Example 6-2: Wechsler Adult Intelligence Scale Section
To illustrate these calculations we will return to the Wechsler Adult Intelligence Scale data.
This dataset includes data on n = 37 subjects taking the Wechsler Adult Intelligence Test. This test is broken up into four components:
- Picture Completion
Recall from the last lesson that the correlation between Information and Similarities was \(r = 0.77153\).
The partial correlation between Information and Similarities given Arithmetic and Picture Completion may be computed using the SAS program shown below.
Download the SAS program: wechsler2.sas
Download the SAS Output: wechsler2.lst
Explore the code below to find the partial correlation of Information and Similarities given Arithmetic and Picture Completion using the Wechsler Adult Intelligence Test data in SAS.
Note: In the upper right-hand corner of the code block you will have the option of copying ( ) the code to your clipboard or downloading ( ) the file to your computer.
options ls=78; title "Partial Correlations - Wechsler Data"; /*The first two lines define the name of the data set with the name 'wechsler' * and specify the path where the contents of the data set are read from. * Since we have a header row, the first observation begins on the 2nd row, * and the delimiter option is needed because columns are separated by commas. * The input statement is where we provide names for the variables in order * of the columns in the data set. If any were categorical (not the case here), * we would need to put a '$' character after its name.*/ data wechsler; infile "D:\Statistics\STAT 505\data\wechsler.csv" firstobs=2 delimiter=','; input id info sim arith pict; run; /*glm stands for 'general linear model' * the model statement specifies info and sim * as responses and arith and pict as predictors * the 'nouni' option suppresses univariate stats * the 'manova' statement requests multivariate * statistics for info and sim jointly * the 'printe' option provides the sum of squares * and cross products matrix for error*/ proc glm; model info sim = arith pict / nouni; manova / printe; run;
Finding the partial correlation
To find the partial correlation of Information and Similarities given Arithmetic and Picture Completion:
- Open the ‘wechsler’ data set in a new worksheet.
- Stat > Regression > Regression > Fit Regression Model
- Highlight and select ‘info’ for the Responses window and both ‘arith’ and ‘pic’ for the Continuous Predictors window.
- Under Storage, choose ‘Residuals’ and then ‘OK’ twice. The residuals are displayed in a new column ‘RESI’ in the worksheet.
- Repeat step 2. above with ‘sim’ substituted for ‘info’ as the response. The new residuals are stored in a new column ‘RESI_1’
- Stat > Basic Statistics > Correlation
- Highlight and select ‘RESI’ and ‘RESI_1’ to move them to the Variables window.
- Select ‘OK’. The partial correlation is displayed in the results area.
The output is in two tables. The first table gives the conditional variance-covariance matrix for Information and Similarities given Arithmetic and Picture Completion. The second table gives the partial correlation. Here we can see that the partial correlation is:
\(r = 0.71188\)
Conclusion: Comparing this to the previous value for the ordinary correlation, we can see that the partial correlation is not much smaller than the ordinary correlation. This suggests that little of the relationship between Information and Similarities can be explained by performance on the Arithmetic and Picture Completion portions of the test.
Partial correlations should be compared to the corresponding ordinary correlations. When interpreting partial correlations, three results can potentially occur. Each of these results yields a different interpretation.
- Partial and ordinary correlations are approximately equal. This occurred in our present setting. This suggests that the relationship between the variables of interest cannot be explained by the remaining explanatory variables upon which we are conditioning.
- Partial correlations are closer to zero than ordinary correlations. This is a common result and often what we anticipate. This suggests that the relationship between the variables of interest might be explained by their common relationships to the explanatory variables upon which we are conditioning. For example, we might find the ordinary correlation between blood pressure and blood cholesterol might be a high, strong positive correlation. We could potentially find a very small partial correlation between these two variables after we have taken into account the age of the subject. If this were the case, this might suggest that both variables are related to age, and the observed correlation is only due to their common relationship to age.
- Partial correlations are farther from zero than ordinary correlations. This rarely happens. This situation would suggest that unless we take into account the explanatory variables upon which we are conditioning, the relationship between the variables of interest is hidden or masked.