{"id":1061,"date":"2020-06-03T05:45:54","date_gmt":"2020-06-03T09:45:54","guid":{"rendered":"http:\/\/pressbooks.library.upei.ca\/montelpare\/?post_type=chapter&p=1061"},"modified":"2020-08-24T14:18:59","modified_gmt":"2020-08-24T18:18:59","slug":"statistical-applications-with-linear-regression-analyses","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.upei.ca\/montelpare\/chapter\/statistical-applications-with-linear-regression-analyses\/","title":{"raw":"Statistical applications with linear regression analyses","rendered":"Statistical applications with linear regression analyses"},"content":{"raw":"
\r\n

Learner Ootcomes<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nAfter reading this chapter you should be able to:\r\n
    \r\n \t
  1. Define and describe simple linear regression<\/li>\r\n \t
  2. Create a SAS program to compute the outcome for a linear regression application including the slope of a line<\/li>\r\n \t
  3. Create a line of best fit<\/li>\r\n \t
  4. Identify the critical components in the output generated from a linear regression application<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

    Calculating the slope of a line<\/h2>\r\nThe first step in understanding linear regression is to review the calculation for the slope of a line.\r\n\r\nAlthough presented early in your introduction to mathematics and algebra, most likely in secondary school, learning about the slope of a line may have been one of those topics that you missed, or forgot, or decided that you would never need in the future. Of course, since your intended vocation was not going to require statistical analyses and only essential math, why bother listening. However, now we are reviewing statistical applications and so understanding the calculation for the slope of a line is actually meaningful.\r\n\r\nProviding an estimate for the slope of a line can be one way to calculate the rate of change<\/strong> in a variable of interest on the vertical axis \u2013 usually denoted as y, in relation to an independent variable, such as time plotted on the horizontal axis \u2013 denoted as x. The slope of a line provides a measure of the steepness of the line as a function of the change in the variable (y) in relation to the change in the variable (x). Consider for example the data plotted in the following graph (Figure 33.1).\r\n\r\n \r\n
    \"\"\r\nFigure 33.1 Line of Best fit or Y (Score) over X (Time)<\/strong><\/h6>\r\nIn the graph above, we observe a distinct positive relationship between the scores for the variable on the y-axis and the scores for the variable on the x-axis. That is, as we move from left to right on the x-axis we observe an increase in the scores on the y-axis. The slope of the line is an estimate or what we refer to as a coefficient, a single number that represents all of the points on the line of the relationship between the variables: x and y.\r\n\r\nThe basic calculation to determine this estimate (i.e. the slope) of this relationship is given here as [latex]slope = {\\Delta{y}\\over\\Delta{x}}[\/latex] which is read as the change in the y variable divided by the change in the x variable.\r\n
    \r\n

    Key Takeaways<\/p>\r\n\r\n<\/header>\r\n

    The slope of a straight line is constant for the entire line and therefore any two points, chosen at random on the line will provide the estimate of the slope (this is not the case for non-linear lines). You might recall that the slope of a line is often represented by the letter m, and the formula for slope is read as: slope = rise over run.<\/div>\r\n
    Using SAS programming as shown below, we see that The slope of the line for the graph above is 2.60<\/strong>, and the y-intercept (the point where the line crosses the y-axis) is 22.40. <\/strong>Both estimates are <\/span>shown in the table below.<\/div>\r\n<\/div>\r\n
