Statistical Methods for Psychology

significant contribution. That does not mean that it would not contribute to any other model

predicting SAT. (In fact, when used as the only predictor, it predicts SAT at better-than-

chance levels. R 52 .440, p 5 .001.) It only means that once we have the other predictors

in our model, Salary does not have any independent (or unique) contribution to make.

Alternative Interpretation of Partial and Semipartial Correlation

There is an alternative way of viewing the meaning of partial and semipartial correlations

that can be very instructive. This method is best presented in terms of what are called

Venn diagrams.The Venn diagram shown in Figure 15.4 is definitely not drawn to scale,

but it does illustrate various aspects of the relationship between SAT and the two predictors.

Suppose that the box in Figure 15.4 is taken to represent all the variability in the crite-

rion (SAT). We will set the area of the box equal to 1.00—the proportion of the variation in

SAT to be explained. The circle labeled LogPctSAT is taken to represent the proportion of

the variation in SAT that is explained by LogPctSAT. In other words, the area of the circle is

equal to 5 .857. Similarly, the area of the circle labeled Expend is the percentage of the

variation in SAT explained by Expend and is equal to 5 .145. Finally, the overlap be-

tween the two circles represents the portion of SAT that both LogPctSAT and Expend have

in common, and equals .116. The area outside of either circle but within the box is the por-

tion of SAT that cannot be explained by either variable and is the residual variation 5 .059.

The areas labeled B, C, and Din Figure 15.4 represent portions of the variation in SAT

that can be accounted for by LogPctSAT and/orExpend. (Area Arepresents the portion

that cannot be explained by either variable or their combination, the residual variation.)

Thus, the two predictors in our example account for 88.6% of the variation of Y: B 1 C 1

D 5 .741 1 .116 1 .029 5 .886. The squared semipartial correlationbetween LogPct-

SAT and SAT, with Expend partialled out of LogPctSAT, is the portion of the variation of

SAT that LogPctSAT accounts for over and abovethe portion accounted for by Expend. As

such, it is .857 and is labeled as B

The semipartial correlation is the square root of this quantity.

The squared partial correlationhas a similar interpretation. Instead of being the addi-

tional percentage of SAT that LocPctSAT explains but that Expend does not, which is the

r0(1.2)= 1 .741=.861

r^2 0(1.2)=R^2 0.12 2 r^201 =.886 2 .145=.741

r^202

r^201

15.8 Partial and Semipartial Correlation 537

Figure 15.4 Venn diagram illustrating partial and semipartial correlation

ABC D

0.886

0.857

0.145

Expend

LogPctSAT

0.741 0.116 0.029

Venn diagrams