very powerful and will often say data are normal when they are not. If you use it, use a p-value cut-off of 0.10 rather than 0.05<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                    \r\n\r\nThe null hypothesis is that data are normal, so:<\/strong>\r\n
                                      \r\n \t
                                    • p > 0.05, data are normal.<\/strong><\/li>\r\n \t
                                    • p < 0.05, data are signif. diff. from normal.<\/strong><\/li>\r\n<\/ul>\r\nThe output is a probability graph (but without the confidence lines to help interpret) and the results of the statistical test in the small box at top right.\r\n\r\nLook for the P-Value <\/strong>to determine if data are normal. If p<0.05, it is non-normal.\r\n\r\nOther information provided: mean and SD values as well as the total \u2018N\u2019 and the test value from the statistical test (in this case, AD for the Anderson Darling). Do not <\/strong>confuse the test statistic with the P-Value. (If you pick the Ryan Joiner test, it will give the RJ value, and KS for Kolgomorov-Smirnov)\r\n\r\n<\/div>\r\n \r\n\r\nInterpreting P-Value Results from all three statistical normality tests:<\/strong>\r\n
                                      \r\n\r\n\r\n\r\n\r\n
                                      Anderson Darling<\/td>\r\nRyan- Joiner<\/td>\r\nKolmogorov\r\n\r\n-Smirnov<\/td>\r\nNote the differences in result here. Recall that the AD test is badly affected by tied values, and it will usually say data are non-normal when they are actually normal if tied values are present. The Ryan Joiner test is better for tied values, and indicates that data are right on the edge of normal (which is similar to what we saw with the skew\/kurtosis calculation). The K-S test can be used as long as the cut-off is 0.10 rather than 0.05, however these data appear non-normal.<\/td>\r\n<\/tr>\r\n
                                      p = 0.014<\/td>\r\n>0.100<\/td>\r\n0.016<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo<\/strong> assess<\/strong> normality,<\/strong> always<\/strong> use<\/strong> more<\/strong> than<\/strong> one<\/strong> method:<\/strong>\r\n\r\nFor data to be normal, the histogram should look like a bell curve, the normal probability plot should have points falling close to the line, the SE of the skew and kurtosis should fall between -1.96 and +1.96, and the statistical tests should have a p value greater than 0.05 (or 0.10 for the K-S test). How did we do?\r\n\r\nInterpretation:<\/strong><\/span>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                      <\/td>\r\n\u00a0<\/strong>\r\n\r\n <\/td>\r\n\u00a0<\/strong>\r\n\r\nConclusion<\/span><\/td>\r\n<\/tr>\r\n
                                      Histogram: looks a bit skewed, but not too far off<\/td>\r\n<\/td>\r\nNormal<\/td>\r\n<\/tr>\r\n
                                      Normal Prob.Plot: the dots fall within the 95% conf.<\/td>\r\n<\/td>\r\nNormal<\/td>\r\n<\/tr>\r\n
                                      Skew & Kurtosis: fall within the range<\/td>\r\n<\/td>\r\nNormal<\/td>\r\n<\/tr>\r\n
                                      Anderson-Darling: p = 0.014<\/td>\r\n<\/td>\r\nNon-normal<\/td>\r\n<\/tr>\r\n
                                      Ryan-Joiner: p > 0.100<\/td>\r\n<\/td>\r\nNormal<\/td>\r\n<\/tr>\r\n
                                      Kolmogorov-Smironov: p = 0.016<\/td>\r\n<\/td>\r\nNon-normal<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0<\/strong>The Anderson Darling and K-S tests give a different result than the others, but since we know it is affected by tied values, we don\u2019t use that one since there are lot of tied values in the plot. The other two indicate that data are normal or very close. Conclusion: Data are normal<\/strong>\r\n\r\n\u00a0<\/strong>\u00a0<\/strong>\r\n\r\nExample using non-normal data<\/strong>\r\n\r\nFor comparison, lets look at some obviously non-normal data: The pulses 2 column:\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
                                      \r\n\r\nInterpretation of the statistical Normality tests:<\/strong>\u00a0<\/strong>\r\n\r\n\r\n\r\n\r\n
                                      Anderson Darling<\/td>\r\nRyan Joiner<\/td>\r\nKolmogorov- Smironov<\/td>\r\nThis time, all statistical tests indicate that data are non-normal, since the p-values are <0.05 in all cases. Even though there are tied values (so we don\u2019t trust the AD test), the RJ and KS tests are clearly non-normal.<\/td>\r\n<\/tr>\r\n
                                      p <0.005<\/td>\r\np<0.01<\/td>\r\nP<0.01<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0<\/strong>Evaluating the SE of the Skew and Kurtosis:<\/strong>\u00a0<\/strong>\r\n\r\n\r\n\r\n\r\n
                                      Skewness1<\/td>\r\nKurtosis1<\/td>\r\nSEskew\u00a0= skew \u00f7 \/(6\/n)<\/td>\r\nSEkurt\u00a0 = kurtosis \u00f7 \/(24\/n)<\/td>\r\n<\/tr>\r\n
                                      1.11<\/td>\r\n1.49<\/td>\r\n=1.11 \u00f7 \u221a (6\/91) = 4.32<\/td>\r\n=1.49 \u00f7 \u221a (24\/91) = 27.67<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0Interpretation:<\/strong>\r\n\r\n<\/div>\r\n
                                      \r\n\r\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Conclusion\r\n\r\n<\/div>\r\n
                                      \r\n\r\nHistogram: quite skewed\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nNormal Prob.Plot: the dots are well off the line\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nSkew & Kurtosis: fall outside the range\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nAnderson-Darling: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nRyan-Joiner: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nKolmogorov-Smironov: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal\r\n\r\nThese data (Pulses 2) are clearly non-normal, as all normality tests confirm this.\r\n\r\n\u00a0<\/strong>\r\n\r\n\u00a0Comments on interpreting normality testing:<\/strong>\r\n\r\n\r\n\r\n
                                      \r\n
                                      \r\n
                                        \r\n \t
                                      • Most statistical tests are \u201crobust\u201d to minor violations of normality, so as long as it is close, data can be considered normal for the purposes of doing the statistical tests to find out if groups differ or are related to each other.<\/li>\r\n \t
                                      • It is usually easy to tell for data that are strongly normal or strongly non-normal<\/li>\r\n \t
                                      • For datasets where it seems \u201cclose\u201d, the best tests are to just look at the probability plot (with confidence limits) and to check the skew and kurtosis.<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0<\/strong>\r\n

                                        Parametric vs Non-parametric testing:<\/h3>\r\n
                                          \r\n \t
                                        • \u00a0 Parametric tests are better at picking up statistical differences than non- parametric tests, so we prefer to use them when we can\r\n<\/strong><\/li>\r\n<\/ul>\r\nAssumptions for parametric testing:<\/strong>\r\n
                                            \r\n \t
                                          • Data are normally distributed (or close to normal) \u2013 test normality<\/strong><\/li>\r\n \t
                                          • The variances of the groups are similar to each other \u2013 test \u201cequal variances\u201d<\/strong><\/li>\r\n \t
                                          • Data are independent of each other \u2013\u00a0 study design point<\/strong><\/li>\r\n \t
                                          • Data were collected in a random fashion \u2013 study design point<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                            \r\n\r\n\u00a0<\/strong>\r\n

                                            Statistical Tests\u00a0<\/strong><\/h3>\r\nWhich do you use? Here is a key to the basic tests<\/strong>\r\n

                                            Dichotomous key to statistical tests<\/h3>\r\n1a. Are you comparing the averages from groups of data?........................................................................................................................................................................................... 2<\/strong>\r\n\r\n1b. Are you looking for a relationship between two variables..................................................................................................................................................................................... 11<\/strong>\r\n\r\n2a. Are you comparing the average from one group of numbers to a single predicted value?......................................................................................................................... 3<\/strong>\r\n\r\n2b. Are you comparing the average from more than one group to averages?........................................................................................................................................................ 4<\/strong>\r\n\r\n3a. Are your data normally distributed.........................................................................................................................................................................................One Sample t-test\r\n<\/strong>\r\n\r\n3b. Are your data non-normal? ......................................................................................................................................................................................One Sample Wilcoxin test<\/strong>\r\n\r\n4a. Are you comparing the averages of two groups?.........................................................................................................................................................................................................5<\/strong>\r\n\r\n4b. Are you comparing the averages of three or more groups?.....................................................................................................................................................................................8<\/strong>\r\n\r\n5a. Are your data paired? (i.e. are you measuring something at time a and b on the same individuals?..................Paired t-test or non-parametric paired test\r\n<\/strong>\r\n\r\n5b. Are your data unpaired (i.e. are you just comparing the average values for your groups?............................................................................................................................6<\/strong>\r\n\r\n6a. Are your data non-normal?.....................................................................................................................................................................................................Mann-Whitney U-test\r\n<\/strong>\r\n\r\n6b. Are your data normally distributed....................................................................................................................................................................................................................................7<\/strong>\r\n\r\n7a. Are your data normal with equal variance................................................................................................................................................................................Student\u2019s t-test<\/strong>\r\n\r\n7b. Are your data normally distributed with unequal variance?...............................................................................................Students t-test, with variance correction<\/strong>\r\n\r\n8a. Are you comparing averages of three or more groups without subgroups?.......................................................................................................................................................9<\/strong>\r\n\r\n8b. Do your data have a subgroup or factor you want to compare (e.g. response of males and females within different treatment groups)?................................10<\/strong>\r\n\r\n9a. Are your data normally distributed with equal variance.......................................................................................................................................................One Way ANOVA<\/strong>\r\n\r\n9b. Are your data QRQ normal or have unequal variance?.........................................................................................................................................................Kruskall-Wallis Test<\/strong>\r\n\r\n10a. Are your data normally distributed .....................................................................................................................................................................Two-Way (factorial) ANOVA<\/strong>\r\n\r\n10b. Are your data QRQ normal or have unequal variance?.............................................................................................................................No simple non-parametric test<\/strong>\r\n\r\n11a. Are your data normally distributed, with error distribution normal and heterogeneous?.......................................................Pearson Correlation and Regression<\/strong>\r\n\r\n11b. Are your data non-normal?................................................................................................................................................................................................Spearman Correlation<\/strong>\r\n\r\n<\/div>\r\n
                                            \r\n\r\n \r\n

                                            What are the stats telling us? Comparing Groups or Relationships<\/h3>\r\nWe could be comparing groups or looking for relationships among variables.\r\n\r\n\u00a0<\/strong>\r\n\r\n\r\n\r\n
                                            \r\n
                                            \r\n\r\nOur first step in doing statistical comparisons should be to plot the data with error bars, to get a visual image of what we are comparing.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                                            Comparing groups<\/h4>\r\nOne of the main types of statistical analyses we do is to compare groups of data. When we do this, we\u2019re really taking the average of the group, and comparing those averages, taking into consideration how variable the data are, and how many samples we have.\r\n\r\n \r\n\r\nThe type of graph we plot depends on the shape of the data (see \u201cFrequency Distributions\u201d, p. 73-75).\r\n
                                              \r\n \t
                                            • If data are normally distributed, use a bar graph with error bars<\/li>\r\n \t
                                            • If data are non-normal, use a box and whisker plot\r\n<\/strong><\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                              \r\n\r\n\"\"\r\n\r\nLooking at the relationship between two variables (plot x against y):\r\n\r\n\"\"\r\n
                                                \r\n \t
                                              • Again, plot the data first, to see what the actual pattern looks like. Then use statistical analysis to see whether the relationship you see with your eye is statistically significant.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                \r\n\r\n\r\n\r\n
                                                \r\n
                                                \r\n\r\nRemember<\/strong> to<\/strong> test<\/strong> data<\/strong> for<\/strong> normality<\/strong> before<\/strong> deciding<\/strong> which test to use. Since you only have one data set here, you do not also need to test equal variance.<\/strong>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

                                                Using statistics to compare groups:<\/h4>\r\nWe use different tests depending on the number of groups, and whether data are parametric.\r\n

                                                Comparing one group to a predicted value:<\/h5>\r\n
                                                  \r\n \t
                                                • Parametric: one-sample t-test (compares the mean)<\/strong><\/li>\r\n \t
                                                • Non-parametric: one-sample Wilcoxin Test (compares the median)<\/strong>\r\n<\/strong><\/li>\r\n<\/ul>\r\nOne sample tests allow us to compare the average and variation in data from an observed group <\/strong>to a predicted value<\/strong>.\r\n
                                                    \r\n \t
                                                  • The Null Hypothesis is that the mean of your observed data is equal to the predicted value<\/li>\r\n \t
                                                  • If P<0.05, then the means are significantly different.<\/li>\r\n<\/ul>\r\nParametric Example:<\/strong> One sample t-test <\/em><\/strong>\r\n\r\nStudy Question: Is the average resting pulse equal to 70 beats per minute?<\/strong>\r\n
                                                      \r\n \t
                                                    • H0: Resting pulse (Pulse 1) = 70 beats per minute<\/li>\r\n \t
                                                    • HA: Resting pulse (Pulse 1) \u2026 70 beats per minute\r\n
                                                        \r\n \t
                                                      • Choose Stat from the property bar, and then Basic Statistics, and then 1 sample t<\/li>\r\n \t
                                                      • Select your variable (e.g., pulse 1<\/strong>)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nNote<\/strong>; if your variable list is blank, just click on the white space in the \u201csamples in columns\r\n
                                                          \r\n \t
                                                        • check the box for \u201cperform hypothesis test<\/strong>\u201d, and type in the value you\u2019re comparing (i.e. 70). Click OK<\/li>\r\n \t
                                                        • Note that we have a simple alternate hypothesis here, of \u201cequal\u201d vs \u201cnot equal\u201d. If you want to be more specific and test if it is greater than or less than your hypothesized mean, click on \u201cOptions\u201d and change it<\/li>\r\n<\/ul>\r\n\"\"\r\n\r\n\r\n\r\n
                                                          \r\n
                                                          \r\n\r\nInterpretation:<\/strong>\r\n\r\np < 0.05, therefore the mean of the observed resting pulses is significantly different from 70.\r\n\r\nTrend: <\/strong>Since the mean (from the output) is 73.14, we can say that pulses are significantly higher than 70 beats\/min.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\nNon parametric Example:<\/strong>\r\n\r\nOne sample Wilcoxin<\/em><\/strong>\r\n\r\nUse this test if data are non-normal.\r\n
                                                            \r\n \t
                                                          • Select Stat<\/strong>, Nonparametrics, <\/strong>then1-sample Wilcoxin<\/strong><\/li>\r\n<\/ul>\r\nSet it up the same way as for the 1-sample t-test, but test a median rather than a mean.\r\n\r\n\"\"\r\n\r\nNote that this non-parametric test also showed a significant p-value (p=0.028), but that this one is much closer to the 0.05 cutoff than the parametric test, reminding us that this is generally a less powerful test.\r\n\r\n\r\n\r\n
                                                            \r\n
                                                            \r\n\r\nAlways<\/strong> choose the parametric test if your data are normal.<\/strong>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                            \r\n
                                                            Comparing 2 Groups: <\/strong><\/h5>\r\nStudy Question: is Pulse 1 different from Pulse 2?<\/em>\r\n\r\n\u00a0<\/strong>Testing Parametric Assumptions: <\/strong>Before doing this test, remember to test each group of data for normality (p. 87-92). The other assumption for the parametric test is that variances in the groups are similar, so you will also have to do an \u201cequal variance\u201d test. We already know that data for Pulses 1 were normal, but the Pulses 2 data were not. Therefore, parametric test will give an invalid result and the correct test to do here is the non-parametric test. We will test for equal variance on p. 98. Examples with both tests are shown here so you can see how to do them.\r\n\r\n<\/div>\r\n
                                                            \r\n
                                                            Parametric test: Student\u2019s t-test <\/em>(AKA 2-sample t-test)<\/h6>\r\nStep<\/strong> 1: <\/strong>Look at plot of data to see pattern<\/strong>\r\n\r\n<\/div>\r\n
                                                            \r\n\r\nStep<\/strong> 2: <\/strong>Setting up the data table<\/strong>\r\n\r\nMinitab has two methods:<\/strong>\r\n
                                                              \r\n \t
                                                            • The t-test allows us to put the data into adjoining columns(spreadsheet fashion) or to have them in one column. For this example, our Pulse 1 and Pulse 2 data are already in adjoining columns so we\u2019ll pick this option.<\/li>\r\n<\/ul>\r\nStep 3: <\/strong>do the test: <\/strong>Choose 2 sample t-test <\/strong>from the Stat\/Basic Statistics drop-down menu.\r\n\r\nNote the box here where it says \u201cassume equal variance\u201d ... if you have tested equal variance and they are statistically equal, then check this box.\r\n\r\nOutput for t-test:<\/strong>\r\n\r\n\"\"\r\n\r\n\r\n\r\n
                                                              \r\n
                                                              \r\n\r\nInterpretation: p < 0.05. Therefore, If assumptions are verified (normal + equal variance), then we would conclude that Pulse 1 is OHVV WKDQ Pulse 2.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u00a0Assumptions:<\/strong>\r\n\r\n<\/div>\r\n
                                                              \r\n\r\n1.\u00a0\u00a0\u00a0 Test Normality: Pulse 2 was not normal (see p. 92)(this<\/strong> means the t-test result was not valid)<\/strong>\r\n\r\n2.\u00a0\u00a0\u00a0 Test Equal Variance: do variance test as described below\r\n\r\n \r\n\r\nTest<\/strong> for equal variance for t-test:<\/strong>\r\n
                                                                \r\n \t
                                                              • Choose \u201c2 Variances\u201d <\/strong>from the Stat\/Basic Statistics drop-down menu.<\/li>\r\n \t
                                                              • Select your variables that you are comparing (in this case, I chose the samples in dif. columns, but if your data were set up in a single column the result would be the same; see p. 100 for example). Click ok<\/strong><\/li>\r\n<\/ul>\r\n\"\"\r\n\r\nMinitab 21 uses Bonnett and Levene Tests and produces some companion graphs. The boxplot shows the range of the data,so we can see that the variation is Pulse 2 is much higher than that for Pulse 1\r\n\r\nThe interval plot shows us the difference in the standard deviation for the two groups, with confidence limits, so we can see that they don\u2019t overlap.\r\n\r\nNote the outputs for the Equal Variance test in the top right corner.\r\n
                                                                  \r\n \t
                                                                • Both show that p < 0.05, so we reject the null hypothesis <\/em><\/strong>that variances are equal<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n
                                                                  \r\n
                                                                  \r\n\r\nImportant:<\/strong>\r\n
                                                                    \r\n \t
                                                                  • If the data passed the normality test, but failed the equal variance test, you may still be able to do a t-test<\/li>\r\n \t
                                                                  • Sample size must be >10<\/li>\r\n \t
                                                                  • Leave \u201cassume equal variance\u201d box unchecked.<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTherefore, we conclude that variances are different\r\n
                                                                    Interpretation:<\/h5>\r\nThe data did not pass the assumption tests, therefore any result from the t-test is not valid and cannot be trusted\r\n\r\nTherefore, we would discard our t-test result, and carry out a Mann-Whitney U-test.\r\n\r\n<\/div>\r\n
                                                                    \r\n

                                                                    Nonparametric test:<\/strong> Mann-Whitney<\/em><\/strong> U-test <\/em><\/strong>(Use this test if data are not normal)<\/h4>\r\nSetting up the data:<\/strong>\r\n
                                                                      \r\n \t
                                                                    • You must have your data in separate columns for this test.<\/li>\r\n \t
                                                                    • No assumption testing is needed for this test.<\/li>\r\n<\/ul>\r\nCarry out the test: <\/strong>Choose the Mann-Whitney test from the Nonparametrics <\/strong>drop down menu (from Basic Statistics<\/strong>)\r\n
                                                                        \r\n \t
                                                                      • Select your variables, and click ok<\/li>\r\n<\/ul>\r\nThis test compares medians (middle values in a list that is ranked from smallest to largest) rather than means. <\/strong>Minitab Output:<\/strong>\r\n\r\n \"\"<\/strong>\r\n\r\nConclusion: <\/strong>no assumptions are required, and there is a signifcant difference among groups, since p < 0.05<\/strong>\r\n\r\nTrend statement: <\/strong>The pulse rate of students after running (Pulse 2) was significantly higher than the pulse rate before running (Mann-Whitney U-test, P=0.0048).\r\n\r\nThis is the correct test, so we can trust our result.<\/strong>\r\n\r\n\r\n\r\n
                                                                        \r\n
                                                                        \r\n\r\nSince the non-parametric test doesn\u2019t need us to test assumptions, why not just use it all the time?<\/strong>\r\n
                                                                          \r\n \t
                                                                        • When sample sizes are large, the non-parametric and parametric tests often come to the same general conclusion (i.e. the differences are significant or they are not), so it seems like more work to have to do all this extra testing. However, when patterns are not as clear (i.e. when p-values are close to the 0.05 cutoff), or for small sample sizes, it can make a major difference. The Parametric tests have greater POWER <\/strong>to see a difference if it actually is present, so we always use these ones if we can. Therefore, we must test assumptions first.<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                          \r\n