    <\/article>
    \r\n\r\n\r\n\r\n\r\n\r\n\r\n
    \u00a0<\/caption>\r\n
    Parameter Estimates for Figure 33.1<\/th>\r\n<\/tr>\r\n
    Variable<\/th>\r\nDF<\/th>\r\nParameter\r\nEstimate<\/th>\r\nStandard\r\nError<\/th>\r\nt\u00a0Value<\/th>\r\nPr\u00a0>\u00a0|t|<\/th>\r\n95% Confidence Limits<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Intercept<\/th>\r\n1<\/th>\r\n22.40<\/strong><\/span><\/td>\r\n0.916<\/td>\r\n24.44<\/td>\r\n<.0001<\/td>\r\n20.29<\/td>\r\n24.51<\/td>\r\n<\/tr>\r\n
    trial<\/th>\r\n1<\/strong><\/th>\r\n2.60<\/span><\/strong><\/td>\r\n0.147<\/td>\r\n17.60<\/td>\r\n<.0001<\/td>\r\n2.26<\/td>\r\n2.94<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/article>\r\n
    \r\n

    Determining the Slope of a line<\/p>\r\n\r\n<\/header>\r\n

    \r\n\r\nThe SAS code for the graph above is shownhere. The program uses two variables: the independent variable (x) called TRIAL<\/strong><\/em> and the outcome variable (y) called SCORE<\/strong><\/em>. Lines 1 to 3 in the code below set up the program environment. Line 4 identifies the program workflow. Line 5 explains the arrangement of the columns of data. Line 6 cues the program that the data will follow. Ten data points are included in this dataset. Lines 7 to 16 are the SAS code to produce the regression estimates and create the graph with both a line and with dots to represent each point.\r\n
      \r\n \t
    1. \r\n
      options pagesize=55 linesize=120 center date;<\/pre>\r\n<\/li>\r\n \t
    2. \r\n
      goptions reset=all cback=white border htitle=12pt htext=10pt;<\/pre>\r\n<\/li>\r\n \t
    3. \r\n
      LIBNAME txtbook '\/home\/username\/textbookExamples\/regression';<\/pre>\r\n<\/li>\r\n \t
    4. \r\n
      data txtbook.slope1;<\/pre>\r\n<\/li>\r\n \t
    5. \r\n
      input trial 1-2 score 4-5;<\/pre>\r\n<\/li>\r\n \t
    6. \r\n
      datalines; \r\n01 24\r\n02 27\r\n03 30\r\n04 33\r\n05 36\r\n06 39\r\n07 42\r\n08 45\r\n09 45\r\n10 46\r\n;<\/pre>\r\n<\/li>\r\n \t
    7. \r\n
      Title 'Estimating the slope';<\/pre>\r\n<\/li>\r\n \t
    8. \r\n
      axis1 label=(\"Trial\"); axis2 label=(angle=90 \"Score\") minor=(n=4); \r\n\r\n\/* Define the symbol characteristics for the plot groups *\/<\/pre>\r\n<\/li>\r\n \t
    9. \r\n
      symbol1 interpol=none value=dot color=depk;<\/pre>\r\n<\/li>\r\n \t
    10. \r\n
      symbol2 interpol=none value=dot color=vibg; \r\n\r\n\/* Define the symbol characteristics for the regression line *\/<\/pre>\r\n<\/li>\r\n \t
    11. \r\n
      symbol3 interpol=rl value=none color=black; \r\n\r\n\/* proc gplot data=txtbook.slope1;<\/pre>\r\n<\/li>\r\n \t
    12. \r\n
      plot score*trial \/ haxis=axis1 vaxis=axis2 ;<\/pre>\r\n<\/li>\r\n \t
    13. \r\n
      plot2 score*trial \/ noaxis; *\/<\/pre>\r\n<\/li>\r\n \t
    14. \r\n
      proc reg; model score=trial \/ clb; \/* command to produce estimates *\/<\/pre>\r\n<\/li>\r\n \t
    15. \r\n
      plot score*trial =\"*\";<\/pre>\r\n<\/li>\r\n \t
    16. \r\n
      run;<\/pre>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n\r\n
      \r\n\r\nLet's consider a research application of simple linear regression. In the following research study, we are interested in the change in pain estimates for horses that undergo castration. Here we collected 4 pain estimates at 30-minute intervals following equine castration surgery. The data for the estimates of pain are represented in the following table. Notice for each 30-minute time point the estimate of pain on a 10 point scale diminishes by one point.\r\n\r\n\r\n\r\n\r\n
