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of math word problems solved is statistically adjusted to take verbal
ability into consideration by using it as a covariate, an ANCOVA now
indicates that the groups do not differ significantly on the dependent
variable.
An ANOVA results in the researchers determining that the groups are
not significantly different; when the dependent variable of number
of math word problems solved is statistically adjusted to take verbal
ability into consideration by using it as a covariate, an ANCOVA still
shows that the groups do not differ significantly on the dependent
variable.
An ANOVA results in the researchers determining that the groups are
not significantly different; when the dependent variable of number
of math word problems solved is statistically adjusted to take verbal
ability into consideration by using it as a covariate, an ANCOVA
now shows the groups to be significantly different.
These four alternative outcomes are all possible. In two of them, the
researchers draw the same conclusion from the analysis (the groups differ
significantly, the groups do not differ significantly). But in two of them
the ANCOVA reverses the conclusion that would have been drawn from
the ANOVA, and these circumstances are much more striking.
In one of these reversal-of-outcome circumstances, the researchers
conclude, based on the ANOVA, that there is a difference between girls
and boys in math word problem ability. However, when verbal ability
is statistically controlled by using ANCOVA, that apparent difference no
longer manifests itself. Thus, the difference in problem solving attributed
to gender in the ANOVA was now explained by verbal ability in the
ANCOVA. When the independent variable of gender was finally evaluated
in the ANCOVA, it was unable to account for a statistically significant
portion of the variance in math score remaining after the effects of verbal
ability were first accounted for.
In the other reversal-of-outcome circumstance, the researchers con-
clude, based on the ANOVA, that girls and boys perform comparably in
math word problem ability. However, when verbal ability is statistically
controlled by using ANCOVA, a statistically significant difference between
the genders is obtained. Here, verbal ability can be thought of as masking
or compensating for the effect of gender. When verbal ability is statistically
controlled by removing the variance in math scores associated with it, the
difference between girls and boys can be detected.
16.4 THE PROCESS OF PERFORMING ANCOVA
The key to understanding ANCOVA is not fundamentally different from
the key to understanding ANOVA: We attempt to explain the variance of
the dependent variable on the basis of the effects of the other variables
in the design. In ANOVA, these other effects are the main effects of the
