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we would have predicted the number of math problems solved based only
on a knowledge of each child's verbal ability score. These predicted values
therefore represent (they have “used up”) all of the information that we
have available to us in the verbal ability variable; that is, verbal ability has
done all that it can do for us in predicting or explaining the variance in
math scores.
There are several ways to express the idea that the predicted num-
ber of math problems solved reflects the absence of their verbal ability
component:
The effect verbal ability has been statistically nullified.
The effect verbal ability has been statistically controlled.
The effect verbal ability has been statistically removed.
The effect verbal ability has been statistically partialled out.
The effect verbal ability has been statistically equated in the groups.
You will likely see one or more of these expressions used in the literature
when the authors are describing a covariance procedure.
The predicted values from the linear regression procedure can be
viewed as scores on the dependent measures that have been adjusted by
removing the effects of the covariate from them. These values are referred
to as
adjusted values
of the dependent variable in that they no longer
contain information related to the covariate. Both the originally observed
as well as these predicted or adjusted values of the dependent variable
exist side by side during the process of computing the ANCOVA; SPSS
and SAS allow the predicted values to be saved to the data file for future
reference.
16.4.3 PERFORMING AN ANOVA ON THE ADJUSTED
DEPENDENT VARIABLE
In the final stage of ANCOVA, we use as the dependent variable not the
scores on the observed dependent variable but rather the adjusted values
of the dependent variable that were generated from the linear regression
analysis.Intermsofourillustration,intheANCOVAweevaluatetheeffect
of gender on the adjusted number of math problems solved. Because the
effect of verbal ability is not reflected in (has been statistically removed
from) these adjusted values, any differences between the groups can be
more confidently attributed to the independent variable.
16.4.4 EXAMINING MEAN DIFFERENCES IN ANCOVA
A simple way of thinking about ANCOVA is that it is just an ANOVA
performed on adjusted rather than the original or observed scores on the
dependent variable. Because the original scores are adjusted, the groups
are compared on the adjusted scores. Therefore, if a statistically significant
F
ratio is obtained for the independent variable in an ANCOVA, it means
that
the groups differ on the adjusted means
, which may be quite different
from a comparison of the means based on the observed dependent variable