      Time in reference to surgery<\/td>\r\n30 minutes after<\/td>\r\n60 minutes after<\/td>\r\n90 minutes after<\/td>\r\n120 minutes after<\/td>\r\n<\/tr>\r\n
      10-point pain scale<\/td>\r\n9<\/td>\r\n8<\/td>\r\n7<\/td>\r\n6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can graph these data using the following SAS code.\r\n
      \r\n
      Example 33.2 Application of Simple Linear Regression<\/p>\r\n\r\n<\/header>\r\n
      \r\nDATA LINE;\r\nINPUT ID TIME PAIN;\r\nDATALINES;\r\n01 \u00a030 9\r\n02 \u00a060 8\r\n03 \u00a090 7\r\n04 120 6\r\n;\r\nPROC SGPLOT NOBORDER NOAUTOLEGEND;\r\nREG Y=PAIN X=TIME;\r\nRUN;<\/span><\/div>\r\n<\/div>\r\nFigure 33.2, below shows the line of best fit that illustrates the relationship between pain ratings over time since surgery.\r\n\r\n\"\"\r\n
      Figure 33.2. Pain Ratings Following Surgery<\/strong><\/p>\r\nThe calculation of the slope of the line for the estimate of pain over time is represented by the following equation.\r\n\r\n[latex]slope = {\\Delta{y}\\over\\Delta{x}}[\/latex]\r\n\r\n[latex]slope = {(y_2 - y_1)\\over (x_2 - x_1)} = [\/latex]\r\n\r\n[latex]slope = {(8 - 9)\\over (60 - 30)} = {(-1)\\over (30)}\u00a0 ={0.033} [\/latex]\r\n\r\nNotice that the slope has a value of\u00a0 m -0.033 which when multiplied by the time value in the equation [latex]y = {mx + b}[\/latex] indicates that as the value of TIME increases from left to right on the x-axis, the value for PAIN\u00a0 decreases on the y-axis.\r\n\r\nWith the following lines of SAS code added to our program we can compute the slope and the y-intercept term for the data above and cofirm that which we calculated by hand.\r\n
      \r\n\r\nPROC REG; MODEL PAIN = TIME;<\/strong>\r\n\r\nRUN;<\/strong>\r\n\r\n<\/div>\r\nThis SAS code produced the following output.\r\n\r\nThe REG Procedure -- <\/strong>Model: MODEL1 for <\/strong>Dependent Variable (Y) : pain<\/strong>\r\n
      \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Parameter Estimates<\/strong><\/td>\r\n<\/tr>\r\n
      Variable<\/strong><\/td>\r\nDF<\/strong><\/td>\r\nParameter\r\nEstimate<\/strong><\/td>\r\nStandard\r\nError<\/strong><\/td>\r\nt\u00a0Value<\/strong><\/td>\r\nPr\u00a0>\u00a0|t|<\/strong><\/td>\r\n<\/tr>\r\n<\/thead>\r\n
      Intercept<\/strong><\/td>\r\n1<\/td>\r\n10.00000<\/td>\r\n0<\/td>\r\nInfty<\/td>\r\n<.0001<\/td>\r\n<\/tr>\r\n
      time (x variable)\r\n<\/strong><\/td>\r\n1<\/td>\r\n-0.03333<\/td>\r\n0<\/td>\r\n-Infty<\/td>\r\n<.0001<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nNotice both the slope score of -0.033 and the y-intercept (10) are included in this table. Likewise, because there are only four scores in the data set the standard error is 0 and the estimates for slope and y-intercept are significantly greater than 0.\r\n
      Using Regression to Compute a Laboratory Standard Curve<\/h2>\r\nIn many laboratory experiments, researchers will create what is known as a curve of standards or a standard curve to establish the relationship between substrates and products.. One way that we can use regression and the calculation of the line of best fit is to test the linearity of a relationship between the concentrations of a substrate (x) and a product (y). In the following example we were measuring the presence of a gene of interest at 5 different concentrations.\r\n\r\nThe Gene of interest was Bactin and the concentrations were: EXPAND..\r\n\r\nThe following SAS program was used to calculate the relationship using regression, and the graphical representation of the relationship.\r\n
      \r\n
      SAS progran to compute line of best fit with PROC REG & PROC SGPLOT<\/p>\r\n\r\n<\/header>\r\n
