Statistical Methods for Psychology

(the covariate), we would have a clearer test of our original hypothesis. This is exactly what

the analysis of covariance is designed to do, and this is precisely the situation in which it

does its job best—its job in this case being to reduce the error term.

A more controversial use of the analysis of covariance concerns situations in which the

treatment groups have different covariate (driving experience) means. Such a situation (us-

ing the same hypothetical experiment) is depicted in Figure 16.3, in which two of the treat-

ments have been displaced along the Xaxis. At the point at which the three regression lines

intersect the vertical line , you can see the values , , and. These are the

adjusted Ymeansand represent our best guess as to what the Ymeans would have been if

the treatments had not differed on the covariate. The analysis of covariance then tests

whether these adjustedmeans differ significantly, again using an error term from which

the variance attributable to the covariate has been partialled out. Notice that the adjusted

performance means are quite different from the unadjusted means. The adjustment has

increased 1 and decreased 3.

Although the structure and procedures of the analysis of covariance are the same re-

gardless of whether the treatment groups differ on the covariate means, the different ways

of visualizing the problem as represented in Figures 16.2 and 16.3 are instructive. In the

first case, we are simply reducing the error term. In the second case, we are both reducing

the error term andadjusting the means on the dependent variable. We will have more to say

about this distinction later in the chapter.

Assumptions of the Analysis of Covariance

Aside from the usual analysis of variance assumptions of normality and homogeneity of

variance, we must add two more assumptions. First we will assume that whatever the rela-

tionship between Yand the covariate (C), this relationship is linear.^5 Second, we will assume

homogeneity of regression—that the regression coefficients are equal across treatments—

. This is merely the assumption that the three lines in Figure
16.2 or 16.3 are parallel, and it is necessary to justify our substitution of one regression line
(the pooled within-groups regression line) for the separate regression lines. As we shall see
shortly, this assumption is testable. Note that no assumption has been made about the
nature of the covariate; it may be either a fixed or a random variable. (It can even be a cate-
gorical variable if we create dummy variables to represent the different levels of the vari-
able, as we did in the earlier parts of this chapter.)

b* 1 =b* 2 =b* 3 = Á =b*

Y Y

X=X Y¿ 1 Y¿ 2 Y¿ 3

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

X

Driving experience

Performance

(X 1 ,Y 1 )

(X 3 ,Y 3 )

Y' 1

(X 2 ,Y 2 )

Y' 2

Y' 3

Figure 16.3 Hypothetical data illustrating mean adjustment in the analysis

of covariance

(^5) Methods for handling nonlinear relationships are available but will not be discussed here.
adjusted Y
means
homogeneity of
regression