Thus,To compare the adjusted means, we haveSince , we can reject and conclude that the Active Smoking group
performs more poorly (overall) than the average of the other two groups.
Another experimenter might be interested in examining the effects of Group only for
the Cognitive task. If we want to examine these simple effects, we will again need to mod-
ify our error term in some way. This is necessary because we will be looking at Groups for
only some of the data, and the covariate mean of the Cognitive task subjects may differ
from the covariate mean for all subjects. Probably the safest route here would be to run a
separate analysis of covariance for only those subjects performing the cognitive task.
Although this method has the disadvantage of costing us degrees of freedom for error, it
has the advantage of simplicity and eliminates the need to make discomforting assumptions
in the adjustment of our error term.
To complete our discussion of the tests we might wish to conduct, consider the experi-
menter who wants to compare two particular adjusted cell means (whether or not they are in
the same row or column). The adjusted error term for this comparison was given by Winer
(1971) aswhere is the sum of squares from an analysis of variance on the covariate.
You may wonder why we continually worry about adjusting the error term in making
comparisons. The general nature of the answer is apparent when you recall what the confi-
dence limits around the regression line looked like in Chapter 9. (They were curved—in
fact they were elliptical.) For X 5 , we were relatively confident about. However, as X
departed more and more from we became less and less confident of our prediction, and
consequently the confidence limits widened. If you now go back to Figure 16.3, you will
see that the problem applies directly to the case of adjusted means. In that figure, is a
long way from , and we would probably have relatively little confidence that we have
estimated it correctly. On the other hand, we can probably have a reasonable degree of
confidence in our estimate of. It is just this type of consideration that causes us con-
stantly to adjust our error term.
The example we have used points up an important feature of the analysis of covariance—
the fact that the covariate is just another variable that happens to receive priority. In designing
the study, we were concerned primarily with evaluating the effects of smoking. However, we
had two variables that we considered it necessary to control: type of task and distractibility.
The first one (Task) we controlled by incorporating it into our design as an independentY¿ 2
Y 1
Y¿ 1
X
X YN
SScells(c)MS–error=2 MS–error
nC 11
SScells(c)
tg 21
SSe(c)S
F.05(1,125)=3.92 H 0
F(1,125)=
nc^2
ga^2 iMS–error=
45( 2 8.697)^2
6(72.019)
=7.88
c=2(15.360) 2 1(19.696) 2 1(19.721)= 2 8.697MS–error=71.538C 11730.015
221
54285.867
S=72.019
SSe(c)=54285.867SSg(c)=730.015MS¿error=71.538620 Chapter 16 Analyses of Variance and Covariance as General Linear Models