      \r\n\r\nPROC FORMAT;\r\nVALUE TRFMT 1='CONC30'\r\n2='CONC60'\r\n3='CONC120'\r\n4='CONC240'\r\n5='CONC480';\r\noptions pagesize=55 linesize=80 center;\r\nLIBNAME LINES20 '\/home\/username\/bioInfomatics\/lnBstFt';\r\nDATA LINES20.BACTIN01;\r\ninput TRIAL CONC1 CONC2 CONC3;\r\nDATALINES;\r\n1 19.63 18.63 20.12\r\n2 18.06 17.23 17.05\r\n3 15.63 15.80 15.59\r\n4 14.58 15.00 14.57\r\n5 13.64 13.70 13.56\r\n;\r\nproc reg;\r\nmodel CONC1=TRIAL;\r\nmodel CONC2=TRIAL;\r\nmodel CONC3=TRIAL;\r\nrun;\r\n\r\nproc sgplot data=LINES20.BACTIN01\r\nnoautolegend;\r\nreg x=TRIAL y=CONC1 \/ CLM\r\nCLMATTRS=(CLMLINEATTRS=\r\n(COLOR=Green PATTERN= ShortDash));\r\nFORMAT TRIAL TRFMT. ;\r\nTITLE ' CONFIDENCE LIMITS FOR TRIAL 1 OF BACTIN';\r\nrun;\r\nproc sgplot data=LINES20.BACTIN01\r\nnoautolegend;\r\nreg x=TRIAL y=CONC2 \/ CLM\r\nCLMATTRS=(CLMLINEATTRS=\r\n(COLOR=Green PATTERN= ShortDash));\r\nFORMAT TRIAL TRFMT. ;\r\nTITLE ' CONFIDENCE LIMITS FOR TRIAL 2 OF BACTIN';\r\nrun;\r\n\r\nproc sgplot data=LINES20.BACTIN01\r\nnoautolegend;\r\nreg x=TRIAL y=CONC3 \/ CLM\r\nCLMATTRS=(CLMLINEATTRS=\r\n(COLOR=Green PATTERN= ShortDash));\r\nFORMAT TRIAL TRFMT. ;\r\nTITLE ' CONFIDENCE LIMITS FOR TRIAL 3 OF BACTIN';\r\nrun;\r\n\r\nproc sgplot data=LINES20.BACTIN01;\r\nxaxis type=discrete;\r\nseries x=TRIAL y=CONC1;\r\nseries x=TRIAL y=CONC2;\r\nseries x=TRIAL y=CONC3;\r\n;\r\nFORMAT TRIAL TRFMT. ;\r\nTITLE ' Simple plot for 3 trials OF BACTIN using overlay';\r\n\r\nrun;\r\n\r\n<\/div>\r\n<\/div>\r\n
      \r\n
      Output from PROC REG<\/p>\r\n\r\n<\/header>\r\n
      \r\n
      \r\n
      The REG Procedure<\/p>\r\n
      Model: MODEL3<\/p>\r\n
      Dependent Variable: CONC3<\/p>\r\n\r\n<\/div>\r\n
      \r\n<\/colgroup>\r\n\r\n\r\n\r\n
      \u00a0<\/caption>  
      Number of Observations Read<\/th>\r\n5<\/td>\r\n<\/tr>\r\n
      Number of Observations Used<\/th>\r\n5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/article>
      \r\n<\/colgroup> <\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      \u00a0<\/caption>      
      Analysis of Variance<\/th>\r\n<\/tr>\r\n
      Source<\/th>\r\nDF<\/th>\r\nSum of\r\nSquares<\/th>\r\nMean\r\nSquare<\/th>\r\nF Value<\/th>\r\nPr\u00a0>\u00a0F<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      Model<\/th>\r\n1<\/td>\r\n24.33600<\/td>\r\n24.33600<\/td>\r\n41.74<\/td>\r\n0.0075<\/td>\r\n<\/tr>\r\n
      Error<\/th>\r\n3<\/td>\r\n1.74908<\/td>\r\n0.58303<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      Corrected Total<\/th>\r\n4<\/td>\r\n26.08508<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/article>
      \r\n<\/colgroup>\r\n\r\n\r\n\r\n\r\n
      \u00a0<\/caption>    
      Root MSE<\/th>\r\n0.76356<\/td>\r\nR-Square<\/th>\r\n0.9329<\/td>\r\n<\/tr>\r\n
      Dependent Mean<\/th>\r\n16.17800<\/td>\r\nAdj R-Sq<\/th>\r\n0.9106<\/td>\r\n<\/tr>\r\n
      Coeff Var<\/th>\r\n4.71975<\/td>\r\n<\/th>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/article>
      \r\n<\/colgroup> <\/colgroup>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      \u00a0<\/caption>      
      Parameter Estimates<\/th>\r\n<\/tr>\r\n
      Variable<\/th>\r\nDF<\/th>\r\nParameter\r\nEstimate<\/th>\r\nStandard\r\nError<\/th>\r\nt\u00a0Value<\/th>\r\nPr\u00a0>\u00a0|t|<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
      Intercept<\/th>\r\n1<\/th>\r\n20.85800<\/td>\r\n0.80083<\/td>\r\n26.05<\/td>\r\n0.0001<\/td>\r\n<\/tr>\r\n
      TRIAL<\/th>\r\n1<\/th>\r\n-1.56000<\/td>\r\n0.24146<\/td>\r\n-6.46<\/td>\r\n0.0075<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/article><\/section><\/section><\/section><\/div>\r\n<\/div>\r\n\"\"\r\n\r\n ","rendered":"
      \n
      \n
      Learner Ootcomes<\/p>\n<\/header>\n
      \n
      After reading this chapter you should be able to:<\/p>\n
        \n
      1. Define and describe simple linear regression<\/li>\n
