Statistical Methods for Psychology

predictor variable. In other words, whereas the partial correlation ( ) has variable 2

partialled out of both the criterion and predictor 1, the semipartial correlation has vari-

able 2 partialled out of only predictor 1. In this case, the semipartial correlation is simply the

correlation between Yand the residual ( ) of predicted on. As such, it

is the correlation of Ywith that part of that is independent of. A different way to view

the semi-partial correlation is in terms of the difference between two models, one of which

contains fewer predictors than the other. It can be shown that

We can use our example of expenditures for education. For those data, , and

. Thus

The preceding formula for affords an opportunity to explore further just what

multiple regression equations and correlations represent. Rearranging the formula we have

This formula illustrates that the squared multiple correlation is the sum of the squared cor-

relation between the criterion and one of the variables plus the squared correlation between

the criterion and the part of the other variable that is independent of the first. Thus, we can

think of Rand R^2 as being based on as much information as possible from one variable, any

additional, nonredundantinformation from a second, and so on. In general

where is the squared correlation between the criterion and variable 3, with variables

1 and 2 partialled out of 3. This way of looking at multiple regression will be particularly

helpful when we consider the role of individual variables in predicting the criterion, and

when we consider the least squares approach to the analysis of variance in Chapter 16. As

an aside, it should be mentioned that when the predictors are independent of one another,

the preceding formula reduces to

because, if the variables are independent, there is no variance in common to be partialled out.

The squared partialcorrelation between SAT and Expend, partialling the LogPctSAT

from both SAT and Expend, by the method discussed next is .198, showing that 20% of the

variation in SAT that could not be explained by LogPctSATcan be accounted for by that

portion of Expend that could not be explained by LogPctSAT. This point will be elaborated

in the next section.

We do not need a separate significance test for semipartial or partial correlations,

because we already have such a test in the test on the regression coefficients. If that test is

significant, then corresponding , partial, and semipartial coefficients are also significant.^6

Therefore, from Exhibit 15.1 we also know that these coefficients for Expend are all

significant. Keep in mind, however, that when we speak about the significance of a coeffi-

cient we are speaking of it within the context of the other variables in the model. For ex-

ample, we saw earlier that when Salary is included in the model it does not make a

b

R^2 0.123... p=r^2011 r^2021 r 03 21 Á 1 r^20 p

r^2 0(3.12)

R^2 0.123... p=r^2011 r^2 0(2.1) 1 r^2 0(3.12) 1... 1 r^2 0(p.123... p 2 1)

R^2 0.12=r^2021 r^2 0(1.2)

r0(1.2)

r0(1.2)= 3 r^2 0(1.2)= 1 .029=.170

r^2 0(1.2)=.886 2 .857=.029

r^202 =.857

R^2 0.12=.886

r^2 0(1.2)=R^2 0.12 2 r^202

X 1 X 2

X 12 XN 1 =X 1 r X 1 X 2

r0(1.2)

r01.2

536 Chapter 15 Multiple Regression

(^6) You will note that we consider both partial and semipartial correlation but only mentioned the partialregression
coefficient ( ). This coefficient could equally well be called the bj semipartialregression coefficient.