                                                                          A second two-sample example, t-test with data in a single column.<\/strong><\/h4>\r\nStudy question: Do males and females have the same average resting pulse?<\/strong>\r\n
                                                                            \r\n \t
                                                                          • \u00a0<\/strong>Choose the 2-sample t-test following the instructions on p. 97 (Stat,<\/strong> Basic Stats, 2-sample t)<\/strong><\/li>\r\n \t
                                                                          • When selecting your variable, you\u2019ll be able to choose the \u201csamples in the same column option\u201d in the t- test menu, but you\u2019ll still have to separate them to check for normality. You can do this manually, by splitting the worksheet (see p. 46), or by \u201cunstacking\u201d the column.<\/em><\/li>\r\n<\/ul>\r\n\u201cUnstacking\u201d a column to test subgroups separately:<\/strong>\r\n
                                                                              \r\n \t
                                                                            • Choose Data<\/strong> from the upper property bar, then Unstack<\/strong> columns <\/strong>from the drop-down menu:<\/li>\r\n<\/ul>\r\n\"\"\r\n\r\n\u201cSuperscripts\u201d refer to your grouping variable, so if you want to separate out the two sexes (male and female), then choose \u201csex\u201d as your subscript.\r\n
                                                                                \r\n \t
                                                                              • Check the boxes for \u201cafter last column in use\u201d and \u201cname the columns\u201d so that the data will appear in your worksheet.<\/li>\r\n<\/ul>\r\nTest normality<\/strong>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
                                                                                \r\n\r\nPulse 1 males: The probability plot has most values in the 95% confidence bands, and the p- Value for the Ryan Joiner statistical test is >0.05 (too many tied values for AD)\r\n\r\nConclusion: Normal<\/strong>\r\n\r\nPulse 1 females: The probability plot has all values in the 95% confidence bands, and the p-Value for the Ryan Joiner statistical test is >0.05\r\n\r\nConclusion: Normal<\/strong>\r\n\r\n<\/div>\r\n
                                                                                \r\n\r\n \r\n\r\nTest equal variance<\/strong>\r\n\r\n\"\"<\/strong>\r\n\r\nConclusion: variance is equal since p>0.05 for both tests<\/strong>\r\n\r\nImportant: do not stop here... these tests only assessed assumptions. Now you must do the test to test your study question.\r\n\r\n<\/div>\r\n \r\n
                                                                                \r\n\r\n \r\n\r\nCarry out the 2-sample t-test<\/strong>, <\/strong>this time with data in one column.\r\n\r\nSince variances are equal, check box for \u201cAssume equal variances\u201d.\r\n\r\n \r\n\r\nTwo-sample T for pulse1<\/strong>\r\n\r\n\"\"\r\n\r\nConclusion: <\/strong>Assumptions are satisfied, so test is valid and groups are significantly different since p<0.05.<\/strong>\r\n\r\n\u00a0<\/strong>Trend statement: <\/strong>Average pulse rate is significantly higher for females (group 2) than for males (group 1) (t-test, p=0.008)<\/strong>\r\n
                                                                                  \r\n \t
                                                                                • \u00a0Remember to always write a trend statement giving the clear trend in the data, with the name of the statistical test and the p-value in brackets.<\/strong>\r\n<\/strong><\/li>\r\n<\/ul>\r\n\u00a0<\/strong>\r\n\r\n\r\n\r\n
                                                                                  \r\n
                                                                                  \r\n\r\nHow should you report p-values?<\/strong>\r\n
                                                                                    \r\n \t
                                                                                  • When possible, report the actual p-value if you have it from the computer test<\/li>\r\n \t
                                                                                  • Rule of thumb: only report to a max. of 3 decimal places<\/strong>:<\/li>\r\n \t
                                                                                  • If the p is given as 0.000 or 0.0000 then write as p<0.001\r\n
                                                                                      \r\n \t
                                                                                    • Even if the computer reports a value like 0.000, NEVER <\/strong>say p=0.000, since probability is always higher than zero<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                      \r\n\r\n\r\n\r\n
                                                                                      \r\n
                                                                                      \r\n\r\nInteresting side note:<\/strong>\r\n\r\nThese questions illustrate how many questions can come out of a single dataset, if the sample size is large enough, and the study design incorporated many variables. In the previous example, we tested whether two different groups of students were different from each other. In this example, we are testing to see whether an individual group of students can show significant differences from one time\r\n\r\nto another. For this to work, the data must be \u201cpaired\u201d... i.e. we must identify each individual and be sure we know his\/her before and after result.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTwo sample testing when values are \u201cPaired\u201d <\/strong>Study<\/em><\/strong> question Did the pulse rates of the individual students go up after running?<\/em><\/strong>\r\n\r\nNote<\/strong> the<\/strong> importance<\/strong> of<\/strong> how the question is worded:<\/strong>\r\n\r\nIf we want to know if the average pulse rate goes up after running, we would do a regular t-test or Mann-Whitney U-test (if non-normal). But because we have data on each individual person, we can see if the individual rates go up by using a test that focuses on individual responses, called a \u201cpaired\u201d test. This is particularly useful if high variability in individuals makes it difficult to see a pattern in the average response, and the paired test will have more power to see the differences than the regular one.\r\n\r\nHow<\/strong> Paired tests work:\u00a0 <\/strong>In paired tests, we look at a measured value for known individuals at two (or more) times.\r\n\r\nThe example below is for the pulses in the group who ran in place for one minute, so you can see that their pulse rates all went up, though not by the same amount.\r\n\r\n\"\"\r\n\r\nWe can test this using a one-sample test, to see if the group of numbers in the difference column is significantly different from zero.\r\n\r\nFor parametric data, there is a t-test that calculates the differences and does the one sample test automatically in a single step. For non-parametric data, we have to do the extra step ourselves.<\/strong>\r\n\r\nStep 1: <\/strong>separate out the runners from the non- runners in both our resting pulse and our after running pulse columns (pulse 1 and pulse 2). This will give us four groups of data as you can see from the graph at right.<\/strong>\r\n\r\nRemember<\/strong> to<\/strong> plot<\/strong> your<\/strong> data<\/strong> so<\/strong> you<\/strong> can<\/strong> see<\/strong> what you are comparing!<\/strong>\r\n
                                                                                        \r\n \t
                                                                                      • We can do this manually (by copying and pasting into additional columns, \u201csplitting\u201d the worksheet into multiple smaller worksheets to work on using Minitab or \u201cunstacking\u201d the columns.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                        \r\n\r\n\"\"\r\n\r\nFigure 1. Comparison of pulse rates for a group of students before and after one group ran for 1 Min.\r\n\r\n<\/div>\r\n
                                                                                        \r\n\r\nStep 2: Compare pulse 1 and pulse 2 for the non-runners (control) and then the runners (test)<\/strong>\r\n\r\nFirst: Test assumptions:<\/strong>\r\n\r\n1. <\/strong>Normality testing<\/strong>: test normality of all four groups, using Ryan Joiner (since many tied values)\r\n\r\n\"\"\r\n\r\n2.\u00a0\u00a0\u00a0 <\/strong>Equal variance<\/strong>: Running group p=0.004\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-running group: p=0.624\r\n\r\n(Variances between before & after running: non-equal for runners, equal for non-runners)\r\n\r\nConclusion from assumptions:<\/strong>\r\n
                                                                                          \r\n \t
                                                                                        • Runners: one group was normal the other non normal; variances were not equal<\/li>\r\n \t
                                                                                        • Non-runners: data were normal, and variances were equal<\/li>\r\n<\/ul>\r\nTherefore, both the parametric and non-parametric paired tests are needed to assess whether there are differences between the individuals pulse rates at the two different times.\r\n\r\n<\/div>\r\n
                                                                                          \r\n

                                                                                          Parametric data - use the Paired t-test<\/strong><\/h4>\r\nRun this test using the Non-runners\u2019 pulse rate data, since these data were normal and had equal variance.\r\n\r\nMake sure the data to be tested are in separate columns, e.g. non-runners at time 1 and non-runners at time 2.<\/strong>\r\n
                                                                                            \r\n \t
                                                                                          • Select Stat <\/strong>from the property bar, and Basic Statistics.<\/strong><\/li>\r\n \t
                                                                                          • Choose the paired t test <\/strong>from the drop down menu.<\/li>\r\n \t
                                                                                          • Choose variables so that your data for the non-runners from time 1 is being compared to non-runners in time 2.<\/li>\r\n<\/ul>\r\nMinitab Output:<\/strong>\r\n\r\n \r\n\r\nPaired T-Test and CI: pulse1nonran, pulse2nonran<\/strong>\u00a0<\/strong>\r\n\r\n\r\n\r\n
                                                                                            \r\n
                                                                                            \r\n\r\nSince p>0.05, we accept the null hypothesis, and say there is no difference in the groups.<\/strong>\r\n\r\nNotice the difference here in what is being tested: instead of comparing one mean to another, it tests whether the mean difference is equal to zero\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                            \r\n

                                                                                            Non-parametric paired two sample test:<\/strong><\/h4>\r\n
                                                                                              \r\n \t
                                                                                            • If data are non-normal, then you must use a non-parametric test. There is no one-step non-parametric paired test, but you can carry it out in two steps.<\/li>\r\n<\/ul>\r\nStep one: <\/strong>Subtract your time 1 column from your time 2 column (as in the table on p. 103). You can do this in Excel, and copy and paste the values into Minitab.\r\n\r\nStep two: <\/strong>Carry out a one-sample Wilcoxin test to test whether your median is different from zero (as shown on p. 96).\r\n\r\nChoose your column with the runner differences, and check the box for testing that the median is zero:\r\n\r\nMinitab Output:\r\n\r\n \"\"\r\n<\/strong>\r\n\r\nWilcoxon Signed Rank Test: run difs<\/strong>\r\n\r\nSince p<0.001, then we reject the Null hypothesis; i.e. the pulse rates are significantly different between time 1 and time 2.<\/strong>\r\n\r\n<\/div>\r\n \r\n
                                                                                              \r\n

                                                                                              Comparing >2 groups of data<\/strong><\/h4>\r\n\r\n\r\n\r\n
                                                                                              \r\n
                                                                                              \r\n\r\nBalanced<\/strong> Designs<\/strong>\r\n\r\nSome packages (e.g. Excel) assume a \u201cbalanced design\u201d; that means that there have to be the same number of observations in each group being studied. In Minitab, a balanced design is not necessary for simple ANOVA (although it is in 2-way ANOVA), but note that if your groups have too few observations, you have very little \u201cpower\u201d to pick out differences.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnalysis of Variance (ANOVA) or the non- parametric equivalent (Kruskall-Wallis) allows us to compare more than 2 groups of data. As with the t-test, we are comparing means, and the null hypothesis is that the means are equal.\r\n
                                                                                                \r\n \t
                                                                                              • If p<0.05, the means are statistically different (assuming the assumptions of ANOVA are met).<\/strong><\/li>\r\n<\/ul>\r\nSetting up your data:<\/strong>\r\n\r\nYour dataset can be set up in the worksheet so that the responses being measured are all in one column, and the groups that you are comparing are given as categories in a separate column (See our pulses dataset on p. 83 as an example). Alternatively, the data can be set up so your groups are in adjoining columns. For the ANOVA, these have special names: If data are in one column, Minitab refers to this as \u201cstacked<\/strong>\u201d. If data are in adjoining columns, Minitab refers to this set-up as \u201cunstacked<\/strong>\u201d.\r\n
                                                                                                Parametric Data: Analysis of Variance (ANOVA)<\/h5>\r\nOne-Way Design <\/strong>(comparing groups without subgroups)\r\n\r\nExample of comparing three or more groups of data; \u201cstacked\u201d data. Study Question: Are resting pulse rates different depending on normal activity level?<\/em><\/strong>\r\n
                                                                                                  \r\n \t
                                                                                                • This will give three groups to compare, with three activity levels.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n
                                                                                                  \r\n
                                                                                                  \r\n\r\nImportant design note: The design of this study is highly unbalanced, and the group reporting low activity level was very small in number; it may be too small to give random and independent data. Therefore, ANOVA result should be treated with caution.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAssumptions:\r\n\r\nANOVA assumes that data are parametric, which means:\r\n
                                                                                                    \r\n \t
                                                                                                  • Normally distributed<\/li>\r\n \t
                                                                                                  • Variances are equal<\/li>\r\n \t
                                                                                                  • Design should be set up so data are independent and random.<\/li>\r\n<\/ul>\r\n1. <\/strong>Test for normality <\/strong>for each group of data as explained in the earlier section\r\n
                                                                                                      \r\n \t
                                                                                                    • When this was done, all groups were normal (p>0.05)<\/li>\r\n<\/ul>\r\n2. <\/strong> Equal Variance Test<\/strong>: Do not use the equal variance test you used for t-tests!<\/strong>\r\n
                                                                                                        \r\n \t
                                                                                                      • To test equal variance on three or more groups of data, we need to use the equal variance test found in the ANOVA menu. The one in the t-test menu only tests variance for two groups of data, not for three or more.<\/li>\r\n<\/ul>\r\nSelect Stat <\/strong>from the property bar, then ANOVA<\/strong>. Look part way down the drop-down menu, and select Test for Equal Variances<\/strong>\r\n\r\nChoose your response variable (in this case, pulses 1 to assess the resting pulse rates) and your factor variable (in this case, activity level), and click okay.\r\n\r\nEqual variances result:<\/strong>\r\n\r\n\"\"\r\n\r\n<\/div>\r\n
                                                                                                        \r\n\r\np>0.05 so variances are equal\r\n\r\nNotice in the graph how the variances are all very similar with a lot of overlap... that shows that the variances are very similar.\r\n\r\nConclusion: <\/strong>Although we need to be cautious about interpreting patterns about the low activity level due to the small and unbalanced sample size, our data meet the assumptions of the ANOVA, so we can run the test.\r\n\r\nMethod for One-Way ANOVA:<\/strong>\r\n
                                                                                                          \r\n \t
                                                                                                        • Choose \u201cstat<\/strong>\u201d from the property bar, then choose \u201cANOV<\/strong>A\u201d, then choose \u201cone-way<\/strong>\u201d. One way analysis of variance means that you are simply comparing 3 or more groups of data, without any subgroups in them.<\/li>\r\n<\/ul>\r\nThis puts you into the menu for the ANOVA with data set up in one column\r\n\r\nThe<\/strong> Null hypothesis <\/strong>is that\r\n\r\nX1 = X2 = X3\r\n\r\ntherefore, a significant p-value (p<0.05) means that at least one group is different from the others.\r\n\r\n<\/div>\r\n
                                                                                                          \r\n
                                                                                                            \r\n \t
                                                                                                          • Select your variables<\/strong>\r\n
                                                                                                              \r\n \t
                                                                                                            • The \u201cresponse<\/strong>\u201d variable is the one where your measurements are, for example one of the columns of pulse rates.<\/li>\r\n \t
                                                                                                            • The \u201cfactor<\/strong>\u201d is the column where the different groupings or levels are shown.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                              \r\n\r\nIn our pulses dataset, we have 3 levels of activity, so we could choose activity level as our factor.\r\n\r\nClick<\/strong> on<\/strong> \u201cokay\u201d,<\/strong> and<\/strong> the<\/strong> output<\/strong> will look like:<\/strong>\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n\"\"<\/strong>\r\n\r\nInterpretation: <\/strong>p>0.05, so the means are not significantly different <\/strong>between the groups. We conclude that activity level does not <\/strong>have an effect on resting pulse in these students.\r\n\r\n<\/div>\r\nRemember to always plot your data <\/strong>to see the patterns. This helps you to determine what you actually want to test, and can help you interpret your data patterns. You can see from the graph at right that although it looks like the people with low activity level had a higher resting pulse rate, the variability in the data means that there is no significant difference\r\n
                                                                                                              <\/div>\r\n
                                                                                                              \r\n\r\n\"\"\r\n\r\nConclusion: we have done the appropriate<\/strong> test, so we can trust our result.<\/strong>\r\n\r\nTrend Statement: <\/strong>There is no significant difference among resting pulses in the three groups (ANOVA, p = 0.155).<\/strong>\r\n
                                                                                                              Non-parametric data: Kruskall-Wallis Test<\/strong><\/h5>\r\n<\/div>\r\n
                                                                                                              \r\n\r\nIf data were not <\/strong>normal [and could not be made normal through transformation], we would use the Kruskal-Wallis test, which compares medians.\r\n\r\nData Setup:<\/strong>\r\n
                                                                                                                \r\n \t
                                                                                                              • Data must be set up so that the response variable (in this case, Pulse rate) is in one column, and the grouping variable is in another column.<\/li>\r\n \t
                                                                                                              • Select \u201cnon-parametrics<\/strong>\u201d from the \u201cstats<\/strong>\u201d menu, and choose the Kruskal-Wallis.<\/li>\r\n \t
                                                                                                              • Select your variables:<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n
                                                                                                                \r\n
                                                                                                                \r\n\r\nTransforming<\/strong>:\r\n
                                                                                                                  \r\n \t
                                                                                                                • If your data are not normal and don\u2019t have equal variance, you can use the non-parametric test, or you can try transforming the data. Common transformations are the log transformation, square root transformation, and inverse tranformation <\/strong>\u2013 To transform, convert all values in each column of data being analysed to the transformation, and try your analyses again.<\/li>\r\n \t
                                                                                                                • Remember to always report your actual data (in the text or in graphs) and not the transformed data<\/li>\r\n<\/ul>\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n\"\"\r\n\r\nTrend statement: <\/strong>There is no significant difference among the pulse rates for the students who have the different activity levels (Kruskal-Wallis test, p=0.153)<\/strong>\r\n\r\n<\/div>\r\n
                                                                                                                  \r\n

