Calculating the Analysis of Covariance
When viewed within the framework of multiple regression, the analysis of covariance is
basically no different from the analysis of variance, except that we wish to partial out the
effects of the covariate. As Cohen (1968) put it, “A covariate is, after all, nothing but an
independent variable which, because of the logic dictated by the substantive issues of the
research, assumes priority among the set of independent variables as a basis for accounting
for Yvariance” (p. 439).
If we want to ask about the variation in Yafter the covariate (C) has been partialled out,
and if the variation in Ycan be associated with only C, the treatment effect (a), and error,
then represents the total amount of accountable variation. If we now compare
with , the difference will be the variation attributable to treatment
effects over and abovethat attributable to the covariate.
We will take as an example a variation on the study by Conti and Musty (1984)
presented in Chapter 11. As you may recall, in that study the authors were interested in
examining the effects of different amounts of THC, the major active ingredient in mari-
juana, injected directly into the brain. The dependent variable was locomotor activity,
which normally increases with the administration of THC by more traditional routes.
Because of the nature of the experimental setting (all animals were observed under base-
line conditions and then again after the administration of THC), activity should decrease in
all animals as they become familiar and more comfortable with the apparatus. If THC has
its effect through the nucleus accumbens, however, the effects of moderate doses of THC
should partially compensate for this anticipated decrease, leading to relatively greater
activity levels in the moderate-dose groups as compared to the low- or high-dose groups.
Conti and Musty (1984) actually analyzed postinjection activity as a percentage of pre-
injection activity, because that is the way such data are routinely analyzed in their field. An
alternative procedure would have been to run an analysis of covariance on the postinjection
scores, partialling out preinjection differences. Such a procedure would adjust for the fact that
much of the variability in postinjection activity could be accounted for by variability in prein-
jection activity. It would also control for the fact that, by chance, there were group differences
in the level of preinjection activity that could contaminate postinjection scores.
As will become clear later, it is important to note here that all animals were assigned at
random to groups. Therefore, we would expectthe group means on the preinjection phase
to be equal. Any differences that do appear on the preinjection phase, then, are due to
chance, and, in the absence of any treatment effect, we would expect that postinjection
means, adjusted for chance preinjection differences, would be equal. The fact that subjects
were assigned at random to treatments is what allows us to expect equal adjusted group
means at postinjection (if is true), and this in turn allows us to interpret group differ-
ences at postinjection to be a result of real treatment differences rather than of some arti-
fact of subject assignment.
The data and the design matrix for the Conti and Musty (1984) study are presented in
Table 16.7. The raw data have been divided by 100 simply to make the resulting sums of
squares manageable.^6 In the design matrix that follows the data, only the first and last subject
in each group are represented. Columns 6 through 9 of Xrepresent the interaction of the co-
variate and the group variables. These columns are used to test the hypothesis of homogene-
ity of regression coefficients across groups:
H 0 : b* 1 =b* 2 =b* 3 =b* 4 =b* 5H 0
SSregressionC,a SSregressionCSSregressionC,aSection 16.5 The One-Way Analysis of Covariance 601(^6) If the data had not been divided by 100, the resulting sums of squares and mean squares would be
1002 =10,000times their present size. The Fand tvalues would be unaffected.