      2. Create a SAS program to compute the outcome for a linear regression application including the slope of a line<\/li>\n
      3. Create a line of best fit<\/li>\n
      4. Identify the critical components in the output generated from a linear regression application<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
        Calculating the slope of a line<\/h2>\n
        The first step in understanding linear regression is to review the calculation for the slope of a line.<\/p>\n
        Although presented early in your introduction to mathematics and algebra, most likely in secondary school, learning about the slope of a line may have been one of those topics that you missed, or forgot, or decided that you would never need in the future. Of course, since your intended vocation was not going to require statistical analyses and only essential math, why bother listening. However, now we are reviewing statistical applications and so understanding the calculation for the slope of a line is actually meaningful.<\/p>\n
        Providing an estimate for the slope of a line can be one way to calculate the rate of change<\/strong> in a variable of interest on the vertical axis \u2013 usually denoted as y, in relation to an independent variable, such as time plotted on the horizontal axis \u2013 denoted as x. The slope of a line provides a measure of the steepness of the line as a function of the change in the variable (y) in relation to the change in the variable (x). Consider for example the data plotted in the following graph (Figure 33.1).<\/p>\n
         <\/p>\n
        \"\"
        \nFigure 33.1 Line of Best fit or Y (Score) over X (Time)<\/strong><\/h6>\n
        In the graph above, we observe a distinct positive relationship between the scores for the variable on the y-axis and the scores for the variable on the x-axis. That is, as we move from left to right on the x-axis we observe an increase in the scores on the y-axis. The slope of the line is an estimate or what we refer to as a coefficient, a single number that represents all of the points on the line of the relationship between the variables: x and y.<\/p>\n
        The basic calculation to determine this estimate (i.e. the slope) of this relationship is given here as [latex]slope = {\\Delta{y}\\over\\Delta{x}}[\/latex] which is read as the change in the y variable divided by the change in the x variable.<\/p>\n
        \n
        \n
        Key Takeaways<\/p>\n<\/header>\n
        The slope of a straight line is constant for the entire line and therefore any two points, chosen at random on the line will provide the estimate of the slope (this is not the case for non-linear lines). You might recall that the slope of a line is often represented by the letter m, and the formula for slope is read as: slope = rise over run.<\/div>\n
        Using SAS programming as shown below, we see that The slope of the line for the graph above is 2.60<\/strong>, and the y-intercept (the point where the line crosses the y-axis) is 22.40. <\/strong>Both estimates are <\/span>shown in the table below.<\/div>\n<\/div>\n
        <\/article>\n
        \n\n\n\n\n\n\n\n
        \u00a0<\/caption>\n
        Parameter Estimates for Figure 33.1<\/th>\n<\/tr>\n
        Variable<\/th>\nDF<\/th>\nParameter
        \nEstimate<\/th>\n        		Standard
        \nError<\/th>\n        		t\u00a0Value<\/th>\nPr\u00a0>\u00a0|t|<\/th>\n95% Confidence Limits<\/th>\n<\/tr>\n<\/thead>\n
        Intercept<\/th>\n1<\/th>\n22.40<\/strong><\/span><\/td>\n0.916<\/td>\n24.44<\/td>\n<.0001<\/td>\n20.29<\/td>\n24.51<\/td>\n<\/tr>\n