                                                                                                                  Example of comparing three or more groups of data; \u201cun-stacked\u201d data<\/strong><\/h4>\r\nStudy Question: <\/strong>Is there a difference among the runners and non-runners, before and after running?<\/em>\r\n\r\nRunning Group\r\n\r\nData Setup: <\/strong>For this method, put the data into 4 separate columns, and run a One-Way ANOVA<\/strong>, unstacked<\/strong>.\r\n\r\n\"\"\r\n\r\nA quick plot of the data shows no dif. in pulse between time 1 and 2 for the non-runners, but there seems to be a big difference for the runners. What we would like to know is whether that difference is significant.\r\n\r\nThe null hypothesis is that there is no difference among groups.<\/strong>\r\n\r\n\u00a0<\/strong>Assumptions<\/strong>:\r\n
                                                                                                                    \r\n \t
                                                                                                                  • Normality: <\/strong>some groups were normal, and others were not. The non-normal groups were relatively close to normal.<\/li>\r\n \t
                                                                                                                  • Variances: <\/strong>were not equal<\/li>\r\n<\/ul>\r\nConclusion from Assumptions:<\/strong>\r\n\r\nData are non-parametric (if<\/strong> even<\/strong> one<\/strong> group<\/strong> you<\/strong> are<\/strong> comparing<\/strong> don\u2019t<\/strong> fit<\/strong> assumptions,<\/strong> then<\/strong> your<\/strong> data<\/strong> are<\/strong> nonparametric)<\/strong>, so should either be transformed or a non-parametric test should be chosen. However, ANOVA is \u201crobust\u201d to minor violations of assumptions, so since data are close to normal, it may be okay to do the parametric test. To be sure, do both the parametric and non-parametric test and compare.\r\n\r\nONE-WAY ANOVA, \u201cunstacked\u201d<\/strong>\r\n
                                                                                                                      \r\n \t
                                                                                                                    • Select Stat <\/strong>from the property bar, then choose ANOVA <\/strong>from the drop down menu.<\/li>\r\n \t
                                                                                                                    • Choose One-Way (Unstacked)<\/strong><\/li>\r\n \t
                                                                                                                    • Select variables (these must be in adjoining columns in your dataset)<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                      \r\n\r\nOutput (ANOVA table)<\/strong>\r\n\r\n \"\"<\/strong>\r\n\r\n\r\n\r\n
                                                                                                                      \r\n
                                                                                                                      \r\n\r\nInterpretation: <\/strong>p<0.05, so there is a significant difference in the groups. However, we don\u2019t know which groups are different.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nImportant<\/strong>\u00a0<\/strong>note:<\/strong>\r\n\r\np<0.05, so groups are significantly different. However, there is no way to know from the simple One-Way ANOVA which group is different from which other(s). We need to do further testing to figure this out.\r\n\r\nMultiple Comparison Tests:\r\n\r\n\u00a0<\/strong>ANOVA is designed to tell us if there are any differences, but not which groups are different. For this: do Multiple Comparison Tests<\/strong>\u00a0<\/strong>\r\n\r\n\r\n\r\n
                                                                                                                      \r\n
                                                                                                                      \r\n\r\nType<\/strong> I Error Inflation:<\/strong>\r\n\r\n\u00a0<\/strong>\r\n\r\nRecall that if we do multiple comparisons on the same data set (e.g. group 1 vs group 2<\/strong>, group 1 vs group 3 <\/strong>and group 2 vs group 3<\/strong>) then our probability of getting an incorrect interpretation goes up. If you have three groups, that probability goes up from 0.05 to 0.14 for all the comparisons taken together.\r\n\r\n\u00a0<\/strong>\r\n\r\nThis means that if we want to compare the individual groups, we need to use a special test that calculates a \u201cfamily error rate\u201d (the error rate for all the groups considered together) rather than individual error rates. These are called \u201cpost-hoc\u201d<\/em> or \u201cmultiple comparison\u201d tests.\r\n\r\n\u00a0<\/strong>\r\n\r\nIf you find a significant result with your ANOVA (i.e. if you run ANOVA and your p<0.05), then you should run a multiple comparison test to see which of the groups are significantly different from each other.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                                                      \r\n\r\nThe most commonly used multiple comparison test is Tukey\u2019s Test. To do a Multiple Comparison test, click on \u201ccomparisons\u201d in the initial ANOVA menu\r\n
                                                                                                                        \r\n \t
                                                                                                                      • Check the box for Tukey\u2019s test, and click OK<\/li>\r\n \t
                                                                                                                      • Then run the ANOVA as before. (You will see the ANOVA output, and some additional information that lets you compare each group)<\/li>\r\n<\/ul>\r\nOutput:\r\n\r\n\"\"\r\n\r\n\r\n\r\n
                                                                                                                        \r\n
                                                                                                                        \r\n\r\nInterpretation: <\/strong>Look for groupings that have a different letter under the \u201cGrouping\u201d heading. In the example above, the pulse 1 and pulse 2 groups for the non-runners, and the pulse 1 group for the runners all have the same letter, so they are not significantly different. However, the pulse 2 data for the runners has a different letter, so that means it is significantly different than all the other groups. To find out how it is different, look at the column with the means: Pulse2ran clearly has a higher mean value than the other ones.\r\n\r\n \r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTherefore, to report this trend, you\u2019d write something like:<\/strong>\r\nThere was a significant difference in pulse rates in the groups of students (ANOVA, p <0.001). There was no difference in resting pulses of the two groups of students (runners vs non runners) prior to running, but there was a significant increase in pulse rate in the running group after running (Tukeys test, p<0.05).\r\n
                                                                                                                        Comparing >2 groups of data with subgroups<\/h5>\r\n<\/div>\r\n
                                                                                                                        \r\n\r\nFactorial Analysis of Variance\u00a0 <\/strong>(This can be a 2-way ANOVA, 3-Way, etc.)<\/strong>\r\n\r\n\u00a0<\/strong>If your data contain distinct subgroups, you can run an ANOVA that lets you test the effects of those subgroups, or factors, on your response at the same time as testing your main grouping factor. This is called a factorial analysis.\r\n\r\n\r\n\r\n
                                                                                                                        \r\n
                                                                                                                        \r\n\r\nImportant note: Minitab requires a \u201cbalanced design\u201d for 2-way ANOVA<\/strong>\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConsider an example where a researcher would like to know whether a particular feed supplement would increase growth in chickens, and whether the sex of the chicken would affect how it worked. This is a standard \u201ctwo-way\u201d design, where we are looking at two factors at the same time.\r\n\r\n\r\n\r\n
                                                                                                                        \r\n
                                                                                                                        \r\n\r\nData<\/strong> Set-up<\/strong>\r\n\r\nSet up the data in Minitab so that the response data (weight) are in one column, and the grouping factors are in other columns:\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n
                                                                                                                        \r\n
                                                                                                                        \r\n\r\nThis is known as a factorial table. If your data can be set up in this way to show groupings, then it is a good candidate for a two-way ANOVA.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTable 1. Weight (grams) after two weeks in a group of bantam chicks on two different diets; the standard diet, and one supplemented by blueberry extract.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                                                                                                         \r\n\r\n \r\n\r\nmales<\/td>\r\nsupplement\r\n\r\n \r\n\r\n590<\/td>\r\ncontrol\r\n\r\n \r\n\r\n440<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n530<\/td>\r\n570<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n550<\/td>\r\n509<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n570<\/td>\r\n510<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n650<\/td>\r\n589<\/td>\r\n<\/tr>\r\n
                                                                                                                        females<\/td>\r\n530<\/td>\r\n550<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n580<\/td>\r\n420<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n520<\/td>\r\n440<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n520<\/td>\r\n520<\/td>\r\n<\/tr>\r\n
                                                                                                                        <\/td>\r\n560<\/td>\r\n370<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                                                        \r\n\r\nAssumption<\/strong> Testing:<\/strong>\r\n
                                                                                                                          \r\n \t
                                                                                                                        • Normality (A-D): <\/strong>\r\n
                                                                                                                            \r\n \t
                                                                                                                          • control, female, p=0.64<\/li>\r\n \t
                                                                                                                          • control, male, p=0.52<\/li>\r\n \t
                                                                                                                          • supplement, female, p=0.21<\/li>\r\n \t
                                                                                                                          • supplement, male, p=0.59<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nAll data are normally distributed<\/strong>\r\n\r\nEqual Variance:<\/strong>\r\n
                                                                                                                              \r\n \t
                                                                                                                            • Use the \u201cTest for equal variances\u201d in the ANOVA menu.<\/li>\r\n \t
                                                                                                                            • Set it up as shown below:<\/li>\r\n<\/ul>\r\n\"\"\r\n\r\nEqual Variance Output:<\/strong>\r\n\r\np>0.05, therefore accept the null <\/strong>that variances are equal.<\/strong>\r\n\r\n<\/div>\r\n
                                                                                                                              \r\n\r\nNow<\/strong> run the test:<\/strong>\r\n
                                                                                                                                \r\n \t
                                                                                                                              • \u00a0<\/strong>Select Stat <\/strong>from the property bar, then choose ANOVA,<\/strong> then Two-Way<\/strong><\/li>\r\n \t
                                                                                                                              • \u00a0<\/strong>Select your variables:\r\n
                                                                                                                                  \r\n \t
                                                                                                                                • Response = <\/strong>the data column, so, weight Row and column factors:<\/strong><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                                                                                                                                • Before doing a 2-way ANOVA, you should set up a table to assess your factors<\/li>\r\n \t
                                                                                                                                • Use that to decide which is a row factor and which is a column factor.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                                  \r\n\r\n \r\n\r\nFirst, plot the data to see the patterns you want to test:<\/strong>\r\n\r\n<\/div>\r\n\"\"\r\n<\/strong>\r\n
                                                                                                                                  \r\n\r\nTwo Way ANOVA Output:<\/strong>\r\n\r\n \"\"<\/strong>\r\n
                                                                                                                                    \r\n \t
                                                                                                                                  • <\/li>\r\n<\/ul>\r\nInterpretation:<\/strong>\r\n\r\nStep 1: look at p-values for individual factors<\/strong>\r\n\r\nSex: p=0.051\r\n
                                                                                                                                      \r\n \t
                                                                                                                                    • Since p>0.05, we accept the null and conclude there is no difference Food: p = 0.011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since p<0.05, we reject the null and conclude there is a difference<\/li>\r\n<\/ul>\r\nStep 2: look at the p-value for interaction<\/strong>\r\n\r\nInteraction: p=0.577,\r\n
                                                                                                                                        \r\n \t
                                                                                                                                      • Since p>0.05, there is no interaction with respect to the weight response among the factors (i.e. weight acted the same way for both sexes (was lower) regardless of the food type).<\/li>\r\n<\/ul>\r\nInteraction<\/strong> refers to whether the response acts the same way for the different levels of the factors.\r\n
                                                                                                                                          \r\n \t
                                                                                                                                        • e.g. If the growth was higher on blueberries for males, and lower for blueberries for females, that would be an example of a different response for the two sexes... This would give a significant interaction.<\/li>\r\n \t
                                                                                                                                        • Since growth was lower for females than males for both food types, it means there was no interaction.<\/li>\r\n<\/ul>\r\nTo see the actual trend, we look at the graph:<\/strong>\r\n\r\nChickens that were fed a diet supplemented by blueberry extract grew significantly larger than those on a standard diet (Two-Way ANOVA, p=0.012), and there was no significant difference in response between male and female chicks (Two-Way ANOVA, p=0.057). There was no significant interaction between food type and sex of chicks (Two-Way ANOVA, p=0.58), indicating that the food supplement affected weight in the same way for male and female chicks.\r\n\r\n\r\n\r\n
                                                                                                                                          \r\n
                                                                                                                                          \r\n\r\nRemember to always say which is higher\/lower than the other, if you can\r\n\r\nTo report the interaction, be sure you explain it in English - do not just say there is a significant interaction or not.\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                                                                          \r\n
                                                                                                                                          Relationships between variables: Regression and Correlation<\/h5>\r\nCorrelation analysis tests whether there is a correlation between two continuous variables Regression analysis takes it a step further to give us the equation of the line and give information on how much of the variation in the points can be statistically related to the other variable.\r\n\r\nStep 1: Plot the data to see if there is a relationship between the variables <\/strong>e.g. Is Weight related to Height of the students in the pulses study? If so, what is the equation of the line.<\/strong>\r\n\r\n\u00a0<\/strong>The graph suggests that the weight of the students increases as their heights increase.\r\n\r\n<\/div>\r\n
                                                                                                                                          <\/div>\r\n
                                                                                                                                          \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
                                                                                                                                          \r\n\r\nWe can run correlation <\/strong>analysis to see how strongly related (correlated) the variables are, and we can run regression <\/strong>analysis to get the equation of the \u201cbest\u201d line that describes the relationship.\r\n\r\nWe can use regression <\/strong>to describe relationships, or to generate an equation (usually called a \u201cmodel\u201d that can be used to predict for values\r\n\r\nAssumptions: Correlation\r\n<\/strong>\r\n
                                                                                                                                            \r\n \t
                                                                                                                                          • Each variable must be normally distributed<\/li>\r\n \t
                                                                                                                                          • The relationship must be linear<\/li>\r\n \t
                                                                                                                                          • The residuals (errors) must be normally distributed<\/li>\r\n<\/ul>\r\nAssumptions: Regression<\/strong>\r\n
                                                                                                                                              \r\n \t
                                                                                                                                            • Each variable must be normally distributed<\/li>\r\n \t
                                                                                                                                            • The relationship must be linear (for linear regression)<\/li>\r\n \t
                                                                                                                                            • The residuals must be evenly distributed along the line<\/li>\r\n<\/ul>\r\nSome<\/strong> important<\/strong> definitions:<\/strong>\r\n
                                                                                                                                                \r\n \t
                                                                                                                                              1. Independent variable: the variable that is fixed for our analysis... what we are comparing the other variable to plot this variable on the x-axis<\/li>\r\n \t
                                                                                                                                              2. Dependent variable: the variable that \u201cdepends on\u201d the x-axis variable; the one we expect to change or vary with our fixed variable. Plot this variable on the y-axis<\/li>\r\n \t
                                                                                                                                              3. Residual: the amount of vertical (y-axis) variation of the observed points from the regression line (i.e. the \u2018y\u2019 value of your point minus the \u2018y\u2019 value on your line)<\/li>\r\n \t
                                                                                                                                              4. Standardized residual: (residual) \u00f7 (standard deviation of residual). Standardized residuals have been standardized to have a variance of 1. Standardized residuals over 2 are usually considered large, and points with that large a residual are often considered to be outliers outside of our observed range.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
                                                                                                                                                \r\n\r\nMethod:<\/strong>\r\n
                                                                                                                                                  \r\n \t
                                                                                                                                                • Test normality as described before<\/li>\r\n \t
                                                                                                                                                • Test linearity by looking at the graph and deciding if the points form a straight line<\/li>\r\n \t
                                                                                                                                                • Test the normality and linearity of the residuals through specialized menus in the Regression menu.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                                                  \r\n
                                                                                                                                                  Correlation<\/strong><\/h5>\r\n\"\"\r\n\r\n<\/div>\r\n
                                                                                                                                                  \r\n\r\nThe graph above shows a relationship between height and weight of the student participants.<\/strong>\r\n
                                                                                                                                                    \r\n \t
                                                                                                                                                  • How related are they?<\/li>\r\n \t
                                                                                                                                                  • Is the relationship significant?<\/li>\r\n<\/ul>\r\nTo test these questions, we run a correlation analysis.\r\n\r\nThe correlation analysis gives us two statistics, <\/strong>the correlation coefficient, and the p-value\r\n\r\n<\/div>\r\n
                                                                                                                                                    \r\n\r\nCorrelation Coeffient <\/strong>( r ) tells you how related the two variables are on a scale of zero to one\r\n
                                                                                                                                                      \r\n \t
                                                                                                                                                    • A negative \u2018r\u2019 means the relationship is negative (slopes downward)<\/li>\r\n \t
                                                                                                                                                    • A positive \u2018r\u2019 means the relationship is positive (slopes upwards)<\/li>\r\n \t
                                                                                                                                                    • A value of 1 or -1 means 100% related<\/li>\r\n \t
                                                                                                                                                    • General rule of thumb is that a \u201cgood\u201d correlation is 0.7 to 1.0 or -0.7 to -1.0<\/li>\r\n<\/ul>\r\nP-value: <\/strong>Tells you whether the slope of your line is significantly different from zero (i.e do you have a significant relationship. If p<0.05, the variables are significantly correlated\r\n\r\nCarry out the test:<\/strong>\r\n
                                                                                                                                                        \r\n \t
                                                                                                                                                      • Select Stat <\/strong>from the property bar, then choose correlation <\/strong>from the Basic statistics <\/strong>drop down menu.<\/li>\r\n<\/ul>\r\n[Note: <\/strong>do correlation analysis only if you don\u2019t need the equation of the line, or the coefficient of determination (r2)]\r\n\r\n \r\n\r\nWith correlation, it doesn\u2019t matter what order you put the variables in, since we\u2019re just looking at a simple relationship.\r\n\r\nOutput from Minitab:<\/strong>\u00a0<\/strong>\r\n\r\n\r\n\r\n
                                                                                                                                                        \r\n
                                                                                                                                                        \r\n\r\nCorrelations: height, weight<\/strong>\r\n\r\nPearson correlation of height and weight = 0.786 P-Value = 0.000\r\n\r\nremember to write as p<0.001\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nInterpretation:<\/strong>\r\n
                                                                                                                                                          \r\n \t
                                                                                                                                                        • The r-value = 0.786, which tells us that we have a positive correlation, and it is a \u201cgood\u201d correlation (since the value is above 0.7) tells us how strong it is.<\/li>\r\n \t
                                                                                                                                                        • The p-value tells us if it is a significant correlation, while the r-value<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                                                          \r\n\r\nAssumptions were satisfied:<\/strong>\r\n\r\nBoth sets of data are normal (based on prob. plots) and the relationship is linear (based on the graph)\r\n\r\n<\/div>\r\n
                                                                                                                                                          \r\n
                                                                                                                                                          Regression:<\/h5>\r\nRegression analysis lets us determine the \u201cmodel\u201d (line) that best describes the data.\r\n\r\nThe regression analysis gives us three statistics:\r\n
                                                                                                                                                            \r\n \t
                                                                                                                                                          • \u00a0r2 =\u201ccoefficient of determination\u201d<\/li>\r\n \t
                                                                                                                                                          • The p-value, and<\/li>\r\n \t
                                                                                                                                                          • The equation of the line.<\/li>\r\n<\/ul>\r\nThe p-value is the same as for correlation\r\n\r\n<\/div>\r\n
                                                                                                                                                            \r\n\r\nThe Coefficient of determination, r<\/strong>2<\/strong>, <\/strong>tells us the proportion of the variation in the \u2018y\u2019 vaues that can be directly related(\"statistically\u201d) to the x-value. It is often referred to as the amount of variation \u201cexplained\u201d by the variation in \u2018x\u2019, but note that that is meant in the statistical sense.\r\n\r\n<\/div>\r\n
                                                                                                                                                            \r\n\r\nIn regression, we assume that one variable is fixed or independent, and that the other variable depends on the first. The independent variable goes on the x-axis and the dependent one goes on the y-axis.\r\n\r\nThe assumptions of linear regression are:<\/strong>\r\n
                                                                                                                                                              \r\n \t
                                                                                                                                                            1. Linearity: data on the graph form a reasonable line, so we say they are linear<\/li>\r\n \t
                                                                                                                                                            2. Data normality: Normality plots of the two variable showed that both were normally distributed<\/li>\r\n \t
                                                                                                                                                            3. Error normality (normality of residuals) - test this after running the regression<\/li>\r\n \t
                                                                                                                                                            4. Equal distribution of residuals along the line - test after running the regression<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n
                                                                                                                                                              \r\n
                                                                                                                                                              \r\n\r\nA note on regression assumptions:<\/strong>\r\n\r\nUnlike the assumptions for the group comparisons (e.g. ANOVA or t-test), many of the Regression are there to guide your interpretation, not to be an absolute guide to whether you can do the test or not.\r\n\r\n\u201cMust<\/strong> have\u201d<\/strong> asummptions:\r\n
                                                                                                                                                                \r\n \t
                                                                                                                                                              • Data must be linear for Linear Regression<\/li>\r\n \t
                                                                                                                                                              • Data must be normal or near normal<\/li>\r\n<\/ul>\r\nAssumptions<\/strong> that<\/strong> guide<\/strong> interpretation:<\/strong>\r\n
                                                                                                                                                                  \r\n \t
                                                                                                                                                                • Spread of residuals:\r\n
                                                                                                                                                                    \r\n \t
                                                                                                                                                                  • Residuals are the distance that each dependent variable point is away from the line (on the y-axis).\r\n
                                                                                                                                                                      \r\n \t
                                                                                                                                                                    • For the regression equation to have good predictive ability (i.e. so you can predict values of y if you know values of x), the residuals have to be normally distributed and distributed equally above and below the line for the entire relationship. For example, if there is low variability (scatter) at the lower part of the graph, and high variability (scatter) at the high part of the graph, it means that the line doesn\u2019t predict as well at the upper parts of the graph.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n- i.e. Even if these assumptions are violoated slightly, you can still do the regression, but have to use caution if predicting \u2018y\u2019 from \u2018x\u2019\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRun<\/strong> the Regression analysis:<\/strong>\r\n
                                                                                                                                                                        \r\n \t
                                                                                                                                                                      • Select Stat<\/strong>from the property bar, then choose Regression <\/strong>and Regression <\/strong>and fit regression model<\/strong><\/li>\r\n \t
                                                                                                                                                                      • In the regression menu, put the dependent variable in the response box, and the independent variable in the predictor box<\/li>\r\n<\/ul>\r\nTo test the assumptions while doing the test, <\/strong>click<\/strong> on<\/strong> the \u201cGraphs\u201d box.<\/strong>\r\n
                                                                                                                                                                        \r\n
                                                                                                                                                                          \r\n \t
                                                                                                                                                                        • In the graphs window, check the box beside Regular<\/strong>\r\n
                                                                                                                                                                            \r\n \t
                                                                                                                                                                          • This gives the actual values for the residuals<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n[\u2018Standardized\u2019 will convert the values based on their standard deviations so you can see them as a function of their variability. This can be useful in identifying outliers, but otherwise, the graph will look very similar]\r\n
                                                                                                                                                                              \r\n \t
                                                                                                                                                                            • For \u201cresidual<\/strong> plots<\/strong>\u201d, check the boxes beside normal probability plot, and the residuals vs the \u201cfits\u201d (the independent var).\r\n
                                                                                                                                                                                \r\n \t
                                                                                                                                                                              • If you think that there may be a bias in the order the data are collected, then also plot the residuals vs the order (e.g. if you collect data over a period of time, so you might get a different response later in the day than earlier, that could be a bias)<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                                                                                \r\n
                                                                                                                                                                                Interpretation of Regression Assumptions:<\/h6>\r\nDistribution of residuals along the line:<\/strong>\r\n\r\n\"\"\r\n
                                                                                                                                                                                  \r\n \t
                                                                                                                                                                                • Here, you are looking for evidence that points are scattered fairly evenly along the horizontal line. This means your \u201cvariance\u201d (variability in the observed values compared to the predicted line) is spread equally along the line... analogous to the \u201cequal variance\u201d assumption in ANOVA.<\/li>\r\n \t
                                                                                                                                                                                • The distribution of points above and below the line isn\u2019t perfect, but is fairly uniform if you don\u2019t count the outlier point in the upper right corner.<\/li>\r\n \t
                                                                                                                                                                                • This distribution is close enough to pass the assumption test.<\/li>\r\n<\/ul>\r\nNormality of residuals:<\/strong>\r\n\r\n \"\"<\/strong>\r\n\r\nThis graph is just like the other normality plots we look at to see if data are normal. The computer will automatically determine the residuals for you, and plot them on the normality plot.\r\n\r\nInterpretation:<\/strong>\r\n
                                                                                                                                                                                    \r\n \t
                                                                                                                                                                                  • The values are mostly on the line, again with one outlier, so conclude that they are normal<\/li>\r\n \t
                                                                                                                                                                                  • [Just to be sure, I did a normality test on the data, and it confirm that they are normal (p>0.05).]<\/li>\r\n<\/ul>\r\nThe output from the Regression analysis:<\/strong>\r\n\r\n \"\"<\/strong>\r\n\r\nRegression Analysis Interpretation:\r\n\r\n<\/div>\r\n
                                                                                                                                                                                    \r\n
                                                                                                                                                                                      \r\n \t
                                                                                                                                                                                    1. Equation of the line. This is in the form of a straight line, y = mx + b<\/li>\r\n \t
                                                                                                                                                                                    2. The r2 value tells us the proportion of the variability in weight that can be \u201cexplained\u201d (or is statistically related to) by height, so 62% of the weight variation is mathematically related to height. 38% is related to other, unmeasured factors.<\/li>\r\n \t
                                                                                                                                                                                    3. The p-value tests whether the slope of the line is significantly different from zero (if there is too much variability in the points, you can\u2019t be sure that you have a relationship).<\/li>\r\n<\/ol>\r\nThe assumptions were satisfied, so we can generally trust our result:<\/strong>\r\n
                                                                                                                                                                                        \r\n \t
                                                                                                                                                                                      • There is a significant positive relationship between height and weight.<\/li>\r\n \t
                                                                                                                                                                                      • The assumption testing tells us we have one outlier value (we could investigate that more closely if we wanted), and that there is a bit more variation at higher levels of \u2018x\u2019 (height) than at lower levels. Therefore, we know that our line does not predict as well for tall people as for short people.<\/li>\r\n \t
                                                                                                                                                                                      • Although there is a significant relationship, there is still quite a bit of unexplained variation, so height is not the only factor that affects weight of the students.<\/li>\r\n<\/ul>\r\n<\/div>\r\n
                                                                                                                                                                                        \"\"<\/div>\r\n
                                                                                                                                                                                        \r\n\r\nFigure 1. Relationship between height and weight for a group of university students taking part in an exercise on how running affects pulse rates.\r\n\r\n\r\n\r\n
                                                                                                                                                                                        \r\n
                                                                                                                                                                                        \r\n\r\nHow do you report results of your regression analysis?<\/strong>\r\n\r\nYour trend statement must give the pattern from your graph, then summarize the statistical results (remember to always plot your data before doing analysis, and include a figure legend). An example from the figure above might read:\r\n\r\nThere was a strong positive relationship between height and weight for the university students taking part in the running exercise, so that as height increased, so did weight (Figure 1, Regression analysis, r2 = 0.617). About 62% of the variation in weight was statistically explained by the variation in height, but the calculated regression line had better predictive ability at lower levels of height than for higher levels of height (Figure 1).\r\n\r\n<\/div><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
                                                                                                                                                                                        \r\n\r\nPulses Dataset used for analyses in this manual<\/strong>.\r\n\r\nA group of students were separated into 2 groups. One group ran on the spot for one minute, and the other group did not. Following the trial, information was gathered on whether they smoked, what their sex was, what their height and weight were, and what their normal activity levels were.\r\n
                                                                                                                                                                                          \r\n \t
                                                                                                                                                                                        • Pulse1 = resting pulse for all.<\/li>\r\n \t
                                                                                                                                                                                        • Pulse2 = pulse following the running. (Beats per minute)<\/li>\r\n \t
                                                                                                                                                                                        • Weight is in pounds, Height is in inches<\/li>\r\n \t
                                                                                                                                                                                        • ran = 1 means they ran,<\/li>\r\n \t
                                                                                                                                                                                        • ran = 2 means they did not.<\/li>\r\n \t
                                                                                                                                                                                        • smokes = 1 means they smoked.<\/li>\r\n \t
                                                                                                                                                                                        • sex = 1 is male<\/li>\r\n \t
                                                                                                                                                                                        • sex = 2 is female<\/li>\r\n \t
                                                                                                                                                                                        • activlev is their normal activity level; 1 is slight, 2 is moderate, and 3 is high.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                                                                                                                                                                          pulse1<\/td>\r\npulse2<\/td>\r\nran<\/td>\r\nsmokes<\/td>\r\nsex<\/td>\r\nheight<\/td>\r\nweight<\/td>\r\nactiv.level<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                           \r\n\r\n64<\/td>\r\n \r\n\r\n78<\/td>\r\n \r\n\r\n1<\/td>\r\n \r\n\r\n2<\/td>\r\n \r\n\r\n1<\/td>\r\n \r\n\r\n66<\/td>\r\n \r\n\r\n140<\/td>\r\n \r\n\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          58<\/td>\r\n75<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n145<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n82<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n73.5<\/td>\r\n160<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          66<\/td>\r\n85<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n73<\/td>\r\n190<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          64<\/td>\r\n82<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n69<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          74<\/td>\r\n84<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n73<\/td>\r\n165<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          84<\/td>\r\n84<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n150<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n72<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n74<\/td>\r\n190<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n75<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n195<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          76<\/td>\r\n88<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n138<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          80<\/td>\r\n104<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n74<\/td>\r\n160<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          80<\/td>\r\n96<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n88<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n70<\/td>\r\n153<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n76<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n67<\/td>\r\n145<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          60<\/td>\r\n76<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n170<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n68<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n175<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          66<\/td>\r\n88<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n69<\/td>\r\n175<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n86<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n73<\/td>\r\n170<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n80<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n74<\/td>\r\n180<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n80<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n66<\/td>\r\n135<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n106<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n170<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          74<\/td>\r\n76<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n70<\/td>\r\n157<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          66<\/td>\r\n102<\/td>\r\n1<\/td>\r\n2<\/td>\r\n1<\/td>\r\n70<\/td>\r\n130<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n98<\/td>\r\n1<\/td>\r\n1<\/td>\r\n1<\/td>\r\n75<\/td>\r\n185<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          96<\/td>\r\n140<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n61<\/td>\r\n140<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n100<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n66<\/td>\r\n120<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          78<\/td>\r\n104<\/td>\r\n1<\/td>\r\n1<\/td>\r\n2<\/td>\r\n68<\/td>\r\n130<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          82<\/td>\r\n100<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n68<\/td>\r\n138<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          88<\/td>\r\n115<\/td>\r\n1<\/td>\r\n1<\/td>\r\n2<\/td>\r\n63<\/td>\r\n121<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n112<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n70<\/td>\r\n125<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          96<\/td>\r\n116<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n68<\/td>\r\n116<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          78<\/td>\r\n118<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n69<\/td>\r\n145<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          88<\/td>\r\n110<\/td>\r\n1<\/td>\r\n1<\/td>\r\n2<\/td>\r\n69<\/td>\r\n150<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n98<\/td>\r\n1<\/td>\r\n1<\/td>\r\n2<\/td>\r\n62.75<\/td>\r\n112<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          80<\/td>\r\n128<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n68<\/td>\r\n125<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n62<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n74<\/td>\r\n190<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          60<\/td>\r\n62<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n70<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n69<\/td>\r\n170<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n66<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n70<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          76<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n215<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                                                                                                                                                                          pulse1<\/td>\r\npulse2<\/td>\r\nran<\/td>\r\nsmokes<\/td>\r\nsex<\/td>\r\nheight<\/td>\r\nweight<\/td>\r\nactiv.level<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n68<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n67<\/td>\r\n150<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          54<\/td>\r\n56<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n69<\/td>\r\n145<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          74<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n73<\/td>\r\n155<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          74<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n73<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n68<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n150<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n73<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n68<\/td>\r\n155<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n64<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n69.5<\/td>\r\n150<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          82<\/td>\r\n83<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n73<\/td>\r\n180<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          64<\/td>\r\n62<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n75<\/td>\r\n160<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          58<\/td>\r\n58<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n66<\/td>\r\n135<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          54<\/td>\r\n50<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n69<\/td>\r\n160<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n71<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n66<\/td>\r\n130<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n61<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n73<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          76<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n74<\/td>\r\n148<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          88<\/td>\r\n84<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n73.5<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n70<\/td>\r\n150<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          90<\/td>\r\n89<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n67<\/td>\r\n140<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          78<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n180<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          70<\/td>\r\n71<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n75<\/td>\r\n190<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          90<\/td>\r\n90<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n68<\/td>\r\n145<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          92<\/td>\r\n94<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n69<\/td>\r\n150<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          60<\/td>\r\n63<\/td>\r\n2<\/td>\r\n1<\/td>\r\n1<\/td>\r\n71.5<\/td>\r\n164<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n71<\/td>\r\n140<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n68<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n72<\/td>\r\n142<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          84<\/td>\r\n84<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n69<\/td>\r\n136<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          74<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n67<\/td>\r\n123<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n66<\/td>\r\n2<\/td>\r\n2<\/td>\r\n1<\/td>\r\n68<\/td>\r\n155<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          84<\/td>\r\n84<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n66<\/td>\r\n130<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          61<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n65.5<\/td>\r\n120<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          64<\/td>\r\n60<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n66<\/td>\r\n130<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          94<\/td>\r\n92<\/td>\r\n2<\/td>\r\n1<\/td>\r\n2<\/td>\r\n62<\/td>\r\n131<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          60<\/td>\r\n66<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n62<\/td>\r\n120<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n70<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n63<\/td>\r\n118<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          58<\/td>\r\n56<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n67<\/td>\r\n125<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          88<\/td>\r\n74<\/td>\r\n2<\/td>\r\n1<\/td>\r\n2<\/td>\r\n65<\/td>\r\n135<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          66<\/td>\r\n72<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n66<\/td>\r\n125<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          84<\/td>\r\n80<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n65<\/td>\r\n118<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          62<\/td>\r\n66<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n65<\/td>\r\n122<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          66<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n65<\/td>\r\n115<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          80<\/td>\r\n74<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n64<\/td>\r\n102<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          78<\/td>\r\n78<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n67<\/td>\r\n115<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n68<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n69<\/td>\r\n150<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          72<\/td>\r\n68<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n68<\/td>\r\n110<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          82<\/td>\r\n80<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n63<\/td>\r\n116<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          76<\/td>\r\n76<\/td>\r\n2<\/td>\r\n1<\/td>\r\n2<\/td>\r\n62<\/td>\r\n108<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          87<\/td>\r\n84<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n63<\/td>\r\n95<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          90<\/td>\r\n92<\/td>\r\n2<\/td>\r\n1<\/td>\r\n2<\/td>\r\n64<\/td>\r\n125<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          78<\/td>\r\n80<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n68<\/td>\r\n133<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          68<\/td>\r\n68<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n62<\/td>\r\n110<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          86<\/td>\r\n84<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n67<\/td>\r\n150<\/td>\r\n3<\/td>\r\n<\/tr>\r\n
                                                                                                                                                                                          76<\/td>\r\n76<\/td>\r\n2<\/td>\r\n2<\/td>\r\n2<\/td>\r\n61.75<\/td>\r\n108<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>","rendered":"
                                                                                                                                                                                          \n

