Statistical Methods for Psychology

On the right of Figure 15.6 you can see that both sets of residuals were reasonably normal,

which is important. Cohen et al. (2003) describe a test of heterogeneity of residuals devised

by Levene. It is basically the same Levene test that we discussed in Chapter 7 when con-

sidering heterogeneity of variance for a ttest on independent samples and focuses on resid-

uals that increase or decrease with increasing values along the Xaxis. Cai and Hayes

(2008) have proposed a test of the regression coefficients themselves that is much more ro-

bust against heterogeneity of regression. Applying their test to our data confirms that the

coefficients for both Expend and LogPctSAT are significant.^9

Comparing Models

Sometimes we have what are called nested modelsor hierarchical modelsin which the

variables in one model represent a subset of the variables in a second model. For example,

we might wonder if we do better predicting SAT from Expend, LogPctSAT, PTratio, and

544 Chapter 15 Multiple Regression

850 900

Fitted values

Residuals vs. Fitted

950 1000

(^4848)
29 34 34 29
1050
–100
–50
Residuals
0
50
–2 –1
Theoretical quantiles
Normal Q-Q
012
–2
–1
Standardized residuals
0
1
2
900
Fitted values
Residuals vs. Fitted
950 1000 1050
–60
–40
Residuals–20
0
20
40
29 29
4
48 4
48
60
–2 –1
Theoretical quantiles
Normal Q-Q
012
–2
–1
Standardized residuals
0
1
2
Figure 15.6 Residual plots with PctSAT and Expend as predictors (top row) and with
LogPctSAT and Expend as predictors (bottom row)
(^9) Cai and Hayes (2008) provide a SAS macro to perform these tests. Although their paper is complex, their macro
is reasonably simple to implement. You simply include it in your SAS program and call it as shown in their paper.
nested models
hierarchical
models