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other group, then the assumption of homogeneity of regression has been
violated.
The way in which this assumption is statistically evaluated makes use
of a concept that we have discussed from Chapter 8 on: the presence of
an interaction effect. We have seen many times in Chapters 8-15 that an
interaction is obtained when the lines for the levels of an independent
variable are not parallel. Another way to express the idea of the lines not
being parallel is to say that the slopes of the lines are different.
Thus, the way in which we test the homogeneity of regression assump-
tion is by setting up an analysis containing the interaction of the experi-
mental effect(s) of the independent variable(s) and the covariate. This can
be done relatively easily in SPSS and SAS. In our simplified example study
shown in Table 16.1, where we have only one independent variable, there
is only one experimental effect, namely, the main effect of group. Keeping
with our example, if the Group
Covariate interaction effect is not sta-
tistically significant, then we presume that the slopes are comparable and
that the assumption of homogeneity of regression is satisfied; if the Group
×
×
Covariate interaction effect is statistically significant, then we presume
that the slopes are not parallel and that the assumption of homogeneity
of regression is violated.
When the assumption of homogeneity of regression is not met,
researchers have some options, three of which are briefly mentioned
here. First, there are some nonparametric ANCOVA procedures that may
be used; a sample of these is described by Bonate (2000) and Maxwell,
Delaney, and O'Callaghan (1993). Second, with moderate to pronounced
departures from the assumption of homogeneity of regression, researchers
can engage in a relatively more complex analysis that assesses the treat-
ment effect(s) as a function of the value of the covariate; this approach is
discussed by Maxwell and Delaney (2000). Third, although it is not neces-
sarily the preferred strategy, researchers can proceed with the ANCOVA,
recognizing that with mild to moderate violations of the assumption (a)
the analysis will be able to withstand such violations and (b) the effects
of the violation will be in the conservative direction (Maxwell & Delaney,
2000).
16.6 NUMERICAL EXAMPLE OF A ONE-WAY ANCOVA
16.6.1 A BRIEF DESCRIPTION OF OUR EXAMPLE
In our hypothetical numerical example, researchers have administered a
test of math word problems to thirty-six school children in a particular
grade who were exposed to one of three different math training programs
during the school year. Children in Group 1 (Traditional) learned math
under the traditional curriculum in place for decades in the school district.
Children in Group 2 (Freeform) were allowed to interact with the material
in whatever way they chose during the math sessions; the teacher was