                                                                                                                                                                                          Part 4: Basic Applied Statistics using Minitab 21<\/h1>\n<\/div>\n
                                                                                                                                                                                          \n

                                                                                                                                                                                          A note about statistical significance:<\/p>\n\n\n\n
                                                                                                                                                                                          \n
                                                                                                                                                                                          \n

                                                                                                                                                                                          All statistical tests are designed to test whether a pattern you see in your data is \u201cstatistically significant\u201d<\/strong>. We say something is statistically significant if our test confirms that the pattern is unlikely to have occurred by chance. To test this we look at the \u201cp-value\u201d<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                          The p-value is related to the hypotheses about the data:<\/strong><\/p>\n

                                                                                                                                                                                            \n
                                                                                                                                                                                          • H<\/strong>0<\/strong>\u00a0<\/strong>(Null Hypothesis) is that there is no difference between groups, or no relationship between variables. This is a \u201cno effect\u201d hypothesis<\/li>\n
                                                                                                                                                                                          • H<\/strong>A<\/strong>\u00a0<\/strong>(Alternate Hypothesis)\u00a0\u00a0 is that there is a difference or a relationship. This is sometimes called the \u201cactive\u201d hypothesis, since it indicates some sort of effect<\/li>\n<\/ul>\n

                                                                                                                                                                                            Therefore, to interpret your data, you need to examine the graph of the data and clearly state an hypothesis, or you won\u2019t know what the p-value means!<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                            \n
                                                                                                                                                                                            \n

                                                                                                                                                                                            If p<0.05, reject your Null Hypothesis<\/strong><\/p>\n

                                                                                                                                                                                            If p>0.05, accept your Null Hypothesis<\/strong><\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                            For each of the tests detailed in the next pages, note what the<\/strong> Null Hypothesis is, so that you can determine how to interpret<\/strong> the p-value from the test.<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                            \"\"<\/p>\n

                                                                                                                                                                                            \n

                                                                                                                                                                                            Getting started\u00a0<\/strong><\/h2>\n

                                                                                                                                                                                            DATA<\/strong> ENTRY<\/strong><\/p>\n

                                                                                                                                                                                            When working in a spreadsheet, the common method of entering your data is in adjoining columns. For example, if you have data such as we looked at in class on the crab temperatures, you would put your crab data for time 1 in one column, and your crab data for time 2 in the next column. However, for advanced stats, you must enter your data so that all the responses for a single variable (in this case, temperature) are in a single column, with another column giving the key for the variable.<\/p>\n

                                                                                                                                                                                            Note that data in the \u201cadvanced stats\u201d format are set up so that the responses are given in the columns, and the variables (e.g. whether they ran or not, what their sex was\/is given as a category number in another column. This method of data entry is necessary for most statistical packages.<\/p>\n\n\n\n
                                                                                                                                                                                            \n
                                                                                                                                                                                            \n

                                                                                                                                                                                            Notes<\/strong> about the data examples<\/strong><\/p>\n

                                                                                                                                                                                            For this manual, most of the examples will be from a dataset on an imaginary group of students that were asked to take a bunch of measurements on themselves before and after running in place for one minute.<\/p>\n

                                                                                                                                                                                            One group of students was asked to run in place for a minute, and another group (the control) did not run.<\/p>\n

                                                                                                                                                                                            Students (in both groups) were asked to take their pulses (in heartbeats per minute) before and after the running exercise, then were asked to indicate whether they were male or female, whether they smoked or not, whether they thought of themselves as active or not, and so on. This data set allows us to illustrate a wide variety of statistical analyses. The dataset is at the back of this manual, and will be placed on your Moodle site, so that you can practice the exercises in this manual, and see what the answers should look like.<\/p>\n

                                                                                                                                                                                              \n
                                                                                                                                                                                            • Ran = 1 means they ran,<\/li>\n
                                                                                                                                                                                            • Ran = 2 are the nonrunners (control)<\/li>\n
                                                                                                                                                                                            • Smokes = 1 means they smoke<\/li>\n
                                                                                                                                                                                            • Sex = 1 means male<\/li>\n
                                                                                                                                                                                            • Sex = 2 means female<\/li>\n
                                                                                                                                                                                            • Activity levels: 1 = slight, 2 = moderate, 3 = high<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
                                                                                                                                                                                              \n

                                                                                                                                                                                              To start in Minitab:<\/p>\n

                                                                                                                                                                                                \n
                                                                                                                                                                                              • Open Minitab 21\n
                                                                                                                                                                                                  \n
                                                                                                                                                                                                • You will see a \u201csession\u201d window on top where you\u2019ll find the record of what you\u2019ve done, as well as the text results of statistical tests.<\/li>\n
                                                                                                                                                                                                • The worksheet on the bottom will contain your data.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                                                                                                                                                  Navigate in Minitab through the menus on the upper property toolbar.<\/strong><\/p>\n

                                                                                                                                                                                                  \"\"<\/p>\n

                                                                                                                                                                                                  Entering data:<\/strong><\/p>\n

                                                                                                                                                                                                  Data can be entered (typed in) directly, or copied from a spreadsheet<\/p>\n

                                                                                                                                                                                                    \n
                                                                                                                                                                                                  • From spreadsheet: open your spreadsheet to the desired database and select and copy the data. O<\/strong>nly copy the numbers<\/strong>… do not copy column headers (that will designate your columns as text columns, and cause problems when doing the statistical analyses)<\/li>\n
                                                                                                                                                                                                  • Reenter Minitab, paste the data into the worksheet.<\/li>\n
                                                                                                                                                                                                  • Name your columns by clicking on the blank space below the column number, and typing in the name<\/li>\n
                                                                                                                                                                                                  • Save your worksheet onto your data disk or personal drive. It will save as a .mpx file<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                    \n

                                                                                                                                                                                                    Simple Column Statistics\u00a0<\/strong><\/p>\n

                                                                                                                                                                                                    Descriptive statistics:<\/strong><\/p>\n

                                                                                                                                                                                                      \n
                                                                                                                                                                                                    • Select Stat <\/strong>from the property bar, and click on \u201cbasic statistics\u201d.<\/li>\n
                                                                                                                                                                                                    • Select \u201cdisplay descriptive statistics<\/strong>\u201d to see the Display window<\/li>\n
                                                                                                                                                                                                    • Click on the box labelled \u201cStatistics\u201d to see the range of statistics available.<\/li>\n
                                                                                                                                                                                                    • The list of variables is on the left and an empty box for the variable(s) to test is on the right.<\/li>\n<\/ul>\n

                                                                                                                                                                                                      Highlight the variable you want to test, and click select<\/strong><\/p>\n

                                                                                                                                                                                                      There is a long list of potential statistics you can have the computer calculate, all in one operation. Check the boxes of all you would like.<\/p>\n

                                                                                                                                                                                                        \n
                                                                                                                                                                                                      • Click ok <\/strong>to return to the display descriptive stats window<\/li>\n
                                                                                                                                                                                                      • Click ok <\/strong>again to obtain the data.<\/li>\n<\/ul>\n

                                                                                                                                                                                                        Your results will appear in the upper Minitab Session Window. You can copy and paste the results into your word processor spreadsheet if you want to.<\/p>\n<\/div>\n

                                                                                                                                                                                                        \n

                                                                                                                                                                                                        Variables<\/strong> to<\/strong> choose:<\/strong> Variables must be ordinal (such as Pulse 1, Pulse 2, Weight, Height in this example; Note that your categorical variables won\u2019t work here.<\/p>\n

