Defining the Correlation Coefficient for Continuous Data: the Pearson Product Moment Correlation Coefficient

Pearson’s Product Moment Correlation Coefficient is aptly named for the mathematical computational steps that are used to produce the outcome.  First, the computation is based on the multiplication of variables, which result in generating a product. Second, the variables, labelled x and y, are computed as independent statistical moments, whereby each score is evaluated against the variable’s algebraic measure of centrality of the group scores, a.k.a. the mean. In this way, the correlation coefficient provides the researcher with an estimate of a relationship between two dependent variables.

The outcome estimate of the Pearson Product Moment Correlation Coefficient does NOT imply cause or causality between the two variables. Rather, the outcome estimate is merely an estimate of how closely two variables describe independent responses for a sample. The correlation coefficient is calculated by combining the estimates of variance on each of the separately measured variables of interest.

Specifically, the correlation coefficient is a ratio score that ranges from -1.00 to +1.00 and is created from a cross product:

as the numerator term and then adjusted through division by the error term in the denominator, which is the square root of the product of the sum of squares for the x variable and the sum of squares for the y variable, as shown here:

The formula to compute the Pearson product-moment correlation coefficient is shown here:

When presented graphically, the paired measures of the x and y variables for the set of scores are represented as a single point within the graphing space. The calculation of the correlation coefficient describes the mathematical relationship between the x and y variable pairs are associated within a geometric space.  The following three graphs illustrate the extremes of the Pearson Product Moment Correlation Coefficients for the relationships between x and y variable pairs. In Figure 1, the relationship illustrates a Pearson Product Moment Correlation Coefficient as extreme positive with an r value of 1.00. In Figure 2, the relationship illustrates an extreme negative r value of -1.00.  In Figure 3, the relationship is shown as a Pearson Product Moment Correlation Coefficient of zero, or no mathematical relationship between each participant’s paired x and y scores.

Figure 1. Representation of the Pearson Product Moment Correlation Coefficient as r=1.00

In the figure above,  the cluster of paired scores on the variable x and the variable y are presented in a positive direction within the graphing space. For each individual’s score on the x variable there is an equal score on the y variable relative to the scale of scores. This means that a correspondingly high score on the y variable matches a high score on the x variable; and similarly a correspondingly low score on the y variable matches a low score on the x variable.  When the x and y variables as represented as a single point in the graphing space the resulting image suggests a positive correlation between the variables x and y for the set of participants’ scores.

Figure 2. Representation of the Pearson Product Moment Correlation Coefficient as r= -1.00

In the figure above, the cluster of paired scores on the variable x and the variable y are presented in a negative direction within the graphing space. For each individual’s score on the x variable, there is an inverse score on the y variable relative to the scale of scores. This means that a correspondingly high score on the y variable matches a low score on the x variable, and similarly, a correspondingly low score on the y variable matches a high score on the x variable.  When the x and y variables as represented as a single point in the graphing space the resulting image suggests a negative correlation between the variables x and y for the set of participants’ scores.

Figure 3. Representation of the Pearson Product Moment Correlation Coefficient as r= 0.00

In Figure 3, above, the cluster of paired scores on the variable x and the variable y are presented as having no definitive direction within the graphing space. For each individual’s score on the x variable, there is no corresponding score on the y variable relative to the scale of scores. In this instance, the x and y variables are simply represented as pairs of scores within the graphing space. No relationship exists when there is complete randomness in the scoring by an individual on the x and on the y variables and thereby do not depict a relationship between the two variables for the set of participants’ scores.

The correlation coefficient provides the researcher with an estimate of a relationship between two variables. The Pearson Product Moment Correlation Coefficient does NOT imply cause or causality between the two variables. Rather, the measure is merely an estimate of how closely two variables describe independent responses for a sample. The correlation coefficient is calculated by combining the measures of variance on each of the separately measured variables, as shown in the following formula:

A SAS Example

Consider the following data set, with 6 participants measured on two separate variables.  Your intention is to determine if the group measures on VAR_1 are similar to the group measures of VAR_2. For those of you that need more tangible labels, let’s consider that VAR_1 represents IQ scores and that VAR_2 represents SHOE SIZE.

So that our query is simply, is there a relationship between IQ scores and SHOE SIZE. In order to analyze these data, you create the following SAS program which produces the output shown below.

Output from PROC CORR with Corresponding Line of Best Fit in a Scatterplot

Correlation between 2 measures in the same person using the SAS PROC CORR Procedure. Table 1. Simple Statistics

Table 2. Matrix of Pearson Correlation Coefficients, N = 6, Prob > |r| under H0: Rho=0

The results indicate that there is a strong correlation between IQ scores and shoe size as indicated by the correlation coefficient value in Table 2 of the SAS output in the scatterplot, above. The reported correlation coefficient (r value) was 0.995. which is considered statistically significant (p<0.0001)^[1]. The r value indicates that as a group the data on each variable are traveling in the same direction. In other words, low IQ scores were associated with small shoe sizes and high IQ scores were associated with larger shoe sizes.

