Semipartial Correlation

A semipartial correlation is a statistic measuring the relationship between a dependent variable (or criterion) with an adjusted independent variable (or predictor) controlling for the effect of other predictors in the analysis. Like partial correlation, the [Page 1019]semipartial correlation is computed in the process of multiple regression analysis.

Figure 1 Diagrams Illustrating Conceptual Differences Between Semipartial, Partial, and Pearson's Correlations

The diagrams illustrate how the semipartial correlation differs conceptually from both partial and Pearson's correlations. Grey circles (As and Bs) in the diagrams represent predictor variables, whereas the black circles, Y s, represent criterion variables. Boxes represent residuals of As and Y s after subtracting the effects that the Bs exert on them. Curved arrows represent correlation (or covariance) between pairs of variables in the diagrams, whereas the straight arrow represents a regression path connecting B to Y.

In Figure 1a, the curved arrow rYA represents a zero-order Pearson's correlation coefficient, which measures the extent to which A and Y are correlated with each other, not taking into account the effect of the third variable B. In Figure 1b, the curved arrow srYA,B represents a semipartial correlation coefficient. It measures the extent to which the criterion, Y, and the residual of A (i.e., dA,B), obtained by subtracting from A its covariance with B, are correlated with each other. In Figure 1c, the curved arrow prYA,B represents a partial correlation coefficient. It measures the extent to which the residual of Y (i.e., dY,B) and the residual of A (i.e., dA,B) are correlated with each other.

Illustratively, in a multiple regression analysis in which Y is the criterion variable that is predicted by A and B (two predictors), both the semipartial correlation and the partial correlation obtained from A and B would be related to specific (or unique) contributions of A and B, respectively. In particular, the squared semipartial correlation for A is the proportion of the total variance of Y uniquely accounted for by A, after removing the shared variance between A and B from A but not from Y. Alternatively, the squared partial correlation for A is the proportion of the residual variance of Y uniquely accounted for by A after removing the effect of B on both A and Y.

Both the partial and semipartial correlations for any given pair of variables can be calculated on the basis of the zero-order Pearson's correlation between a pair of variables according to the following algebraic rules:

Comparing equations (1) and (2), the computation of a semipartial correlation (i.e., srYA,B) differs from that of a partial correlation (i.e., prYA,B) in the composition of their denominators. The partial correlation equation has a smaller denominator than does the semipartial correlation equation (i.e., for the partial correlation, the denominator is a product of two terms each smaller than 1.00, whereas the semipartial correlation includes only one of those terms). These differences [Page 1020]make a semipartial correlation always lower than a partial correlation, except for the cases where A and B are independent or uncorrelated.