                                                                                                                                                                                                        This section also allows you to see some basic graphs for your variables, by clicking on \u201cgraphs\u201d rather than\u201cstatistics\u201d<\/p>\n

                                                                                                                                                                                                          \n
                                                                                                                                                                                                        • Click on all of these options to see how these graphs can help you get a feel for what your data looks like.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                          \n

                                                                                                                                                                                                          Descriptive statistics for sub-groups within each column of data (e.g. males and females in your group; age groups, etc.)<\/h3>\n

                                                                                                                                                                                                          Options:<\/p>\n

                                                                                                                                                                                                          a)\u00a0 cut and paste in your spreadsheet, then copy into Minitab and run analysis twice<\/p>\n

                                                                                                                                                                                                          b)\u00a0 Split the data <\/strong>in Minitab (see page 46 for method) and run analysis twice<\/p>\n

                                                                                                                                                                                                          c)\u00a0 use the \u201cBy variables<\/strong>\u201d<\/strong> option and run the analyses simultaneously.<\/p>\n

                                                                                                                                                                                                          e.g. In the pulses dataset, each of the two pulse columns (Pulse 1, Pulse 2) include groups (runners & non-runners, males and females, smokers and non-smokers). If you want to compare one subgroup to the other within a single column of data, you will need descriptive statistics and normality testing on each subgroup.<\/p>\n

                                                                                                                                                                                                          Example: <\/strong>calculate the descriptive statistics for the runners and the non-runners separately, in the pulses2 column (the second pulse rate, measured after running).<\/p>\n

                                                                                                                                                                                                            \n
                                                                                                                                                                                                          • Go into stat <\/strong>on the property bar, and click on \u201cbasic statistics<\/strong>\u201d, then \u201cdisplay descriptive statistics<\/strong>\u201d. Select the Pulse2 column, and click select. Then place your cursor in the By variables <\/strong>window, and select the group variable, Ran<\/li>\n<\/ul>\n

                                                                                                                                                                                                            Your output (in the Session box) will have information for both subgroups in your variable (i.e. Ran 1 & 2), and your graphs will also show both subgroups.<\/p>\n

                                                                                                                                                                                                            You can see how this lets you look at multiple subgroups separately without having to do a lot of cutting and pasting. you can choose this option in your \u201cstore descriptive statistics\u201d section as well.<\/p>\n<\/div>\n

                                                                                                                                                                                                            \"\"<\/p>\n

                                                                                                                                                                                                            \n

                                                                                                                                                                                                            Normality Testing<\/h3>\n

                                                                                                                                                                                                            Many statistical tests depend on data being parametric (one part of which is normality). Normality testing is a first step for most statistical testing.<\/p>\n\n\n\n
                                                                                                                                                                                                            \n
                                                                                                                                                                                                            \n

                                                                                                                                                                                                            The normal distribution is a type of frequency distribution which has a characteristic bell curve shape with a particular height and width for its mean (average) and Standard Deviation. We can assess normality (by eye) by plotting the frequency distribution of the data, and comparing it to the normal curve that is calculated for a data set with this mean and standard deviation. However, it can be hard to see from the frequency plot so there are other methods to assess normality.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                            Several methods to test normality: Generally use at least two, since not all work well for all data<\/strong><\/p>\n

                                                                                                                                                                                                            a.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 frequency histogram<\/em><\/strong>: <\/strong>important to see what data look like, but not very accurate for assessing whether they fit the normal distribution<\/p>\n

                                                                                                                                                                                                            b.\u00a0\u00a0\u00a0\u00a0\u00a0 normal probability plot<\/em><\/strong>:<\/strong> This modifies the frequency scale, so that if data are normal, they fall on a straight line. **This is usually the best method to assess normality<\/em><\/strong><\/p>\n

                                                                                                                                                                                                            c.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 determining whether the shape of the curve fits a mathematical range <\/em><\/strong>(based on \u201cskew\u201d (tails) and \u201ckurtosis\u201d (height of curve)<\/strong>). This is a great method to do as a check on the other methods, in case you just aren\u2019t sure of the interpretation.<\/p>\n

                                                                                                                                                                                                            d.\u00a0\u00a0\u00a0\u00a0\u00a0 Statistical methods: <\/em><\/strong>These give some comfort because they seem quantitative, but in fact, they are not accurate in many cases so should be used with caution. Several do not work well with small sample sizes, and several don\u2019t work well if there are many \u201ctied values\u201d (the same number repeated frequently in the dataset)<\/p>\n

                                                                                                                                                                                                            Frequency Distributions<\/h3>\n

                                                                                                                                                                                                            First, assess the frequency distribution as a first step to see what the data look like.<\/p>\n

                                                                                                                                                                                                              \n
                                                                                                                                                                                                            • Plot the histogram through the Column Statistics <\/strong>menu as described on p. 83, or through Graph, Histogram, <\/strong>as described on pp. 73-75.<\/li>\n
                                                                                                                                                                                                            • In the graphing menu, choose \u201cwith fit\u201d <\/strong>for the histogram with the normal curve.<\/li>\n
                                                                                                                                                                                                            • Select your variables as before, then click on \u201cdataview<\/strong>\u201d. Choose the \u201cdistribution tab, and make sure your distribution says \u201cnormal\u201d and click ok<\/li>\n<\/ul>\n

                                                                                                                                                                                                              \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                              \n

                                                                                                                                                                                                              Does your graph match the bell?<\/strong><\/p>\n

                                                                                                                                                                                                              The normal probability plot<\/h3>\n
                                                                                                                                                                                                                \n
                                                                                                                                                                                                              • Select Graph <\/strong>from the property bar, and then choose Probability plot from the drop-down menu. When prompted, choose the \u201csingle\u201d graph, and click okay<\/li>\n
                                                                                                                                                                                                              • Choose your variable as before. The normal probability plot is the default, but if you want to test another distribution (e.g. random), you can click on \u201cdistribution\u201d.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                This is produces a plot of your frequency histogram on a special probability scale, so that if the data are normal, your points should fall on a straight line. (Remember that this is for the entire column of data; if you need a subset of the data, such as males vs females, you\u2019ll need to separate them)<\/p>\n

                                                                                                                                                                                                                You can check this by eye, or you can do a statistical normality test on the data.<\/p>\n

                                                                                                                                                                                                                \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                This plot includes the 95% confidence limit for the line. If the points generally fall along the line and are within the confidence limits lines, then you can assume normality<\/p>\n

                                                                                                                                                                                                                <\/div>\n

                                                                                                                                                                                                                The conclusion from this <\/strong>graph is that the data are normal. <\/strong>The dots deviate from the line very slightly, but not by much, and most fall within the confidence limits. We can run through other methods to see if they confirm our impression from the graph.<\/p>\n

                                                                                                                                                                                                                \n

                                                                                                                                                                                                                Using skew and kurtosis calculations<\/h3>\n

                                                                                                                                                                                                                \"\"Data are normally distributed if the standard error of the skew (SE<\/strong>skew<\/strong>)<\/strong> and the standard error of the kurtosis (SE<\/strong>kurtosis<\/strong>)<\/strong> fall between -1.96 and +1.96.<\/strong><\/p>\n

                                                                                                                                                                                                                Method: Determine the Standard Error (SE) of the kurtosis and skew from the skew and kurtosis values in the Descriptive stats analysis (see p. 84) <\/strong>(note that some statistical packages do this calculation for you). The equations below provide an approximation of the SE values.<\/p>\n

                                                                                                                                                                                                                \"\"<\/p>\n

                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                • SEskew\u00a0= Skew \u00f7 \u221a (6\/n)<\/li>\n
                                                                                                                                                                                                                • SEkurtosis\u00a0= kurtosis \u00f7 \u221a (24\/n)<\/li>\n
                                                                                                                                                                                                                • n=no. of obs.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                  For the Pulse 1 column:<\/p>\n

                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                  • SEskew\u00a0= skew \u00f7 \u221a(6\/n)\u00a0\u00a0 = .43 \u00f7 \u221a (6\/91) = 1.67<\/li>\n
                                                                                                                                                                                                                  • SEkurt = kurtosis \u00f7 \u221a (24\/n\u00a0 = -.58 \u00f7 \u221a (24\/91) = -1.13<\/li>\n<\/ul>\n

                                                                                                                                                                                                                    These both fall between 1.96 and -1.96, so data are statistically normal<\/strong><\/p>\n

                                                                                                                                                                                                                    Note: the SE skew value is close to non-normal, as you can see by the bars in the figure looking a bit crowded towards the left side, but it still falls in the statistical range.<\/strong><\/p>\n

                                                                                                                                                                                                                    \u00a0<\/strong>This test confirms that the Pulse 1 data are normally distributed<\/strong><\/p>\n

                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                    • \u00a0Note: \u201cData is\u201d ??\u00a0\u00a0 \u201cData are\u201d ?? The word \u201cdata\u201d is plural (the singular version is \u201cdatum\u201d) so should always be given with the plural form of the verb when written.
                                                                                                                                                                                                                      \n<\/strong><\/li>\n<\/ul>\n\n\n\n
                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                      Using Statistical Normality tests:<\/strong><\/p>\n

                                                                                                                                                                                                                      One big problem with statistical normality tests is that they are adversely affected by both small sample sizes and large numbers of \u201ctied values\u201d, i.e. when we have a number of duplicate values in our list of numbers. Tied values often occur if we have a large data set. Therefore the normality test must always be treated with caution, and results should ALWAYS be checked against the normal probability curve.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                      \u00a0<\/strong>Using statistical normality tests<\/h3>\n<\/div>\n
                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                      • Minitab provides 3 statistical normality tests. All three will plot the \u201cnormal probability curve\u201d as part of the analysis so you can compare the test result to the graph.<\/li>\n
                                                                                                                                                                                                                      • The purpose of the tests is to see whether the dots are significantly different from the line.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                        Method:<\/strong><\/p>\n

                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                        • Choose Stat<\/strong>, Basic Statistics, then Normality Test <\/strong>(near the bottom of the drop down menu).<\/li>\n
                                                                                                                                                                                                                        • In the Normality Test window, select your variable, choose your test, and click ok.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                          \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                          Which test to choose?<\/strong><\/p>\n

                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                          • Anderson Darling<\/strong>: quite strongly affected by tied values, which are often encountered in large sample sizes. If you pick the AD test, then make sure you view the probability plot to see if there are tied values.<\/li>\n
                                                                                                                                                                                                                          • Ryan<\/strong> Joiner (Shapiro Wilk<\/strong>): this test is useful for large sample sizes (>) as it doesn\u2019t react as strongly to tied values<\/li>\n
                                                                                                                                                                                                                          • The Kolmogorov-Smirnov test<\/strong>: Avoid this test unless it is \/illefors corrected: It is not very powerful and will often say data are normal when they are not. If you use it, use a p-value cut-off of 0.10 rather than 0.05<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                            \n

                                                                                                                                                                                                                            The null hypothesis is that data are normal, so:<\/strong><\/p>\n

                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                            • p > 0.05, data are normal.<\/strong><\/li>\n
                                                                                                                                                                                                                            • p < 0.05, data are signif. diff. from normal.<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                              The output is a probability graph (but without the confidence lines to help interpret) and the results of the statistical test in the small box at top right.<\/p>\n

                                                                                                                                                                                                                              Look for the P-Value <\/strong>to determine if data are normal. If p<0.05, it is non-normal.<\/p>\n

                                                                                                                                                                                                                              Other information provided: mean and SD values as well as the total \u2018N\u2019 and the test value from the statistical test (in this case, AD for the Anderson Darling). Do not <\/strong>confuse the test statistic with the P-Value. (If you pick the Ryan Joiner test, it will give the RJ value, and KS for Kolgomorov-Smirnov)<\/p>\n<\/div>\n

                                                                                                                                                                                                                               <\/p>\n

                                                                                                                                                                                                                              Interpreting P-Value Results from all three statistical normality tests:<\/strong><\/p>\n

                                                                                                                                                                                                                              \n\n\n\n\n
                                                                                                                                                                                                                              Anderson Darling<\/td>\nRyan- Joiner<\/td>\nKolmogorov<\/p>\n

                                                                                                                                                                                                                              -Smirnov<\/td>\n

                                                                                                                                                                                                                              		Note the differences in result here. Recall that the AD test is badly affected by tied values, and it will usually say data are non-normal when they are actually normal if tied values are present. The Ryan Joiner test is better for tied values, and indicates that data are right on the edge of normal (which is similar to what we saw with the skew\/kurtosis calculation). The K-S test can be used as long as the cut-off is 0.10 rather than 0.05, however these data appear non-normal.<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              p = 0.014<\/td>\n>0.100<\/td>\n0.016<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                              To<\/strong> assess<\/strong> normality,<\/strong> always<\/strong> use<\/strong> more<\/strong> than<\/strong> one<\/strong> method:<\/strong><\/p>\n

                                                                                                                                                                                                                              For data to be normal, the histogram should look like a bell curve, the normal probability plot should have points falling close to the line, the SE of the skew and kurtosis should fall between -1.96 and +1.96, and the statistical tests should have a p value greater than 0.05 (or 0.10 for the K-S test). How did we do?<\/p>\n

                                                                                                                                                                                                                              Interpretation:<\/strong><\/span><\/p>\n\n\n\n\n\n\n\n\n\n
                                                                                                                                                                                                                              <\/td>\n\u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                               <\/td>\n

                                                                                                                                                                                                                              		\u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                              Conclusion<\/span><\/td>\n<\/tr>\n

                                                                                                                                                                                                                              Histogram: looks a bit skewed, but not too far off<\/td>\n<\/td>\nNormal<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              Normal Prob.Plot: the dots fall within the 95% conf.<\/td>\n<\/td>\nNormal<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              Skew & Kurtosis: fall within the range<\/td>\n<\/td>\nNormal<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              Anderson-Darling: p = 0.014<\/td>\n<\/td>\nNon-normal<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              Ryan-Joiner: p > 0.100<\/td>\n<\/td>\nNormal<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              Kolmogorov-Smironov: p = 0.016<\/td>\n<\/td>\nNon-normal<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                              \u00a0<\/strong>The Anderson Darling and K-S tests give a different result than the others, but since we know it is affected by tied values, we don\u2019t use that one since there are lot of tied values in the plot. The other two indicate that data are normal or very close. Conclusion: Data are normal<\/strong><\/p>\n

                                                                                                                                                                                                                              \u00a0<\/strong>\u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                              Example using non-normal data<\/strong><\/p>\n

                                                                                                                                                                                                                              For comparison, lets look at some obviously non-normal data: The pulses 2 column:<\/p>\n

                                                                                                                                                                                                                              \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                              Interpretation of the statistical Normality tests:<\/strong>\u00a0<\/strong><\/p>\n\n\n\n\n
                                                                                                                                                                                                                              Anderson Darling<\/td>\nRyan Joiner<\/td>\nKolmogorov- Smironov<\/td>\nThis time, all statistical tests indicate that data are non-normal, since the p-values are <0.05 in all cases. Even though there are tied values (so we don\u2019t trust the AD test), the RJ and KS tests are clearly non-normal.<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              p <0.005<\/td>\np<0.01<\/td>\nP<0.01<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                              \u00a0<\/strong>Evaluating the SE of the Skew and Kurtosis:<\/strong>\u00a0<\/strong><\/p>\n\n\n\n\n
                                                                                                                                                                                                                              Skewness1<\/td>\nKurtosis1<\/td>\nSEskew\u00a0= skew \u00f7 \/(6\/n)<\/td>\nSEkurt\u00a0 = kurtosis \u00f7 \/(24\/n)<\/td>\n<\/tr>\n
                                                                                                                                                                                                                              1.11<\/td>\n1.49<\/td>\n=1.11 \u00f7 \u221a (6\/91) = 4.32<\/td>\n=1.49 \u00f7 \u221a (24\/91) = 27.67<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                              \u00a0Interpretation:<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                              \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Conclusion<\/p>\n<\/div>\n

                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                              Histogram: quite skewed\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              Normal Prob.Plot: the dots are well off the line\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              Skew & Kurtosis: fall outside the range\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              Anderson-Darling: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              Ryan-Joiner: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              Kolmogorov-Smironov: p << 0.05\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-normal<\/p>\n

                                                                                                                                                                                                                              These data (Pulses 2) are clearly non-normal, as all normality tests confirm this.<\/p>\n

                                                                                                                                                                                                                              \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                              \u00a0Comments on interpreting normality testing:<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                              • Most statistical tests are \u201crobust\u201d to minor violations of normality, so as long as it is close, data can be considered normal for the purposes of doing the statistical tests to find out if groups differ or are related to each other.<\/li>\n
                                                                                                                                                                                                                              • It is usually easy to tell for data that are strongly normal or strongly non-normal<\/li>\n
                                                                                                                                                                                                                              • For datasets where it seems \u201cclose\u201d, the best tests are to just look at the probability plot (with confidence limits) and to check the skew and kurtosis.<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                                Parametric vs Non-parametric testing:<\/h3>\n
                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                • \u00a0 Parametric tests are better at picking up statistical differences than non- parametric tests, so we prefer to use them when we can
                                                                                                                                                                                                                                  \n<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                  Assumptions for parametric testing:<\/strong><\/p>\n

                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                  • Data are normally distributed (or close to normal) \u2013 test normality<\/strong><\/li>\n
                                                                                                                                                                                                                                  • The variances of the groups are similar to each other \u2013 test \u201cequal variances\u201d<\/strong><\/li>\n
                                                                                                                                                                                                                                  • Data are independent of each other \u2013\u00a0 study design point<\/strong><\/li>\n
                                                                                                                                                                                                                                  • Data were collected in a random fashion \u2013 study design point<\/strong><\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                    \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                                    Statistical Tests\u00a0<\/strong><\/h3>\n

                                                                                                                                                                                                                                    Which do you use? Here is a key to the basic tests<\/strong><\/p>\n

                                                                                                                                                                                                                                    Dichotomous key to statistical tests<\/h3>\n

                                                                                                                                                                                                                                    1a. Are you comparing the averages from groups of data?……………………………………………………………………………………………………………………………………………………………………. 2<\/strong><\/p>\n

                                                                                                                                                                                                                                    1b. Are you looking for a relationship between two variables………………………………………………………………………………………………………………………………………………………………. 11<\/strong><\/p>\n

                                                                                                                                                                                                                                    2a. Are you comparing the average from one group of numbers to a single predicted value?…………………………………………………………………………………………………………. 3<\/strong><\/p>\n

                                                                                                                                                                                                                                    2b. Are you comparing the average from more than one group to averages?…………………………………………………………………………………………………………………………………….. 4<\/strong><\/p>\n

                                                                                                                                                                                                                                    3a. Are your data normally distributed……………………………………………………………………………………………………………………..……………………………………………One Sample t-test
                                                                                                                                                                                                                                    \n<\/strong><\/p>\n

                                                                                                                                                                                                                                    3b. Are your data non-normal? ………………………………………………………………………………………………………………………………………………………………..One Sample Wilcoxin test<\/strong><\/p>\n

                                                                                                                                                                                                                                    4a. Are you comparing the averages of two groups?…………………………………………………………………………………………………………………………………………………………………………………5<\/strong><\/p>\n

                                                                                                                                                                                                                                    4b. Are you comparing the averages of three or more groups?……………………………………………………………………………………………………………………………………………………………….8<\/strong><\/p>\n

                                                                                                                                                                                                                                    5a. Are your data paired? (i.e. are you measuring something at time a and b on the same individuals?………………Paired t-test or non-parametric paired test
                                                                                                                                                                                                                                    \n<\/strong><\/p>\n

                                                                                                                                                                                                                                    5b. Are your data unpaired (i.e. are you just comparing the average values for your groups?…………………………………………………………………………………………………………….6<\/strong><\/p>\n

                                                                                                                                                                                                                                    6a. Are your data non-normal?…………………………………………………..…………………………………………………………………………………………………………………………Mann-Whitney U-test
                                                                                                                                                                                                                                    \n<\/strong><\/p>\n

                                                                                                                                                                                                                                    6b. Are your data normally distributed…………………………………………………………………………………………………………………………………………………………………………………………………………7<\/strong><\/p>\n

                                                                                                                                                                                                                                    7a. Are your data normal with equal variance…………………………………………………………………………………………………………………………………………………………..Student\u2019s t-test<\/strong><\/p>\n

