Analysis of Covariance - Analysis of Variance Designs - page 456

independent variables and their interactions. In ANCOVA, we still have
effects of the independent variables and their interactions, but now we
also have the effect of the covariate (we will limit our discussion to the
situation where we have a single covariate). As is true for the effects of
the independent variables and their interactions, we now also use the
covariate to explain variance of the dependent variable.
TherearethreestepsthatareinvolvedinperforminganANCOVA:
(a) using the covariate to predict the dependent variable, (b) adjusting
the dependent variable to remove the effects of the covariate, and (c)
performing an ANOVA on the adjusted dependent variable scores.
16.4.1 PREDICTING THE DEPENDENT VARIABLE FROMTHE COVARIATE
We use a covariate in the design because we have hypothesized that it may
bear a relationship to (covary with) the dependent variable. We thus want
to“remove”itseffectbeforeweevaluatetheeffectofourindependent
variable.
To say that the covariate may be related to the dependent variable is
to say that the covariate may be correlated to the dependent variable. To
the extent that two variables are correlated, we may predict the values of
one from the values of the other. Prediction is evaluated by means of a
linear regression procedure, which results in a prediction equation often
referred to as a linear regression model.
In this first major step of ANCOVA, we use the covariate as a predictor
of the dependent variable without taking into account group membership,
that is, without involving the independent variable. Hence, this analysis
is performed on the sample as a whole; cases in the analysis are pooled
regardless of their group. Because our focus is on the covariate and not on
the independent variable, the covariate is thus given the first opportunity
to account for the variance of the dependent variable; the independent
variable is not yet involved in the analysis.
If the covariate and the dependent variable are correlated, the lin-
ear regression model (using the covariate as the predictor) will account
for a significant amount of the variance of the dependent variable. The
strength of this relationship can be indexed by a squared multiple correla-
tion coefficient computed by dividing the sum of squares associated with
the regression model by the sum of squares of the total variance of the
dependent variable; it is ordinarily shown in the ANCOVA summary table
as an R squared value.
16.4.2 ADJUSTING THE DEPENDENT VARIABLE VALUES
The regression model uses the scores on the covariate of all participants
to predict their scores on the dependent variable. At the completion of the
regression procedure, each case in the data file has a predicted dependent
variable score. These predicted values will be different from the original
scores for most if not all of the cases in that they reflect only what we are able
to predict from the covariate. Using the variables in our earlier illustration,