The CORR Procedure

Example 2.9 Computing Partial Correlations

A partial correlation measures the strength of the linear relationship between two variables, while adjusting for the effect of other variables.

The following statements request a partial correlation analysis of variables Height and Width while adjusting for the variables Length3 and Weight. The latter variables, which are said to be partialled out of the analysis, are specified with the PARTIAL statement.

ods graphics on;
title 'Fish Measurement Data';
proc corr data=fish1 plots=scatter(alpha=.20 .30);
   var Height Width;
   partial Length3 Weight3;
ods graphics off;

Output 2.9.1 displays descriptive statistics for all the variables. The partial variance and partial standard deviation for the variables in the VAR statement are also displayed.

Output 2.9.1: Descriptive Statistics

Fish Measurement Data

The CORR Procedure

2 Partial Variables: Length3 Weight3
2 Variables: Height Width

Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum Partial
Std Dev
Length3 34 38.38529 4.21628 1305 30.00000 46.50000    
Weight3 34 8.44751 0.97574 287.21524 6.23168 10.00000    
Height 34 15.22057 1.98159 517.49950 11.52000 18.95700 0.26607 0.51582
Width 34 5.43805 0.72967 184.89370 4.02000 6.74970 0.07315 0.27047

When you specify a PARTIAL statement, observations with missing values are excluded from the analysis. Output 2.9.2 displays partial correlations for the variables in the VAR statement.

Output 2.9.2: Pearson Partial Correlation Coefficients

Pearson Partial Correlation Coefficients, N = 34
Prob > |r| under H0: Partial Rho=0
  Height Width

The partial correlation between the variables Height and Width is 0.25692, which is much less than the unpartialled correlation, 0.92632 (in Output 2.9.2). The $p$-value for the partial correlation is 0.1558.

The PLOTS=SCATTER option displays (in Output 2.9.3) a scatter plot of the residuals for the variables Height and Width after controlling for the effect of variables Length3 and Weight. The ALPHA=.20 .30 suboption requests $80\% $ and $70\% $ prediction ellipses, respectively.

Output 2.9.3: Partial Residual Scatter Plot

In Output 2.9.3, a standard deviation of Height has roughly the same length on the X axis as a standard deviation of Width on the Y axis. The major axis length is not significantly larger than the minor axis length, indicating a weak partial correlation between Height and Width.