                                                                                                                                                                                                                                    7b. Are your data normally distributed with unequal variance?…………………………………………………………………………………..Students t-test, with variance correction<\/strong><\/p>\n

                                                                                                                                                                                                                                    8a. Are you comparing averages of three or more groups without subgroups?…………………………………………………………………………………………………………………………………….9<\/strong><\/p>\n

                                                                                                                                                                                                                                    8b. Do your data have a subgroup or factor you want to compare (e.g. response of males and females within different treatment groups)?…………………………..10<\/strong><\/p>\n

                                                                                                                                                                                                                                    9a. Are your data normally distributed with equal variance…………………………………………………………………………………………………………………………………….One Way ANOVA<\/strong><\/p>\n

                                                                                                                                                                                                                                    9b. Are your data QRQ normal or have unequal variance?………………………………………………………………………………………………………………………………………Kruskall-Wallis Test<\/strong><\/p>\n

                                                                                                                                                                                                                                    10a. Are your data normally distributed …………………………………………………………………………………………………………………………………………………Two-Way (factorial) ANOVA<\/strong><\/p>\n

                                                                                                                                                                                                                                    10b. Are your data QRQ normal or have unequal variance?……………………………………………………………………………………………………………..No simple non-parametric test<\/strong><\/p>\n

                                                                                                                                                                                                                                    11a. Are your data normally distributed, with error distribution normal and heterogeneous?……………………………………………….Pearson Correlation and Regression<\/strong><\/p>\n

                                                                                                                                                                                                                                    11b. Are your data non-normal?…………………………………………………………………………………………………………………………………………………………………………Spearman Correlation<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                     <\/p>\n

                                                                                                                                                                                                                                    What are the stats telling us? Comparing Groups or Relationships<\/h3>\n

                                                                                                                                                                                                                                    We could be comparing groups or looking for relationships among variables.<\/p>\n

                                                                                                                                                                                                                                    \u00a0<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                    Our first step in doing statistical comparisons should be to plot the data with error bars, to get a visual image of what we are comparing.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                    Comparing groups<\/h4>\n

                                                                                                                                                                                                                                    One of the main types of statistical analyses we do is to compare groups of data. When we do this, we\u2019re really taking the average of the group, and comparing those averages, taking into consideration how variable the data are, and how many samples we have.<\/p>\n

                                                                                                                                                                                                                                     <\/p>\n

                                                                                                                                                                                                                                    The type of graph we plot depends on the shape of the data (see \u201cFrequency Distributions\u201d, p. 73-75).<\/p>\n

                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                    • If data are normally distributed, use a bar graph with error bars<\/li>\n
                                                                                                                                                                                                                                    • If data are non-normal, use a box and whisker plot
                                                                                                                                                                                                                                      \n<\/strong><\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                      \"\"<\/p>\n

                                                                                                                                                                                                                                      Looking at the relationship between two variables (plot x against y):<\/p>\n

                                                                                                                                                                                                                                      \"\"<\/p>\n

                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                      • Again, plot the data first, to see what the actual pattern looks like. Then use statistical analysis to see whether the relationship you see with your eye is statistically significant.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                        \n\n\n\n
                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                        Remember<\/strong> to<\/strong> test<\/strong> data<\/strong> for<\/strong> normality<\/strong> before<\/strong> deciding<\/strong> which test to use. Since you only have one data set here, you do not also need to test equal variance.<\/strong><\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                        Using statistics to compare groups:<\/h4>\n

                                                                                                                                                                                                                                        We use different tests depending on the number of groups, and whether data are parametric.<\/p>\n

                                                                                                                                                                                                                                        Comparing one group to a predicted value:<\/h5>\n
                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                        • Parametric: one-sample t-test (compares the mean)<\/strong><\/li>\n
                                                                                                                                                                                                                                        • Non-parametric: one-sample Wilcoxin Test (compares the median)<\/strong>
                                                                                                                                                                                                                                          \n<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                          One sample tests allow us to compare the average and variation in data from an observed group <\/strong>to a predicted value<\/strong>.<\/p>\n

                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                          • The Null Hypothesis is that the mean of your observed data is equal to the predicted value<\/li>\n
                                                                                                                                                                                                                                          • If P<0.05, then the means are significantly different.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                            Parametric Example:<\/strong> One sample t-test <\/em><\/strong><\/p>\n

                                                                                                                                                                                                                                            Study Question: Is the average resting pulse equal to 70 beats per minute?<\/strong><\/p>\n

                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                            • H0: Resting pulse (Pulse 1) = 70 beats per minute<\/li>\n
                                                                                                                                                                                                                                            • HA: Resting pulse (Pulse 1) \u2026 70 beats per minute\n
                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                              • Choose Stat from the property bar, and then Basic Statistics, and then 1 sample t<\/li>\n
                                                                                                                                                                                                                                              • Select your variable (e.g., pulse 1<\/strong>)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                Note<\/strong>; if your variable list is blank, just click on the white space in the \u201csamples in columns<\/p>\n

                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                • check the box for \u201cperform hypothesis test<\/strong>\u201d, and type in the value you\u2019re comparing (i.e. 70). Click OK<\/li>\n
                                                                                                                                                                                                                                                • Note that we have a simple alternate hypothesis here, of \u201cequal\u201d vs \u201cnot equal\u201d. If you want to be more specific and test if it is greater than or less than your hypothesized mean, click on \u201cOptions\u201d and change it<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                  \"\"<\/p>\n\n\n\n
                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                  Interpretation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                  p < 0.05, therefore the mean of the observed resting pulses is significantly different from 70.<\/p>\n

                                                                                                                                                                                                                                                  Trend: <\/strong>Since the mean (from the output) is 73.14, we can say that pulses are significantly higher than 70 beats\/min.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                   <\/p>\n

                                                                                                                                                                                                                                                  Non parametric Example:<\/strong><\/p>\n

                                                                                                                                                                                                                                                  One sample Wilcoxin<\/em><\/strong><\/p>\n

                                                                                                                                                                                                                                                  Use this test if data are non-normal.<\/p>\n

                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                  • Select Stat<\/strong>, Nonparametrics, <\/strong>then1-sample Wilcoxin<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                    Set it up the same way as for the 1-sample t-test, but test a median rather than a mean.<\/p>\n

                                                                                                                                                                                                                                                    \"\"<\/p>\n

                                                                                                                                                                                                                                                    Note that this non-parametric test also showed a significant p-value (p=0.028), but that this one is much closer to the 0.05 cutoff than the parametric test, reminding us that this is generally a less powerful test.<\/p>\n\n\n\n
                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                    Always<\/strong> choose the parametric test if your data are normal.<\/strong><\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                    Comparing 2 Groups: <\/strong><\/h5>\n

                                                                                                                                                                                                                                                    Study Question: is Pulse 1 different from Pulse 2?<\/em><\/p>\n

                                                                                                                                                                                                                                                    \u00a0<\/strong>Testing Parametric Assumptions: <\/strong>Before doing this test, remember to test each group of data for normality (p. 87-92). The other assumption for the parametric test is that variances in the groups are similar, so you will also have to do an \u201cequal variance\u201d test. We already know that data for Pulses 1 were normal, but the Pulses 2 data were not. Therefore, parametric test will give an invalid result and the correct test to do here is the non-parametric test. We will test for equal variance on p. 98. Examples with both tests are shown here so you can see how to do them.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                    Parametric test: Student\u2019s t-test <\/em>(AKA 2-sample t-test)<\/h6>\n

                                                                                                                                                                                                                                                    Step<\/strong> 1: <\/strong>Look at plot of data to see pattern<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                    Step<\/strong> 2: <\/strong>Setting up the data table<\/strong><\/p>\n

                                                                                                                                                                                                                                                    Minitab has two methods:<\/strong><\/p>\n

                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                    • The t-test allows us to put the data into adjoining columns(spreadsheet fashion) or to have them in one column. For this example, our Pulse 1 and Pulse 2 data are already in adjoining columns so we\u2019ll pick this option.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                      Step 3: <\/strong>do the test: <\/strong>Choose 2 sample t-test <\/strong>from the Stat\/Basic Statistics drop-down menu.<\/p>\n

                                                                                                                                                                                                                                                      Note the box here where it says \u201cassume equal variance\u201d … if you have tested equal variance and they are statistically equal, then check this box.<\/p>\n

                                                                                                                                                                                                                                                      Output for t-test:<\/strong><\/p>\n

                                                                                                                                                                                                                                                      \"\"<\/p>\n\n\n\n
                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                      Interpretation: p < 0.05. Therefore, If assumptions are verified (normal + equal variance), then we would conclude that Pulse 1 is OHVV WKDQ Pulse 2.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                      \u00a0Assumptions:<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                      1.\u00a0\u00a0\u00a0 Test Normality: Pulse 2 was not normal (see p. 92)(this<\/strong> means the t-test result was not valid)<\/strong><\/p>\n

                                                                                                                                                                                                                                                      2.\u00a0\u00a0\u00a0 Test Equal Variance: do variance test as described below<\/p>\n

                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                      Test<\/strong> for equal variance for t-test:<\/strong><\/p>\n

                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                      • Choose \u201c2 Variances\u201d <\/strong>from the Stat\/Basic Statistics drop-down menu.<\/li>\n
                                                                                                                                                                                                                                                      • Select your variables that you are comparing (in this case, I chose the samples in dif. columns, but if your data were set up in a single column the result would be the same; see p. 100 for example). Click ok<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                        \"\"<\/p>\n

                                                                                                                                                                                                                                                        Minitab 21 uses Bonnett and Levene Tests and produces some companion graphs. The boxplot shows the range of the data,so we can see that the variation is Pulse 2 is much higher than that for Pulse 1<\/p>\n

                                                                                                                                                                                                                                                        The interval plot shows us the difference in the standard deviation for the two groups, with confidence limits, so we can see that they don\u2019t overlap.<\/p>\n

                                                                                                                                                                                                                                                        Note the outputs for the Equal Variance test in the top right corner.<\/p>\n

                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                        • Both show that p < 0.05, so we reject the null hypothesis <\/em><\/strong>that variances are equal<\/li>\n<\/ul>\n\n\n\n
                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                          Important:<\/strong><\/p>\n

                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                          • If the data passed the normality test, but failed the equal variance test, you may still be able to do a t-test<\/li>\n
                                                                                                                                                                                                                                                          • Sample size must be >10<\/li>\n
                                                                                                                                                                                                                                                          • Leave \u201cassume equal variance\u201d box unchecked.<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                            Therefore, we conclude that variances are different<\/p>\n

                                                                                                                                                                                                                                                            Interpretation:<\/h5>\n

                                                                                                                                                                                                                                                            The data did not pass the assumption tests, therefore any result from the t-test is not valid and cannot be trusted<\/p>\n

                                                                                                                                                                                                                                                            Therefore, we would discard our t-test result, and carry out a Mann-Whitney U-test.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                            \n

                                                                                                                                                                                                                                                            Nonparametric test:<\/strong> Mann-Whitney<\/em><\/strong> U-test <\/em><\/strong>(Use this test if data are not normal)<\/h4>\n

                                                                                                                                                                                                                                                            Setting up the data:<\/strong><\/p>\n

                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                            • You must have your data in separate columns for this test.<\/li>\n
                                                                                                                                                                                                                                                            • No assumption testing is needed for this test.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                              Carry out the test: <\/strong>Choose the Mann-Whitney test from the Nonparametrics <\/strong>drop down menu (from Basic Statistics<\/strong>)<\/p>\n

                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                              • Select your variables, and click ok<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                This test compares medians (middle values in a list that is ranked from smallest to largest) rather than means. <\/strong>Minitab Output:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                Conclusion: <\/strong>no assumptions are required, and there is a signifcant difference among groups, since p < 0.05<\/strong><\/p>\n

                                                                                                                                                                                                                                                                Trend statement: <\/strong>The pulse rate of students after running (Pulse 2) was significantly higher than the pulse rate before running (Mann-Whitney U-test, P=0.0048).<\/p>\n

                                                                                                                                                                                                                                                                This is the correct test, so we can trust our result.<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                Since the non-parametric test doesn\u2019t need us to test assumptions, why not just use it all the time?<\/strong><\/p>\n

                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                • When sample sizes are large, the non-parametric and parametric tests often come to the same general conclusion (i.e. the differences are significant or they are not), so it seems like more work to have to do all this extra testing. However, when patterns are not as clear (i.e. when p-values are close to the 0.05 cutoff), or for small sample sizes, it can make a major difference. The Parametric tests have greater POWER <\/strong>to see a difference if it actually is present, so we always use these ones if we can. Therefore, we must test assumptions first.<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                  A second two-sample example, t-test with data in a single column.<\/strong><\/h4>\n

                                                                                                                                                                                                                                                                  Study question: Do males and females have the same average resting pulse?<\/strong><\/p>\n

                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                  • \u00a0<\/strong>Choose the 2-sample t-test following the instructions on p. 97 (Stat,<\/strong> Basic Stats, 2-sample t)<\/strong><\/li>\n
                                                                                                                                                                                                                                                                  • When selecting your variable, you\u2019ll be able to choose the \u201csamples in the same column option\u201d in the t- test menu, but you\u2019ll still have to separate them to check for normality. You can do this manually, by splitting the worksheet (see p. 46), or by \u201cunstacking\u201d the column.<\/em><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                    \u201cUnstacking\u201d a column to test subgroups separately:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                    • Choose Data<\/strong> from the upper property bar, then Unstack<\/strong> columns <\/strong>from the drop-down menu:<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                      \"\"<\/p>\n

                                                                                                                                                                                                                                                                      \u201cSuperscripts\u201d refer to your grouping variable, so if you want to separate out the two sexes (male and female), then choose \u201csex\u201d as your subscript.<\/p>\n

                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                      • Check the boxes for \u201cafter last column in use\u201d and \u201cname the columns\u201d so that the data will appear in your worksheet.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                        Test normality<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                                                        Pulse 1 males: The probability plot has most values in the 95% confidence bands, and the p- Value for the Ryan Joiner statistical test is >0.05 (too many tied values for AD)<\/p>\n

                                                                                                                                                                                                                                                                        Conclusion: Normal<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        Pulse 1 females: The probability plot has all values in the 95% confidence bands, and the p-Value for the Ryan Joiner statistical test is >0.05<\/p>\n

                                                                                                                                                                                                                                                                        Conclusion: Normal<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                                                         <\/p>\n

                                                                                                                                                                                                                                                                        Test equal variance<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        Conclusion: variance is equal since p>0.05 for both tests<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        Important: do not stop here… these tests only assessed assumptions. Now you must do the test to test your study question.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                         <\/p>\n

                                                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                                                         <\/p>\n

                                                                                                                                                                                                                                                                        Carry out the 2-sample t-test<\/strong>, <\/strong>this time with data in one column.<\/p>\n

                                                                                                                                                                                                                                                                        Since variances are equal, check box for \u201cAssume equal variances\u201d.<\/p>\n

                                                                                                                                                                                                                                                                         <\/p>\n

                                                                                                                                                                                                                                                                        Two-sample T for pulse1<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        \"\"<\/p>\n

                                                                                                                                                                                                                                                                        Conclusion: <\/strong>Assumptions are satisfied, so test is valid and groups are significantly different since p<0.05.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                        \u00a0<\/strong>Trend statement: <\/strong>Average pulse rate is significantly higher for females (group 2) than for males (group 1) (t-test, p=0.008)<\/strong><\/p>\n

                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                        • \u00a0Remember to always write a trend statement giving the clear trend in the data, with the name of the statistical test and the p-value in brackets.<\/strong>
                                                                                                                                                                                                                                                                          \n<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                          \u00a0<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                          How should you report p-values?<\/strong><\/p>\n

                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                          • When possible, report the actual p-value if you have it from the computer test<\/li>\n
                                                                                                                                                                                                                                                                          • Rule of thumb: only report to a max. of 3 decimal places<\/strong>:<\/li>\n
                                                                                                                                                                                                                                                                          • If the p is given as 0.000 or 0.0000 then write as p<0.001\n
                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                            • Even if the computer reports a value like 0.000, NEVER <\/strong>say p=0.000, since probability is always higher than zero<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
                                                                                                                                                                                                                                                                              \n\n\n\n
                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                                                                              Interesting side note:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                              These questions illustrate how many questions can come out of a single dataset, if the sample size is large enough, and the study design incorporated many variables. In the previous example, we tested whether two different groups of students were different from each other. In this example, we are testing to see whether an individual group of students can show significant differences from one time<\/p>\n

                                                                                                                                                                                                                                                                              to another. For this to work, the data must be \u201cpaired\u201d… i.e. we must identify each individual and be sure we know his\/her before and after result.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                              Two sample testing when values are \u201cPaired\u201d <\/strong>Study<\/em><\/strong> question Did the pulse rates of the individual students go up after running?<\/em><\/strong><\/p>\n

                                                                                                                                                                                                                                                                              Note<\/strong> the<\/strong> importance<\/strong> of<\/strong> how the question is worded:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                              If we want to know if the average pulse rate goes up after running, we would do a regular t-test or Mann-Whitney U-test (if non-normal). But because we have data on each individual person, we can see if the individual rates go up by using a test that focuses on individual responses, called a \u201cpaired\u201d test. This is particularly useful if high variability in individuals makes it difficult to see a pattern in the average response, and the paired test will have more power to see the differences than the regular one.<\/p>\n

                                                                                                                                                                                                                                                                              How<\/strong> Paired tests work:\u00a0 <\/strong>In paired tests, we look at a measured value for known individuals at two (or more) times.<\/p>\n

                                                                                                                                                                                                                                                                              The example below is for the pulses in the group who ran in place for one minute, so you can see that their pulse rates all went up, though not by the same amount.<\/p>\n

                                                                                                                                                                                                                                                                              \"\"<\/p>\n

                                                                                                                                                                                                                                                                              We can test this using a one-sample test, to see if the group of numbers in the difference column is significantly different from zero.<\/p>\n

                                                                                                                                                                                                                                                                              For parametric data, there is a t-test that calculates the differences and does the one sample test automatically in a single step. For non-parametric data, we have to do the extra step ourselves.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                              Step 1: <\/strong>separate out the runners from the non- runners in both our resting pulse and our after running pulse columns (pulse 1 and pulse 2). This will give us four groups of data as you can see from the graph at right.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                              Remember<\/strong> to<\/strong> plot<\/strong> your<\/strong> data<\/strong> so<\/strong> you<\/strong> can<\/strong> see<\/strong> what you are comparing!<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                              • We can do this manually (by copying and pasting into additional columns, \u201csplitting\u201d the worksheet into multiple smaller worksheets to work on using Minitab or \u201cunstacking\u201d the columns.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                \"\"<\/p>\n

                                                                                                                                                                                                                                                                                Figure 1. Comparison of pulse rates for a group of students before and after one group ran for 1 Min.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                Step 2: Compare pulse 1 and pulse 2 for the non-runners (control) and then the runners (test)<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                First: Test assumptions:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                1. <\/strong>Normality testing<\/strong>: test normality of all four groups, using Ryan Joiner (since many tied values)<\/p>\n

                                                                                                                                                                                                                                                                                \"\"<\/p>\n

                                                                                                                                                                                                                                                                                2.\u00a0\u00a0\u00a0 <\/strong>Equal variance<\/strong>: Running group p=0.004\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Non-running group: p=0.624<\/p>\n

                                                                                                                                                                                                                                                                                (Variances between before & after running: non-equal for runners, equal for non-runners)<\/p>\n

                                                                                                                                                                                                                                                                                Conclusion from assumptions:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                • Runners: one group was normal the other non normal; variances were not equal<\/li>\n
                                                                                                                                                                                                                                                                                • Non-runners: data were normal, and variances were equal<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                  Therefore, both the parametric and non-parametric paired tests are needed to assess whether there are differences between the individuals pulse rates at the two different times.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                                  Parametric data – use the Paired t-test<\/strong><\/h4>\n

                                                                                                                                                                                                                                                                                  Run this test using the Non-runners\u2019 pulse rate data, since these data were normal and had equal variance.<\/p>\n

                                                                                                                                                                                                                                                                                  Make sure the data to be tested are in separate columns, e.g. non-runners at time 1 and non-runners at time 2.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                  • Select Stat <\/strong>from the property bar, and Basic Statistics.<\/strong><\/li>\n
                                                                                                                                                                                                                                                                                  • Choose the paired t test <\/strong>from the drop down menu.<\/li>\n
                                                                                                                                                                                                                                                                                  • Choose variables so that your data for the non-runners from time 1 is being compared to non-runners in time 2.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                    Minitab Output:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                     <\/p>\n

