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we will find at least one of these two-way interactions to be statistically
significant; in interpreting the three-way interaction, we narrate each of
the two-way analyses.
Applying our interpretation strategy to the present numerical exam-
ple, we must first split apart one of the independent variables. This is
already done in Figure 9.1 as we have separate matrices of means for those
with children in their home and for those who do not have children in
their home. It is therefore convenient for us to use this breakdown here.
This split has us examining the two-way interaction of Voting
×
Political
Preference for each level of children in home. Thus, we are looking at two
two-way interactions. In your own research, the choice of which variable
to use as a “breaking” variable will likely be governed by the theory on
which the research is based. If there is no theoretical reason for selecting
the breaking variable, then the choice should be based on the ease of
explication needed to communicate the results.
In the process of visually examining the three-way interaction we
next look at the pattern of means (the two-way interaction of Voting
×
Political Preference) in each portion of the split. The fact that the three-
way interaction was statistically significant informs us in advance that the
patterns we will see (i.e., the two-way interactions) will be different. This
examination can be done either by looking at the values of the means
in Figure 9.1 or by graphing the means. We will work with the graphic
representation, and we show the plots in Figure 9.3.
Figure 9.3 shows both two-way interactions, one for those who have
children in the home (top graph) and another for those with no children in
the home (bottom graph). As you can see, the patterns are different, some-
thing that is to be expected based on the significant three-way interaction.
By the way, such pattern differences would be seen regardless of which
independent variable we chose to split; once we examined the two-way
interactions, we would always see different patterns in the plots.
As was the case with the two-way interaction discussed in the last
chapter, visual inspection of the plots is useful to gain an impression of
the pattern of the means, but it needs to be supplemented with the simple
effects analyses to determine which means in the plots were and were not
significantly different from each other.
9.3 COMPUTING THE PORTIONS OF THE SUMMARY TABLE
The degrees of freedom, mean squares,
F
ratios, and eta squared values
are computed in the same manner that we described in Chapters 6 and
8. Thus, the degrees of freedom for the triple interaction, as is true for all
interactions, is computed by multiplying the degrees of freedom of those
main effects contained in it. All
F
ratios involve dividing the mean square
of the respective effect by the mean square of the error term. Eta squared
values represent the division of the sum of squares for the effect by the
sum of squares for the total variance.
