Statistical Methods for Psychology

616 Chapter 16 Analyses of Variance and Covariance as General Linear Models

Table 16.13 Regression results for various models for data in Table 16.11

Model SSregression MSresidual R^2

C, T, G, TG 36,389.60175 75.53859 0.8027

C, T, G 34,763.09104 0.7669

C, G, TG 12,519.11654 0.2762

C, T, TG 35,826.34433 0.7903

T, G, TG 31,744.72593 0.7003

simpler full model against which to compare subsequent reduced models. Our revised full

model is

or, in more traditional analysis of variance terms,

The results of the several multiple regression solutions needed for the analysis of co-

variance are shown in Table 16.13. By calculating and testing the differences between full

and reduced models, you will be able to compute the complete analysis of covariance.

Exhibit 16.5 contains the results of an SPSS analysis of these data. You should com-

pare your results with the results in that exhibit.

For purposes of comparison I have presented the analysis of variance from Exhibit 13.1.

This is the analysis on the same data, but without the covariate.

Notice that in this analysis we have a significant effect due to Task, which is uninter-

esting because the tasks were quite different and we would expect that some tasks would

lead to more errors than others. We also have a Task 3 Group interaction, which was what

we were seeking because it tells us that smoking makes a difference in certain kinds of sit-

uations (which require a lot of cognitive processing) but not in others. Notice that we did

not have an overall effect due to Group. Notice also that our was 107.834, whereas

in the analysis of covariance it was 71.539.

When we look at our analysis of covariance, one of the first things we see is that

(71.539) is about one-third smaller than it was in the analysis of variance. This is due to the

fact that the covariate (Distract) was able to explain much of the variability in Errors that

had been left unexplained in the analysis of variance.

In Exhibit 16.5 we see that we have a significant effect for Groups. This is in part a

function of the smaller error term, and in part a function of adjustments of group means

because of small differences in mean Distract scores across groups. Unless we are willing

to assume that smokers in general are more distractable (and perhaps they are), then it is

MSerror

MSerror

Source df SS MS F

Task 2 28,661.526 14,330.763 132.895*

Group 2 354.548 177.274 1.644

Task 3 Group 4 2728.652 682.213 6.326*

Error 126 13,587.084 107.834

Total 134 45,331.810

*p,0.05

Yijk=m1Ck1ai1bj1abij1 ́ijk

YN=b 01 b 1 C 1 b 2 T 11 b 3 T 21 b 4 G 11 b 5 G 21 b 6 TG 111 b 7 TG 121 b 8 TG 211 b 9 TG 22