                                                                                                                                                                                                                                                                                    Paired T-Test and CI: pulse1nonran, pulse2nonran<\/strong>\u00a0<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                                                    Since p>0.05, we accept the null hypothesis, and say there is no difference in the groups.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                    Notice the difference here in what is being tested: instead of comparing one mean to another, it tests whether the mean difference is equal to zero<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                                                    Non-parametric paired two sample test:<\/strong><\/h4>\n
                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                    • If data are non-normal, then you must use a non-parametric test. There is no one-step non-parametric paired test, but you can carry it out in two steps.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                      Step one: <\/strong>Subtract your time 1 column from your time 2 column (as in the table on p. 103). You can do this in Excel, and copy and paste the values into Minitab.<\/p>\n

                                                                                                                                                                                                                                                                                      Step two: <\/strong>Carry out a one-sample Wilcoxin test to test whether your median is different from zero (as shown on p. 96).<\/p>\n

                                                                                                                                                                                                                                                                                      Choose your column with the runner differences, and check the box for testing that the median is zero:<\/p>\n

                                                                                                                                                                                                                                                                                      Minitab Output:<\/p>\n

                                                                                                                                                                                                                                                                                      \"\"
                                                                                                                                                                                                                                                                                      \n<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                      Wilcoxon Signed Rank Test: run difs<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                      Since p<0.001, then we reject the Null hypothesis; i.e. the pulse rates are significantly different between time 1 and time 2.<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                      Comparing >2 groups of data<\/strong><\/h4>\n\n\n\n
                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                      Balanced<\/strong> Designs<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                      Some packages (e.g. Excel) assume a \u201cbalanced design\u201d; that means that there have to be the same number of observations in each group being studied. In Minitab, a balanced design is not necessary for simple ANOVA (although it is in 2-way ANOVA), but note that if your groups have too few observations, you have very little \u201cpower\u201d to pick out differences.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                      Analysis of Variance (ANOVA) or the non- parametric equivalent (Kruskall-Wallis) allows us to compare more than 2 groups of data. As with the t-test, we are comparing means, and the null hypothesis is that the means are equal.<\/p>\n

                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                      • If p<0.05, the means are statistically different (assuming the assumptions of ANOVA are met).<\/strong><\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                        Setting up your data:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                        Your dataset can be set up in the worksheet so that the responses being measured are all in one column, and the groups that you are comparing are given as categories in a separate column (See our pulses dataset on p. 83 as an example). Alternatively, the data can be set up so your groups are in adjoining columns. For the ANOVA, these have special names: If data are in one column, Minitab refers to this as \u201cstacked<\/strong>\u201d. If data are in adjoining columns, Minitab refers to this set-up as \u201cunstacked<\/strong>\u201d.<\/p>\n

                                                                                                                                                                                                                                                                                        Parametric Data: Analysis of Variance (ANOVA)<\/h5>\n

                                                                                                                                                                                                                                                                                        One-Way Design <\/strong>(comparing groups without subgroups)<\/p>\n

                                                                                                                                                                                                                                                                                        Example of comparing three or more groups of data; \u201cstacked\u201d data. Study Question: Are resting pulse rates different depending on normal activity level?<\/em><\/strong><\/p>\n

                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                        • This will give three groups to compare, with three activity levels.<\/li>\n<\/ul>\n\n\n\n
                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                                          Important design note: The design of this study is highly unbalanced, and the group reporting low activity level was very small in number; it may be too small to give random and independent data. Therefore, ANOVA result should be treated with caution.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                          Assumptions:<\/p>\n

                                                                                                                                                                                                                                                                                          ANOVA assumes that data are parametric, which means:<\/p>\n

                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                          • Normally distributed<\/li>\n
                                                                                                                                                                                                                                                                                          • Variances are equal<\/li>\n
                                                                                                                                                                                                                                                                                          • Design should be set up so data are independent and random.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                            1. <\/strong>Test for normality <\/strong>for each group of data as explained in the earlier section<\/p>\n

                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                            • When this was done, all groups were normal (p>0.05)<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                              2. <\/strong> Equal Variance Test<\/strong>: Do not use the equal variance test you used for t-tests!<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                              • To test equal variance on three or more groups of data, we need to use the equal variance test found in the ANOVA menu. The one in the t-test menu only tests variance for two groups of data, not for three or more.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                Select Stat <\/strong>from the property bar, then ANOVA<\/strong>. Look part way down the drop-down menu, and select Test for Equal Variances<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                Choose your response variable (in this case, pulses 1 to assess the resting pulse rates) and your factor variable (in this case, activity level), and click okay.<\/p>\n

                                                                                                                                                                                                                                                                                                Equal variances result:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                p>0.05 so variances are equal<\/p>\n

                                                                                                                                                                                                                                                                                                Notice in the graph how the variances are all very similar with a lot of overlap… that shows that the variances are very similar.<\/p>\n

                                                                                                                                                                                                                                                                                                Conclusion: <\/strong>Although we need to be cautious about interpreting patterns about the low activity level due to the small and unbalanced sample size, our data meet the assumptions of the ANOVA, so we can run the test.<\/p>\n

                                                                                                                                                                                                                                                                                                Method for One-Way ANOVA:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                • Choose \u201cstat<\/strong>\u201d from the property bar, then choose \u201cANOV<\/strong>A\u201d, then choose \u201cone-way<\/strong>\u201d. One way analysis of variance means that you are simply comparing 3 or more groups of data, without any subgroups in them.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                  This puts you into the menu for the ANOVA with data set up in one column<\/p>\n

                                                                                                                                                                                                                                                                                                  The<\/strong> Null hypothesis <\/strong>is that<\/p>\n

                                                                                                                                                                                                                                                                                                  X1 = X2 = X3<\/p>\n

                                                                                                                                                                                                                                                                                                  therefore, a significant p-value (p<0.05) means that at least one group is different from the others.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                  • Select your variables<\/strong>\n
                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                    • The \u201cresponse<\/strong>\u201d variable is the one where your measurements are, for example one of the columns of pulse rates.<\/li>\n
                                                                                                                                                                                                                                                                                                    • The \u201cfactor<\/strong>\u201d is the column where the different groupings or levels are shown.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                                      In our pulses dataset, we have 3 levels of activity, so we could choose activity level as our factor.<\/p>\n

                                                                                                                                                                                                                                                                                                      Click<\/strong> on<\/strong> \u201cokay\u201d,<\/strong> and<\/strong> the<\/strong> output<\/strong> will look like:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                                       <\/p>\n

                                                                                                                                                                                                                                                                                                      \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                      Interpretation: <\/strong>p>0.05, so the means are not significantly different <\/strong>between the groups. We conclude that activity level does not <\/strong>have an effect on resting pulse in these students.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                      Remember to always plot your data <\/strong>to see the patterns. This helps you to determine what you actually want to test, and can help you interpret your data patterns. You can see from the graph at right that although it looks like the people with low activity level had a higher resting pulse rate, the variability in the data means that there is no significant difference<\/p>\n

                                                                                                                                                                                                                                                                                                      <\/div>\n
                                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                                      \"\"<\/p>\n

                                                                                                                                                                                                                                                                                                      Conclusion: we have done the appropriate<\/strong> test, so we can trust our result.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                      Trend Statement: <\/strong>There is no significant difference among resting pulses in the three groups (ANOVA, p = 0.155).<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                      Non-parametric data: Kruskall-Wallis Test<\/strong><\/h5>\n<\/div>\n
                                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                                      If data were not <\/strong>normal [and could not be made normal through transformation], we would use the Kruskal-Wallis test, which compares medians.<\/p>\n

                                                                                                                                                                                                                                                                                                      Data Setup:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                      • Data must be set up so that the response variable (in this case, Pulse rate) is in one column, and the grouping variable is in another column.<\/li>\n
                                                                                                                                                                                                                                                                                                      • Select \u201cnon-parametrics<\/strong>\u201d from the \u201cstats<\/strong>\u201d menu, and choose the Kruskal-Wallis.<\/li>\n
                                                                                                                                                                                                                                                                                                      • Select your variables:<\/li>\n<\/ul>\n\n\n\n
                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                                                                                        Transforming<\/strong>:<\/p>\n

                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                        • If your data are not normal and don\u2019t have equal variance, you can use the non-parametric test, or you can try transforming the data. Common transformations are the log transformation, square root transformation, and inverse tranformation <\/strong>\u2013 To transform, convert all values in each column of data being analysed to the transformation, and try your analyses again.<\/li>\n
                                                                                                                                                                                                                                                                                                        • Remember to always report your actual data (in the text or in graphs) and not the transformed data<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                           <\/p>\n

                                                                                                                                                                                                                                                                                                          \"\"<\/p>\n

                                                                                                                                                                                                                                                                                                          Trend statement: <\/strong>There is no significant difference among the pulse rates for the students who have the different activity levels (Kruskal-Wallis test, p=0.153)<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                                                          Example of comparing three or more groups of data; \u201cun-stacked\u201d data<\/strong><\/h4>\n

                                                                                                                                                                                                                                                                                                          Study Question: <\/strong>Is there a difference among the runners and non-runners, before and after running?<\/em><\/p>\n

                                                                                                                                                                                                                                                                                                          Running Group<\/p>\n

                                                                                                                                                                                                                                                                                                          Data Setup: <\/strong>For this method, put the data into 4 separate columns, and run a One-Way ANOVA<\/strong>, unstacked<\/strong>.<\/p>\n

                                                                                                                                                                                                                                                                                                          \"\"<\/p>\n

                                                                                                                                                                                                                                                                                                          A quick plot of the data shows no dif. in pulse between time 1 and 2 for the non-runners, but there seems to be a big difference for the runners. What we would like to know is whether that difference is significant.<\/p>\n

                                                                                                                                                                                                                                                                                                          The null hypothesis is that there is no difference among groups.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                          \u00a0<\/strong>Assumptions<\/strong>:<\/p>\n

                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                          • Normality: <\/strong>some groups were normal, and others were not. The non-normal groups were relatively close to normal.<\/li>\n
                                                                                                                                                                                                                                                                                                          • Variances: <\/strong>were not equal<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                            Conclusion from Assumptions:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                            Data are non-parametric (if<\/strong> even<\/strong> one<\/strong> group<\/strong> you<\/strong> are<\/strong> comparing<\/strong> don\u2019t<\/strong> fit<\/strong> assumptions,<\/strong> then<\/strong> your<\/strong> data<\/strong> are<\/strong> nonparametric)<\/strong>, so should either be transformed or a non-parametric test should be chosen. However, ANOVA is \u201crobust\u201d to minor violations of assumptions, so since data are close to normal, it may be okay to do the parametric test. To be sure, do both the parametric and non-parametric test and compare.<\/p>\n

                                                                                                                                                                                                                                                                                                            ONE-WAY ANOVA, \u201cunstacked\u201d<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                            • Select Stat <\/strong>from the property bar, then choose ANOVA <\/strong>from the drop down menu.<\/li>\n
                                                                                                                                                                                                                                                                                                            • Choose One-Way (Unstacked)<\/strong><\/li>\n
                                                                                                                                                                                                                                                                                                            • Select variables (these must be in adjoining columns in your dataset)<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                                                                                                              Output (ANOVA table)<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              \"\"<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                                                                                                              Interpretation: <\/strong>p<0.05, so there is a significant difference in the groups. However, we don\u2019t know which groups are different.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                              Important<\/strong>\u00a0<\/strong>note:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              p<0.05, so groups are significantly different. However, there is no way to know from the simple One-Way ANOVA which group is different from which other(s). We need to do further testing to figure this out.<\/p>\n

                                                                                                                                                                                                                                                                                                              Multiple Comparison Tests:<\/p>\n

                                                                                                                                                                                                                                                                                                              \u00a0<\/strong>ANOVA is designed to tell us if there are any differences, but not which groups are different. For this: do Multiple Comparison Tests<\/strong>\u00a0<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                                                                                                              Type<\/strong> I Error Inflation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              Recall that if we do multiple comparisons on the same data set (e.g. group 1 vs group 2<\/strong>, group 1 vs group 3 <\/strong>and group 2 vs group 3<\/strong>) then our probability of getting an incorrect interpretation goes up. If you have three groups, that probability goes up from 0.05 to 0.14 for all the comparisons taken together.<\/p>\n

                                                                                                                                                                                                                                                                                                              \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              This means that if we want to compare the individual groups, we need to use a special test that calculates a \u201cfamily error rate\u201d (the error rate for all the groups considered together) rather than individual error rates. These are called \u201cpost-hoc\u201d<\/em> or \u201cmultiple comparison\u201d tests.<\/p>\n

                                                                                                                                                                                                                                                                                                              \u00a0<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                              If you find a significant result with your ANOVA (i.e. if you run ANOVA and your p<0.05), then you should run a multiple comparison test to see which of the groups are significantly different from each other.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                                                                              \n

                                                                                                                                                                                                                                                                                                              The most commonly used multiple comparison test is Tukey\u2019s Test. To do a Multiple Comparison test, click on \u201ccomparisons\u201d in the initial ANOVA menu<\/p>\n

                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                              • Check the box for Tukey\u2019s test, and click OK<\/li>\n
                                                                                                                                                                                                                                                                                                              • Then run the ANOVA as before. (You will see the ANOVA output, and some additional information that lets you compare each group)<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                Output:<\/p>\n

                                                                                                                                                                                                                                                                                                                \"\"<\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                Interpretation: <\/strong>Look for groupings that have a different letter under the \u201cGrouping\u201d heading. In the example above, the pulse 1 and pulse 2 groups for the non-runners, and the pulse 1 group for the runners all have the same letter, so they are not significantly different. However, the pulse 2 data for the runners has a different letter, so that means it is significantly different than all the other groups. To find out how it is different, look at the column with the means: Pulse2ran clearly has a higher mean value than the other ones.<\/p>\n

                                                                                                                                                                                                                                                                                                                 <\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                                Therefore, to report this trend, you\u2019d write something like:<\/strong>
                                                                                                                                                                                                                                                                                                                \nThere was a significant difference in pulse rates in the groups of students (ANOVA, p <0.001). There was no difference in resting pulses of the two groups of students (runners vs non runners) prior to running, but there was a significant increase in pulse rate in the running group after running (Tukeys test, p<0.05).<\/p>\n

                                                                                                                                                                                                                                                                                                                Comparing >2 groups of data with subgroups<\/h5>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                Factorial Analysis of Variance\u00a0 <\/strong>(This can be a 2-way ANOVA, 3-Way, etc.)<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                \u00a0<\/strong>If your data contain distinct subgroups, you can run an ANOVA that lets you test the effects of those subgroups, or factors, on your response at the same time as testing your main grouping factor. This is called a factorial analysis.<\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                Important note: Minitab requires a \u201cbalanced design\u201d for 2-way ANOVA<\/strong><\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                                Consider an example where a researcher would like to know whether a particular feed supplement would increase growth in chickens, and whether the sex of the chicken would affect how it worked. This is a standard \u201ctwo-way\u201d design, where we are looking at two factors at the same time.<\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                Data<\/strong> Set-up<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                Set up the data in Minitab so that the response data (weight) are in one column, and the grouping factors are in other columns:<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n
                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                This is known as a factorial table. If your data can be set up in this way to show groupings, then it is a good candidate for a two-way ANOVA.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                                Table 1. Weight (grams) after two weeks in a group of bantam chicks on two different diets; the standard diet, and one supplemented by blueberry extract.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                                                                                                                                                                                                                                                                 <\/p>\n

                                                                                                                                                                                                                                                                                                                 <\/p>\n

                                                                                                                                                                                                                                                                                                                males<\/td>\n

                                                                                                                                                                                                                                                                                                                		supplement<\/p>\n

                                                                                                                                                                                                                                                                                                                 <\/p>\n

                                                                                                                                                                                                                                                                                                                590<\/td>\n

                                                                                                                                                                                                                                                                                                                		control<\/p>\n

                                                                                                                                                                                                                                                                                                                 <\/p>\n

                                                                                                                                                                                                                                                                                                                440<\/td>\n<\/tr>\n

                                                                                                                                                                                                                                                                                                                <\/td>\n530<\/td>\n570<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n550<\/td>\n509<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n570<\/td>\n510<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n650<\/td>\n589<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                females<\/td>\n530<\/td>\n550<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n580<\/td>\n420<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n520<\/td>\n440<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n520<\/td>\n520<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                <\/td>\n560<\/td>\n370<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                Assumption<\/strong> Testing:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                • Normality (A-D): <\/strong>\n
                                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                                  • control, female, p=0.64<\/li>\n
                                                                                                                                                                                                                                                                                                                  • control, male, p=0.52<\/li>\n
                                                                                                                                                                                                                                                                                                                  • supplement, female, p=0.21<\/li>\n
                                                                                                                                                                                                                                                                                                                  • supplement, male, p=0.59<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                    All data are normally distributed<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                    Equal Variance:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                                    • Use the \u201cTest for equal variances\u201d in the ANOVA menu.<\/li>\n
                                                                                                                                                                                                                                                                                                                    • Set it up as shown below:<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                      \"\"<\/p>\n

                                                                                                                                                                                                                                                                                                                      Equal Variance Output:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                      p>0.05, therefore accept the null <\/strong>that variances are equal.<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                                                      Now<\/strong> run the test:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                                      • \u00a0<\/strong>Select Stat <\/strong>from the property bar, then choose ANOVA,<\/strong> then Two-Way<\/strong><\/li>\n
                                                                                                                                                                                                                                                                                                                      • \u00a0<\/strong>Select your variables:\n
                                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                                        • Response = <\/strong>the data column, so, weight Row and column factors:<\/strong><\/li>\n<\/ul>\n<\/li>\n
                                                                                                                                                                                                                                                                                                                        • Before doing a 2-way ANOVA, you should set up a table to assess your factors<\/li>\n
                                                                                                                                                                                                                                                                                                                        • Use that to decide which is a row factor and which is a column factor.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                                                                           <\/p>\n

                                                                                                                                                                                                                                                                                                                          First, plot the data to see the patterns you want to test:<\/strong><\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                          \"\"
                                                                                                                                                                                                                                                                                                                          \n<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                                                                          Two Way ANOVA Output:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                          \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                                          • <\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                            Interpretation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                            Step 1: look at p-values for individual factors<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                            Sex: p=0.051<\/p>\n

                                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                                            • Since p>0.05, we accept the null and conclude there is no difference Food: p = 0.011\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 since p<0.05, we reject the null and conclude there is a difference<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                              Step 2: look at the p-value for interaction<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                              Interaction: p=0.577,<\/p>\n

                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                              • Since p>0.05, there is no interaction with respect to the weight response among the factors (i.e. weight acted the same way for both sexes (was lower) regardless of the food type).<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                Interaction<\/strong> refers to whether the response acts the same way for the different levels of the factors.<\/p>\n

                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                • e.g. If the growth was higher on blueberries for males, and lower for blueberries for females, that would be an example of a different response for the two sexes… This would give a significant interaction.<\/li>\n
                                                                                                                                                                                                                                                                                                                                • Since growth was lower for females than males for both food types, it means there was no interaction.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                  To see the actual trend, we look at the graph:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                  Chickens that were fed a diet supplemented by blueberry extract grew significantly larger than those on a standard diet (Two-Way ANOVA, p=0.012), and there was no significant difference in response between male and female chicks (Two-Way ANOVA, p=0.057). There was no significant interaction between food type and sex of chicks (Two-Way ANOVA, p=0.58), indicating that the food supplement affected weight in the same way for male and female chicks.<\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                                                                                  Remember to always say which is higher\/lower than the other, if you can<\/p>\n

                                                                                                                                                                                                                                                                                                                                  To report the interaction, be sure you explain it in English – do not just say there is a significant interaction or not.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                  Relationships between variables: Regression and Correlation<\/h5>\n

                                                                                                                                                                                                                                                                                                                                  Correlation analysis tests whether there is a correlation between two continuous variables Regression analysis takes it a step further to give us the equation of the line and give information on how much of the variation in the points can be statistically related to the other variable.<\/p>\n