Another SAS Working Example

Consider the following comparison of VO₂ maximum tests to illustrate the computations for the Pearson product-moment correlation coefficient.

An individual’s ability to perform maximal re-synthesis of ATP is often determined from their estimated VO₂ maximum. Yet the true estimate of VO₂ maximum requires an individual to undergo an extensive performance test in a “clinical or laboratory” setting.

Further, because physical fitness is often established from the VO₂ maximum, researchers strive to develop approaches that can provide a suitable estimate for VO₂ maximum without requiring the laboratory test. To this end, several “field tests” have been presented as reliable and valid proxies for the laboratory estimate of VO₂ maximum.

The validity of field-tests for VO₂ maximum as effective measures of true physiological functioning within this system are based on the linear relationship between heart rate and oxygen consumption. Since the changes in heart rate match the changes in aerobic metabolism, an individual’s ability to respire can be predicted indirectly by heart rate rather than the direct determination of oxygen utilization from a controlled laboratory setting.

Laboratory tests for VO₂ maximum typically require that the individual runs or walks on a treadmill, or cycles on a standard bicycle ergometer while expired air is collected and the oxygen concentration in the expired air, is measured. The field tests, however, can use several stimulus modalities (e.g. walking, running, stepping, and cycling) and rather than measure the oxygen concentration in expired air, the technician simply records heart rate at specific stages of the exercise stress test. The relationship between field-tests and a declared “gold standard” is determined by comparing the statistical probability of the association between the predictive nature of the field test and a baseline reference standard measured in the laboratory.

In determining the accuracy of a clinical or diagnostic test, researchers compare the performance of subjects on the selected diagnostic (field) test against the “gold standard”. In the following example, this clinical epidemiological approach was used to determine the accuracy of field tests for predicting maximal oxygen consumption against a laboratory standard treadmill test (Bruce protocol). The field tests included the Cooper’s 12-minute run, the Astrand-Rhyming bicycle test, the Canadian Aerobic Fitness Test, and the one-mile walking test. A sample of the results for each test is presented in the table, below.

Table of VO2 Maximum Estimates For A Sample Of Ten Individuals On 5 Different Procedures

The following computations illustrate the application of the Pearson product-moment correlation coefficient to the oxygen consumption data shown above.

STEP 1: Select two variables to compareà, for example, we will begin with VO2 measured with the treadmill test versus VO2 measured with the 1-mile walk test. The mean for the scores on the treadmill, which in this example we designate as the x variable is 44.3 (± 9.23), and the scores on the walk test, which in this example we designate as the y variable and is 45.2 (± 3.04).

The mean score for the variable x and the mean score for the variable y are used independently in Step 2: computing variance for each selected variable.

STEP 2: Compute the sum of squares for the variance elements of each selected variable, separately, as shown below:

STEP 3: Compute the cross products of  x and y using:    [latex](x_i - \overline{x})(y_i - \overline{y})[/latex]

participant	[latex](x_i - \overline{x})(y_i - \overline{y})[/latex]
01	2.7 * -0.2 = -0.54
02	-8.3 * -2.2 = 18.26
03	10.7 * 1.8 = 19.26
04	-5.3 * -1.2 = 6.36
05	12.7 * -0.2 = -2.54
06	1.7 * 7.8 = 13.26
07	-1.3 * -1.2 = 1.56
08	-15.3 * -3.2 = 48.96
09	9.7 * -1.2 = -11.64
10	-7.3 * -0.2 =1.46
= 94.4

STEP 4: Compute the denominator of the correlation coefficient using the square root of the product of the sum of squares for the x and the y variables.

The specific values can be taken from the computations above where the = 766.01 and the  = 83.6   The denominator term is thus the square root of:  SQRT(766.01 x 83.6) = SQRT(64038.44) = 253

STEP 5: Compute the correlation coefficient by substituting the values from that which was calculated, above.

In our example, the correlation coefficient computed for the relationship between the VO2 on the treadmill and the VO2 on the 1 mile walk test = 0.37

Verifying Our Computations with SAS

The data set used in the example above included two columns of data: the VO2 max scores recorded for each participant on the treadmill test and the 1-mile walk test. The columns were separated by a tab character which we denote in our INFILE statement with the command: delimiter=’09’x . The data set has three variables: id, treadmill scores and 1-mile walk test scores.

The SAS program written for SAS Studio to compute the correlation coefficient for the relationship between the scores on the treadmill and the 1-mile walk test is shown here:

Output from the SAS PROC CORR PROCEDURE