        trial<\/th>\n1<\/strong><\/th>\n2.60<\/span><\/strong><\/td>\n0.147<\/td>\n17.60<\/td>\n<.0001<\/td>\n2.26<\/td>\n2.94<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/article>\n
        \n
        \n
        Determining the Slope of a line<\/p>\n<\/header>\n
        \n
        The SAS code for the graph above is shownhere. The program uses two variables: the independent variable (x) called TRIAL<\/strong><\/em> and the outcome variable (y) called SCORE<\/strong><\/em>. Lines 1 to 3 in the code below set up the program environment. Line 4 identifies the program workflow. Line 5 explains the arrangement of the columns of data. Line 6 cues the program that the data will follow. Ten data points are included in this dataset. Lines 7 to 16 are the SAS code to produce the regression estimates and create the graph with both a line and with dots to represent each point.<\/p>\n
          \n
        1. \n
          options pagesize=55 linesize=120 center date;<\/pre>\n<\/li>\n
        2. \n
          goptions reset=all cback=white border htitle=12pt htext=10pt;<\/pre>\n<\/li>\n
        3. \n
          LIBNAME txtbook '\/home\/username\/textbookExamples\/regression';<\/pre>\n<\/li>\n
        4. \n
          data txtbook.slope1;<\/pre>\n<\/li>\n
        5. \n
          input trial 1-2 score 4-5;<\/pre>\n<\/li>\n
        6. \n
          datalines; \r\n01 24\r\n02 27\r\n03 30\r\n04 33\r\n05 36\r\n06 39\r\n07 42\r\n08 45\r\n09 45\r\n10 46\r\n;<\/pre>\n<\/li>\n
        7. \n
          Title 'Estimating the slope';<\/pre>\n<\/li>\n
        8. \n
          axis1 label=(\"Trial\"); axis2 label=(angle=90 \"Score\") minor=(n=4); \r\n\r\n\/* Define the symbol characteristics for the plot groups *\/<\/pre>\n<\/li>\n
        9. \n
          symbol1 interpol=none value=dot color=depk;<\/pre>\n<\/li>\n
        10. \n
          symbol2 interpol=none value=dot color=vibg; \r\n\r\n\/* Define the symbol characteristics for the regression line *\/<\/pre>\n<\/li>\n
        11. \n
          symbol3 interpol=rl value=none color=black; \r\n\r\n\/* proc gplot data=txtbook.slope1;<\/pre>\n<\/li>\n
        12. \n
          plot score*trial \/ haxis=axis1 vaxis=axis2 ;<\/pre>\n<\/li>\n
        13. \n
          plot2 score*trial \/ noaxis; *\/<\/pre>\n<\/li>\n
        14. \n
          proc reg; model score=trial \/ clb; \/* command to produce estimates *\/<\/pre>\n<\/li>\n
        15. \n
          plot score*trial =\"*\";<\/pre>\n<\/li>\n
        16. \n
          run;<\/pre>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
          \n
          Let’s consider a research application of simple linear regression. In the following research study, we are interested in the change in pain estimates for horses that undergo castration. Here we collected 4 pain estimates at 30-minute intervals following equine castration surgery. The data for the estimates of pain are represented in the following table. Notice for each 30-minute time point the estimate of pain on a 10 point scale diminishes by one point.<\/p>\n\n\n\n\n
          Time in reference to surgery<\/td>\n30 minutes after<\/td>\n60 minutes after<\/td>\n90 minutes after<\/td>\n120 minutes after<\/td>\n<\/tr>\n
          10-point pain scale<\/td>\n9<\/td>\n8<\/td>\n7<\/td>\n6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
          We can graph these data using the following SAS code.<\/p>\n
          \n
          \n
          Example 33.2 Application of Simple Linear Regression<\/p>\n<\/header>\n

          \nDATA LINE;
          \nINPUT ID TIME PAIN;
          \nDATALINES;
          \n01 \u00a030 9
          \n02 \u00a060 8
          \n03 \u00a090 7
          \n04 120 6
          \n;
          \nPROC SGPLOT NOBORDER NOAUTOLEGEND;
          \nREG Y=PAIN X=TIME;
          \nRUN;<\/span><\/div>\n<\/div>\n
