the two groups to have the same mean on the posttest in the absence of a real treatment
effect. Huitema (1980, pp. 149ff) gives an excellent demonstration that when the groups
differ at the beginning of the experiment, the phenomenon of regression to the mean could
lead to posttest differences even in the absence of a treatment effect. For alternative analy-
ses that are useful under certain conditions, see Huitema (1980). Maris (1998) takes a dif-
ferent view of the issue.
The problems of interpreting results of designs in which subjects are not randomly
assigned to the treatment groups are not easily overcome. This is one of the reasons why
random assignment is even more important than random selection of subjects. It is difficult
to overestimate the virtues of random assignment, both for interpreting data and for mak-
ing causal statements about the relationship between variables. In what is probably only a
slight overstatement of the issue, Lord (1967) remarked, “In the writer’s opinion, the ex-
planation is that with the data usually available for such studies, there is simply no logical
or statistical procedure that can be counted on to make proper allowances for uncontrolled
pre-existing differences between groups” (p. 305). (Lord was notreferring to differences
that arise by chance through random assignment.) Anderson (1963) made a similar point
by stating, “One may well wonder exactly what it means to ask what the data would be like
if they were not what they are” (p. 170). All of this is not to say that the analysis of covari-
ance has no place in the analysis of data in which the treatments differ on the covariate.
Anyone using covariance analysis, however, must think carefully about her data and the
practical validity of the conclusions she draws.16.8 Reporting the Results of an Analysis of Covariance
The only difference between describing the results of an analysis of covariance and an
analysis of variance is that we must refer to the covariate and to adjusted means. For the
experiment by Conti and Musty we could write
Conti and Musty (1984) examined the effect of THC on locomotor activity in rats.
They predicted that moderate doses of THC should show the greatest increase in activ-
ity (or the least decrease due to adaptation). After a pretesting session five different
groups of rats were randomly assigned to receive 0, .1 mg, .5 mg, 1 mg, or 2 mg of THC.
Activity level was measured in a 10-minute postinjection interval. Because there was
considerable variability in pretest activity, the pretest measure was used as a covariate
in the analysis.
The analysis of covariance was significant (F(4, 41) 5 4.694, p 5 .003), with inter-
mediate doses showing greater effect. Eta-squared was .09 using a SStotalthat has not
been adjusted for the covariate. A contrast of the means of the control group and the
.5mg group revealed a significant difference (F(1, 41) 5 16.69, p,.05), with a stan-
dardized effect size (d) of 1.23.16.9 The Factorial Analysis of Covariance
The analysis of covariance applies to factorial designs just as well as it does to single-
variable designs. Once again, the covariate may be treated as a variable that, because of
methodological considerations, assumes priority in the analysis. In this section we will deal
only with the case of equal cell sizes, but the generalization to unequal ns is immediate.612 Chapter 16 Analyses of Variance and Covariance as General Linear Models