variable. The second (Distractibility) we controlled by measuring it and treating it as a
covariate. In many respects, these are two ways of treating the same problem. Although there
are obvious differences in the way these two variables are treated, there are also important
similarities. In obtaining , we are actually partialling out bothtask and covariate. It is
true that in the case of equal ns task is orthogonal to group, leaving nothing to partial out; but
that is merely a technicality. In the case of unequal ns, the partialling out of both variables is
a very real procedure. Although it is important not to lose sight of the fact that the analysis of
covariance is a unique technique with its own additional assumptions, it is equally important
to keep in mind that a covariate is just another variable.16.10 Using Multiple Covariates
We have been concerned with the use of a single covariate. There is no theoretical or prac-
tical reason, however, why we must restrict ourselves in this way. For example, a study on
the effectiveness of several different teaching methods might wish to treat IQ, Age, and
Type of School (progressive or conservative) as covariates. When viewed from the point of
view of multiple regression, this presents no particular problem, whereas when viewed
within the traditional framework of the analysis of variance, the computational complexi-
ties for only a very few covariates would be overwhelming.
In the expression , bis really only a shorthand way of representing a set of pre-
dictors (e.g., ). By the same token, ccan be used to stand for a set of covariates
( ). Thus, in terms of the more specific notation, might really representWhen seen in this light, the use of multiple covariates is no different from that of single
covariates. If Crepresents the covariates IQ, Age, and School, then remainsIt should be apparent from the previous example that no restriction is placed on the
nature of the covariate, other than that it is assumed to be linearly related to the criterion.
It can be a continuous variable, as in the case of IQ and Age, or a discrete variable, as in
the dichotomous classification of Schools as progressive and conservative.
A word of warning: Just because it is possible (and in fact easy) to use multiple covari-
ates is not a good reason for adopting this procedure. Interpreting an analysis of covariance
may be difficult enough (if not impossible) with only one covariate. The problems increase
rapidly with the addition of multiple covariates. Thus, it might be easy to say, in evaluating
several methods of teaching English, that such and such a method is better if groups are
equated for age, IQ, type of school, parents’ occupation, and so on. But the experimenter
must then ask himself if such equated groups actually exist in the population. If they do
not, he has just answered a question about what would happen in groups that could never
exist, and it is unlikely that he will receive much applause for his efforts. Moreover, even if
it is possible to form such groups, will they behave in the expected manner? The very fact
that the students are now in homogeneous classes may itself have an effect on the dependent
variable that could not have been predicted.16.11 Alternative Experimental Designs
The analysis of covariance is not the only way to handle data in which a covariate is im-
portant. Two common alternative procedures are also available: stratification(matched
samples) and difference scores.SSAB(adj)=SSregression(IQ, Age, School,A 1 ,B 1 ,B 2 ,AB 11 ,AB 12 ) 2 SSregression(IQ, Age, School,A 1 ,B 1 ,B 2 )SSAB(adj)R^20 .IQ, Age, School,A 1 ,B 1 ,B 2 ,AB 11 ,AB 12C 1 ,C 2 ,... , Ck R^2 c,a,b,abB 1 ,B 2 ,... , BbR^2 c,a,b,abSSgroupSection 16.11 Alternative Experimental Designs 621stratification