          Figure 33.2, below shows the line of best fit that illustrates the relationship between pain ratings over time since surgery.<\/p>\n
          \"\"<\/p>\n
          Figure 33.2. Pain Ratings Following Surgery<\/strong><\/p>\n
          The calculation of the slope of the line for the estimate of pain over time is represented by the following equation.<\/p>\n
          [latex]slope = {\\Delta{y}\\over\\Delta{x}}[\/latex]<\/p>\n
          [latex]slope = {(y_2 - y_1)\\over (x_2 - x_1)} =[\/latex]<\/p>\n
          [latex]slope = {(8 - 9)\\over (60 - 30)} = {(-1)\\over (30)}\u00a0 ={0.033}[\/latex]<\/p>\n
          Notice that the slope has a value of\u00a0 m -0.033 which when multiplied by the time value in the equation [latex]y = {mx + b}[\/latex] indicates that as the value of TIME increases from left to right on the x-axis, the value for PAIN\u00a0 decreases on the y-axis.<\/p>\n
          With the following lines of SAS code added to our program we can compute the slope and the y-intercept term for the data above and cofirm that which we calculated by hand.<\/p>\n
          \n
          PROC REG; MODEL PAIN = TIME;<\/strong><\/p>\n
          RUN;<\/strong><\/p>\n<\/div>\n
          This SAS code produced the following output.<\/p>\n
          The REG Procedure — <\/strong>Model: MODEL1 for <\/strong>Dependent Variable (Y) : pain<\/strong><\/p>\n
          \n\n\n\n\n\n\n\n
          Parameter Estimates<\/strong><\/td>\n<\/tr>\n
          Variable<\/strong><\/td>\nDF<\/strong><\/td>\nParameter
          \nEstimate<\/strong><\/td>\n          		Standard
          \nError<\/strong><\/td>\n          		t\u00a0Value<\/strong><\/td>\nPr\u00a0>\u00a0|t|<\/strong><\/td>\n<\/tr>\n<\/thead>\n
          Intercept<\/strong><\/td>\n1<\/td>\n10.00000<\/td>\n0<\/td>\nInfty<\/td>\n<.0001<\/td>\n<\/tr>\n
          time (x variable)
          \n<\/strong><\/td>\n          		1<\/td>\n-0.03333<\/td>\n0<\/td>\n-Infty<\/td>\n<.0001<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
          Notice both the slope score of -0.033 and the y-intercept (10) are included in this table. Likewise, because there are only four scores in the data set the standard error is 0 and the estimates for slope and y-intercept are significantly greater than 0.<\/p>\n
          Using Regression to Compute a Laboratory Standard Curve<\/h2>\n
          In many laboratory experiments, researchers will create what is known as a curve of standards or a standard curve to establish the relationship between substrates and products.. One way that we can use regression and the calculation of the line of best fit is to test the linearity of a relationship between the concentrations of a substrate (x) and a product (y). In the following example we were measuring the presence of a gene of interest at 5 different concentrations.<\/p>\n
          The Gene of interest was Bactin and the concentrations were: EXPAND..<\/p>\n
          The following SAS program was used to calculate the relationship using regression, and the graphical representation of the relationship.<\/p>\n
          \n
          \n
          SAS progran to compute line of best fit with PROC REG & PROC SGPLOT<\/p>\n<\/header>\n
          \n
          PROC FORMAT;
          \nVALUE TRFMT 1=’CONC30′
          \n2=’CONC60′
          \n3=’CONC120′
          \n4=’CONC240′
          \n5=’CONC480′;
          \noptions pagesize=55 linesize=80 center;
          \nLIBNAME LINES20 ‘\/home\/username\/bioInfomatics\/lnBstFt’;
          \nDATA LINES20.BACTIN01;
          \ninput TRIAL CONC1 CONC2 CONC3;
          \nDATALINES;
          \n1 19.63 18.63 20.12
          \n2 18.06 17.23 17.05
          \n3 15.63 15.80 15.59
          \n4 14.58 15.00 14.57
          \n5 13.64 13.70 13.56