                                                                                                                                                                                                                                                                                                                                  Step 1: Plot the data to see if there is a relationship between the variables <\/strong>e.g. Is Weight related to Height of the students in the pulses study? If so, what is the equation of the line.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                  \u00a0<\/strong>The graph suggests that the weight of the students increases as their heights increase.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                  <\/div>\n
                                                                                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                                                                                  \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                                                                                  We can run correlation <\/strong>analysis to see how strongly related (correlated) the variables are, and we can run regression <\/strong>analysis to get the equation of the \u201cbest\u201d line that describes the relationship.<\/p>\n

                                                                                                                                                                                                                                                                                                                                  We can use regression <\/strong>to describe relationships, or to generate an equation (usually called a \u201cmodel\u201d that can be used to predict for values<\/p>\n

                                                                                                                                                                                                                                                                                                                                  Assumptions: Correlation
                                                                                                                                                                                                                                                                                                                                  \n<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                                                  • Each variable must be normally distributed<\/li>\n
                                                                                                                                                                                                                                                                                                                                  • The relationship must be linear<\/li>\n
                                                                                                                                                                                                                                                                                                                                  • The residuals (errors) must be normally distributed<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                    Assumptions: Regression<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                                                    • Each variable must be normally distributed<\/li>\n
                                                                                                                                                                                                                                                                                                                                    • The relationship must be linear (for linear regression)<\/li>\n
                                                                                                                                                                                                                                                                                                                                    • The residuals must be evenly distributed along the line<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                      Some<\/strong> important<\/strong> definitions:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                                                      1. Independent variable: the variable that is fixed for our analysis… what we are comparing the other variable to plot this variable on the x-axis<\/li>\n
                                                                                                                                                                                                                                                                                                                                      2. Dependent variable: the variable that \u201cdepends on\u201d the x-axis variable; the one we expect to change or vary with our fixed variable. Plot this variable on the y-axis<\/li>\n
                                                                                                                                                                                                                                                                                                                                      3. Residual: the amount of vertical (y-axis) variation of the observed points from the regression line (i.e. the \u2018y\u2019 value of your point minus the \u2018y\u2019 value on your line)<\/li>\n
                                                                                                                                                                                                                                                                                                                                      4. Standardized residual: (residual) \u00f7 (standard deviation of residual). Standardized residuals have been standardized to have a variance of 1. Standardized residuals over 2 are usually considered large, and points with that large a residual are often considered to be outliers outside of our observed range.<\/li>\n<\/ol>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                                        \n

                                                                                                                                                                                                                                                                                                                                        Method:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                                                        • Test normality as described before<\/li>\n
                                                                                                                                                                                                                                                                                                                                        • Test linearity by looking at the graph and deciding if the points form a straight line<\/li>\n
                                                                                                                                                                                                                                                                                                                                        • Test the normality and linearity of the residuals through specialized menus in the Regression menu.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                                                          Correlation<\/strong><\/h5>\n

                                                                                                                                                                                                                                                                                                                                          \"\"<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                          \n

                                                                                                                                                                                                                                                                                                                                          The graph above shows a relationship between height and weight of the student participants.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                                                          • How related are they?<\/li>\n
                                                                                                                                                                                                                                                                                                                                          • Is the relationship significant?<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                            To test these questions, we run a correlation analysis.<\/p>\n

                                                                                                                                                                                                                                                                                                                                            The correlation analysis gives us two statistics, <\/strong>the correlation coefficient, and the p-value<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                            \n

                                                                                                                                                                                                                                                                                                                                            Correlation Coeffient <\/strong>( r ) tells you how related the two variables are on a scale of zero to one<\/p>\n

                                                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                                                            • A negative \u2018r\u2019 means the relationship is negative (slopes downward)<\/li>\n
                                                                                                                                                                                                                                                                                                                                            • A positive \u2018r\u2019 means the relationship is positive (slopes upwards)<\/li>\n
                                                                                                                                                                                                                                                                                                                                            • A value of 1 or -1 means 100% related<\/li>\n
                                                                                                                                                                                                                                                                                                                                            • General rule of thumb is that a \u201cgood\u201d correlation is 0.7 to 1.0 or -0.7 to -1.0<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                              P-value: <\/strong>Tells you whether the slope of your line is significantly different from zero (i.e do you have a significant relationship. If p<0.05, the variables are significantly correlated<\/p>\n

                                                                                                                                                                                                                                                                                                                                              Carry out the test:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                              • Select Stat <\/strong>from the property bar, then choose correlation <\/strong>from the Basic statistics <\/strong>drop down menu.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                [Note: <\/strong>do correlation analysis only if you don\u2019t need the equation of the line, or the coefficient of determination (r2)]<\/p>\n

                                                                                                                                                                                                                                                                                                                                                 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                With correlation, it doesn\u2019t matter what order you put the variables in, since we\u2019re just looking at a simple relationship.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                Output from Minitab:<\/strong>\u00a0<\/strong><\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                                                Correlations: height, weight<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                Pearson correlation of height and weight = 0.786 P-Value = 0.000<\/p>\n

                                                                                                                                                                                                                                                                                                                                                remember to write as p<0.001<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                                                                                                                                                                                                                                                                                                                Interpretation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                                • The r-value = 0.786, which tells us that we have a positive correlation, and it is a \u201cgood\u201d correlation (since the value is above 0.7) tells us how strong it is.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                • The p-value tells us if it is a significant correlation, while the r-value<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                                                  \n

                                                                                                                                                                                                                                                                                                                                                  Assumptions were satisfied:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                  Both sets of data are normal (based on prob. plots) and the relationship is linear (based on the graph)<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                                  Regression:<\/h5>\n

                                                                                                                                                                                                                                                                                                                                                  Regression analysis lets us determine the \u201cmodel\u201d (line) that best describes the data.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                  The regression analysis gives us three statistics:<\/p>\n

                                                                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                                                                  • \u00a0r2 =\u201ccoefficient of determination\u201d<\/li>\n
                                                                                                                                                                                                                                                                                                                                                  • The p-value, and<\/li>\n
                                                                                                                                                                                                                                                                                                                                                  • The equation of the line.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                    The p-value is the same as for correlation<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                                                                                                                    The Coefficient of determination, r<\/strong>2<\/strong>, <\/strong>tells us the proportion of the variation in the \u2018y\u2019 vaues that can be directly related(“statistically\u201d) to the x-value. It is often referred to as the amount of variation \u201cexplained\u201d by the variation in \u2018x\u2019, but note that that is meant in the statistical sense.<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                    \n

                                                                                                                                                                                                                                                                                                                                                    In regression, we assume that one variable is fixed or independent, and that the other variable depends on the first. The independent variable goes on the x-axis and the dependent one goes on the y-axis.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                    The assumptions of linear regression are:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                                                                    1. Linearity: data on the graph form a reasonable line, so we say they are linear<\/li>\n
                                                                                                                                                                                                                                                                                                                                                    2. Data normality: Normality plots of the two variable showed that both were normally distributed<\/li>\n
                                                                                                                                                                                                                                                                                                                                                    3. Error normality (normality of residuals) – test this after running the regression<\/li>\n
                                                                                                                                                                                                                                                                                                                                                    4. Equal distribution of residuals along the line – test after running the regression<\/li>\n<\/ol>\n\n\n\n
                                                                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                                                                      \n

                                                                                                                                                                                                                                                                                                                                                      A note on regression assumptions:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                      Unlike the assumptions for the group comparisons (e.g. ANOVA or t-test), many of the Regression are there to guide your interpretation, not to be an absolute guide to whether you can do the test or not.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                      \u201cMust<\/strong> have\u201d<\/strong> asummptions:<\/p>\n

                                                                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                                                                      • Data must be linear for Linear Regression<\/li>\n
                                                                                                                                                                                                                                                                                                                                                      • Data must be normal or near normal<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                        Assumptions<\/strong> that<\/strong> guide<\/strong> interpretation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                                                                        • Spread of residuals:\n
                                                                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                                                                          • Residuals are the distance that each dependent variable point is away from the line (on the y-axis).\n
                                                                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                                                                            • For the regression equation to have good predictive ability (i.e. so you can predict values of y if you know values of x), the residuals have to be normally distributed and distributed equally above and below the line for the entire relationship. For example, if there is low variability (scatter) at the lower part of the graph, and high variability (scatter) at the high part of the graph, it means that the line doesn\u2019t predict as well at the upper parts of the graph.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                              – i.e. Even if these assumptions are violoated slightly, you can still do the regression, but have to use caution if predicting \u2018y\u2019 from \u2018x\u2019<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                              Run<\/strong> the Regression analysis:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                                              • Select Stat<\/strong>from the property bar, then choose Regression <\/strong>and Regression <\/strong>and fit regression model<\/strong><\/li>\n
                                                                                                                                                                                                                                                                                                                                                              • In the regression menu, put the dependent variable in the response box, and the independent variable in the predictor box<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                                To test the assumptions while doing the test, <\/strong>click<\/strong> on<\/strong> the \u201cGraphs\u201d box.<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                                                • In the graphs window, check the box beside Regular<\/strong>\n
                                                                                                                                                                                                                                                                                                                                                                    \n
                                                                                                                                                                                                                                                                                                                                                                  • This gives the actual values for the residuals<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                                    [\u2018Standardized\u2019 will convert the values based on their standard deviations so you can see them as a function of their variability. This can be useful in identifying outliers, but otherwise, the graph will look very similar]<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                      \n
                                                                                                                                                                                                                                                                                                                                                                    • For \u201cresidual<\/strong> plots<\/strong>\u201d, check the boxes beside normal probability plot, and the residuals vs the \u201cfits\u201d (the independent var).\n
                                                                                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                                                                                      • If you think that there may be a bias in the order the data are collected, then also plot the residuals vs the order (e.g. if you collect data over a period of time, so you might get a different response later in the day than earlier, that could be a bias)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                                                                        \n
                                                                                                                                                                                                                                                                                                                                                                        Interpretation of Regression Assumptions:<\/h6>\n

                                                                                                                                                                                                                                                                                                                                                                        Distribution of residuals along the line:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                        \"\"<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                          \n
                                                                                                                                                                                                                                                                                                                                                                        • Here, you are looking for evidence that points are scattered fairly evenly along the horizontal line. This means your \u201cvariance\u201d (variability in the observed values compared to the predicted line) is spread equally along the line… analogous to the \u201cequal variance\u201d assumption in ANOVA.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                        • The distribution of points above and below the line isn\u2019t perfect, but is fairly uniform if you don\u2019t count the outlier point in the upper right corner.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                        • This distribution is close enough to pass the assumption test.<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                                          Normality of residuals:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                          \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                          This graph is just like the other normality plots we look at to see if data are normal. The computer will automatically determine the residuals for you, and plot them on the normality plot.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                          Interpretation:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                                                                                          • The values are mostly on the line, again with one outlier, so conclude that they are normal<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                          • [Just to be sure, I did a normality test on the data, and it confirm that they are normal (p>0.05).]<\/li>\n<\/ul>\n

                                                                                                                                                                                                                                                                                                                                                                            The output from the Regression analysis:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                            \"\"<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                            Regression Analysis Interpretation:<\/p>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                                            \n
                                                                                                                                                                                                                                                                                                                                                                              \n
                                                                                                                                                                                                                                                                                                                                                                            1. Equation of the line. This is in the form of a straight line, y = mx + b<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                            2. The r2 value tells us the proportion of the variability in weight that can be \u201cexplained\u201d (or is statistically related to) by height, so 62% of the weight variation is mathematically related to height. 38% is related to other, unmeasured factors.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                            3. The p-value tests whether the slope of the line is significantly different from zero (if there is too much variability in the points, you can\u2019t be sure that you have a relationship).<\/li>\n<\/ol>\n

                                                                                                                                                                                                                                                                                                                                                                              The assumptions were satisfied, so we can generally trust our result:<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                                                              • There is a significant positive relationship between height and weight.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                              • The assumption testing tells us we have one outlier value (we could investigate that more closely if we wanted), and that there is a bit more variation at higher levels of \u2018x\u2019 (height) than at lower levels. Therefore, we know that our line does not predict as well for tall people as for short people.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                              • Although there is a significant relationship, there is still quite a bit of unexplained variation, so height is not the only factor that affects weight of the students.<\/li>\n<\/ul>\n<\/div>\n
                                                                                                                                                                                                                                                                                                                                                                                \"\"<\/div>\n
                                                                                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                                                                                Figure 1. Relationship between height and weight for a group of university students taking part in an exercise on how running affects pulse rates.<\/p>\n\n\n\n
                                                                                                                                                                                                                                                                                                                                                                                \n
                                                                                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                                                                                How do you report results of your regression analysis?<\/strong><\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                Your trend statement must give the pattern from your graph, then summarize the statistical results (remember to always plot your data before doing analysis, and include a figure legend). An example from the figure above might read:<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                There was a strong positive relationship between height and weight for the university students taking part in the running exercise, so that as height increased, so did weight (Figure 1, Regression analysis, r2 = 0.617). About 62% of the variation in weight was statistically explained by the variation in height, but the calculated regression line had better predictive ability at lower levels of height than for higher levels of height (Figure 1).<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

                                                                                                                                                                                                                                                                                                                                                                                \n

                                                                                                                                                                                                                                                                                                                                                                                Pulses Dataset used for analyses in this manual<\/strong>.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                A group of students were separated into 2 groups. One group ran on the spot for one minute, and the other group did not. Following the trial, information was gathered on whether they smoked, what their sex was, what their height and weight were, and what their normal activity levels were.<\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  \n
                                                                                                                                                                                                                                                                                                                                                                                • Pulse1 = resting pulse for all.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • Pulse2 = pulse following the running. (Beats per minute)<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • Weight is in pounds, Height is in inches<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • ran = 1 means they ran,<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • ran = 2 means they did not.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • smokes = 1 means they smoked.<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • sex = 1 is male<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • sex = 2 is female<\/li>\n
                                                                                                                                                                                                                                                                                                                                                                                • activlev is their normal activity level; 1 is slight, 2 is moderate, and 3 is high.<\/li>\n<\/ul>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                                                                                                                                                                                                                                                                                                                                  pulse1<\/td>\npulse2<\/td>\nran<\/td>\nsmokes<\/td>\nsex<\/td>\nheight<\/td>\nweight<\/td>\nactiv.level<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                   <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  64<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  1<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  2<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  1<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  140<\/td>\n

                                                                                                                                                                                                                                                                                                                                                                                  		 <\/p>\n

                                                                                                                                                                                                                                                                                                                                                                                  2<\/td>\n<\/tr>\n

                                                                                                                                                                                                                                                                                                                                                                                  58<\/td>\n75<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n145<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n82<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n73.5<\/td>\n160<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n85<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n73<\/td>\n190<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  64<\/td>\n82<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n69<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  74<\/td>\n84<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n73<\/td>\n165<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  84<\/td>\n84<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n150<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n72<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n74<\/td>\n190<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n75<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n195<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  76<\/td>\n88<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n138<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  80<\/td>\n104<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n74<\/td>\n160<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  80<\/td>\n96<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n88<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n70<\/td>\n153<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n76<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n67<\/td>\n145<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  60<\/td>\n76<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n170<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n68<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n175<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n88<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n69<\/td>\n175<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n86<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n73<\/td>\n170<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n80<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n74<\/td>\n180<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n80<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n66<\/td>\n135<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n106<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n170<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  74<\/td>\n76<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n70<\/td>\n157<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n102<\/td>\n1<\/td>\n2<\/td>\n1<\/td>\n70<\/td>\n130<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n98<\/td>\n1<\/td>\n1<\/td>\n1<\/td>\n75<\/td>\n185<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  96<\/td>\n140<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n61<\/td>\n140<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n100<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n66<\/td>\n120<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n104<\/td>\n1<\/td>\n1<\/td>\n2<\/td>\n68<\/td>\n130<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  82<\/td>\n100<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n68<\/td>\n138<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  88<\/td>\n115<\/td>\n1<\/td>\n1<\/td>\n2<\/td>\n63<\/td>\n121<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n112<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n70<\/td>\n125<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  96<\/td>\n116<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n68<\/td>\n116<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n118<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n69<\/td>\n145<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  88<\/td>\n110<\/td>\n1<\/td>\n1<\/td>\n2<\/td>\n69<\/td>\n150<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n98<\/td>\n1<\/td>\n1<\/td>\n2<\/td>\n62.75<\/td>\n112<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  80<\/td>\n128<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n68<\/td>\n125<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n62<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n74<\/td>\n190<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  60<\/td>\n62<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n70<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n69<\/td>\n170<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n66<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n70<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  76<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n215<\/td>\n2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                                                                                                                                                                                                                                                                                                                                  pulse1<\/td>\npulse2<\/td>\nran<\/td>\nsmokes<\/td>\nsex<\/td>\nheight<\/td>\nweight<\/td>\nactiv.level<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n68<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n67<\/td>\n150<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  54<\/td>\n56<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n69<\/td>\n145<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  74<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n73<\/td>\n155<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  74<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n73<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n68<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n150<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n73<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n68<\/td>\n155<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n64<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n69.5<\/td>\n150<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  82<\/td>\n83<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n73<\/td>\n180<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  64<\/td>\n62<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n75<\/td>\n160<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  58<\/td>\n58<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n66<\/td>\n135<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  54<\/td>\n50<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n69<\/td>\n160<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n71<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n66<\/td>\n130<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n61<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n73<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  76<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n74<\/td>\n148<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  88<\/td>\n84<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n73.5<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n70<\/td>\n150<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  90<\/td>\n89<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n67<\/td>\n140<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n180<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  70<\/td>\n71<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n75<\/td>\n190<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  90<\/td>\n90<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n68<\/td>\n145<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  92<\/td>\n94<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n69<\/td>\n150<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  60<\/td>\n63<\/td>\n2<\/td>\n1<\/td>\n1<\/td>\n71.5<\/td>\n164<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n71<\/td>\n140<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n68<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n72<\/td>\n142<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  84<\/td>\n84<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n69<\/td>\n136<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  74<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n67<\/td>\n123<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n66<\/td>\n2<\/td>\n2<\/td>\n1<\/td>\n68<\/td>\n155<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  84<\/td>\n84<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n66<\/td>\n130<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  61<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n65.5<\/td>\n120<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  64<\/td>\n60<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n66<\/td>\n130<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  94<\/td>\n92<\/td>\n2<\/td>\n1<\/td>\n2<\/td>\n62<\/td>\n131<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  60<\/td>\n66<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n62<\/td>\n120<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n70<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n63<\/td>\n118<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  58<\/td>\n56<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n67<\/td>\n125<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  88<\/td>\n74<\/td>\n2<\/td>\n1<\/td>\n2<\/td>\n65<\/td>\n135<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n72<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n66<\/td>\n125<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  84<\/td>\n80<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n65<\/td>\n118<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  62<\/td>\n66<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n65<\/td>\n122<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  66<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n65<\/td>\n115<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  80<\/td>\n74<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n64<\/td>\n102<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n78<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n67<\/td>\n115<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n68<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n69<\/td>\n150<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  72<\/td>\n68<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n68<\/td>\n110<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  82<\/td>\n80<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n63<\/td>\n116<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  76<\/td>\n76<\/td>\n2<\/td>\n1<\/td>\n2<\/td>\n62<\/td>\n108<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  87<\/td>\n84<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n63<\/td>\n95<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  90<\/td>\n92<\/td>\n2<\/td>\n1<\/td>\n2<\/td>\n64<\/td>\n125<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  78<\/td>\n80<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n68<\/td>\n133<\/td>\n1<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  68<\/td>\n68<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n62<\/td>\n110<\/td>\n2<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  86<\/td>\n84<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n67<\/td>\n150<\/td>\n3<\/td>\n<\/tr>\n
                                                                                                                                                                                                                                                                                                                                                                                  76<\/td>\n76<\/td>\n2<\/td>\n2<\/td>\n2<\/td>\n61.75<\/td>\n108<\/td>\n2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"author":116,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-50","chapter","type-chapter","status-publish","hentry"],"part":3,"_links":{"self":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapters\/50","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/wp\/v2\/users\/116"}],"version-history":[{"count":21,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions"}],"predecessor-version":[{"id":438,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions\/438"}],"part":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapters\/50\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/wp\/v2\/media?parent=50"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/pressbooks\/v2\/chapter-type?post=50"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/wp\/v2\/contributor?post=50"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.library.upei.ca\/bio3310\/wp-json\/wp\/v2\/license?post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}