          \n;
          \nproc reg;
          \nmodel CONC1=TRIAL;
          \nmodel CONC2=TRIAL;
          \nmodel CONC3=TRIAL;
          \nrun;<\/p>\n
          proc sgplot data=LINES20.BACTIN01
          \nnoautolegend;
          \nreg x=TRIAL y=CONC1 \/ CLM
          \nCLMATTRS=(CLMLINEATTRS=
          \n(COLOR=Green PATTERN= ShortDash));
          \nFORMAT TRIAL TRFMT. ;
          \nTITLE ‘ CONFIDENCE LIMITS FOR TRIAL 1 OF BACTIN’;
          \nrun;
          \nproc sgplot data=LINES20.BACTIN01
          \nnoautolegend;
          \nreg x=TRIAL y=CONC2 \/ CLM
          \nCLMATTRS=(CLMLINEATTRS=
          \n(COLOR=Green PATTERN= ShortDash));
          \nFORMAT TRIAL TRFMT. ;
          \nTITLE ‘ CONFIDENCE LIMITS FOR TRIAL 2 OF BACTIN’;
          \nrun;<\/p>\n
          proc sgplot data=LINES20.BACTIN01
          \nnoautolegend;
          \nreg x=TRIAL y=CONC3 \/ CLM
          \nCLMATTRS=(CLMLINEATTRS=
          \n(COLOR=Green PATTERN= ShortDash));
          \nFORMAT TRIAL TRFMT. ;
          \nTITLE ‘ CONFIDENCE LIMITS FOR TRIAL 3 OF BACTIN’;
          \nrun;<\/p>\n
          proc sgplot data=LINES20.BACTIN01;
          \nxaxis type=discrete;
          \nseries x=TRIAL y=CONC1;
          \nseries x=TRIAL y=CONC2;
          \nseries x=TRIAL y=CONC3;
          \n;
          \nFORMAT TRIAL TRFMT. ;
          \nTITLE ‘ Simple plot for 3 trials OF BACTIN using overlay’;<\/p>\n
          run;<\/p>\n<\/div>\n<\/div>\n
          \n
          \n
          Output from PROC REG<\/p>\n<\/header>\n
          \n
          \n
          The REG Procedure<\/p>\n
          Model: MODEL3<\/p>\n
          Dependent Variable: CONC3<\/p>\n<\/div>\n
          \n
          \n
          \n
          \n\n\n\n<\/colgroup>\n\n\n\n
          \u00a0<\/caption>\n
          Number of Observations Read<\/th>\n5<\/td>\n<\/tr>\n
          Number of Observations Used<\/th>\n5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/article>\n
          \n\n\n<\/colgroup>\n\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n
          \u00a0<\/caption>\n
          Analysis of Variance<\/th>\n<\/tr>\n
          Source<\/th>\nDF<\/th>\nSum of
          \nSquares<\/th>\n          		Mean
          \nSquare<\/th>\n          		F Value<\/th>\nPr\u00a0>\u00a0F<\/th>\n<\/tr>\n<\/thead>\n
          Model<\/th>\n1<\/td>\n24.33600<\/td>\n24.33600<\/td>\n41.74<\/td>\n0.0075<\/td>\n<\/tr>\n
          Error<\/th>\n3<\/td>\n1.74908<\/td>\n0.58303<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
          Corrected Total<\/th>\n4<\/td>\n26.08508<\/td>\n<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/article>\n
          \n\n\n\n\n\n<\/colgroup>\n\n\n\n\n
          \u00a0<\/caption>\n
          Root MSE<\/th>\n0.76356<\/td>\nR-Square<\/th>\n0.9329<\/td>\n<\/tr>\n
          Dependent Mean<\/th>\n16.17800<\/td>\nAdj R-Sq<\/th>\n0.9106<\/td>\n<\/tr>\n
          Coeff Var<\/th>\n4.71975<\/td>\n<\/th>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/article>\n
          \n\n\n\n<\/colgroup>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n
          \u00a0<\/caption>\n
          Parameter Estimates<\/th>\n<\/tr>\n
          Variable<\/th>\nDF<\/th>\nParameter
          \nEstimate<\/th>\n          		Standard
          \nError<\/th>\n          		t\u00a0Value<\/th>\nPr\u00a0>\u00a0|t|<\/th>\n<\/tr>\n<\/thead>\n
          Intercept<\/th>\n1<\/th>\n20.85800<\/td>\n0.80083<\/td>\n26.05<\/td>\n0.0001<\/td>\n<\/tr>\n
          TRIAL<\/th>\n1<\/th>\n-1.56000<\/td>\n0.24146<\/td>\n-6.46<\/td>\n0.0075<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/article>\n<\/section>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n
          \"\"<\/p>\n
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