Introduction to Categorical Data Analysis

218 LOGLINEAR MODELS FOR CONTINGENCY TABLES
which translates to (0.42, 0.47) for the odds ratio. The odds of injury for passengers
wearing seat belts were less than half the odds for passengers not wearing them, for
each gender–location combination. The fitted odds ratios in Table 7.11 also suggest
that, other factors being fixed, injury was more likely in rural than urban accidents
and more likely for females than males. Also, the estimated odds that males used seat
belts are only 0.63 times the estimated odds for females.
7.2.7 Three-Factor Interaction
Interpretations are more complicated when a model contains three-factor terms. Such
terms refer to interactions, the association between two variables varying across levels
of the third variable. Table 7.10 shows results of adding a single three-factor term to
model (GI, GL, GS, IL, IS, LS). Of the four possible models, (GLS, GI, IL, IS) fits
best. Table 7.9 also displays its fit.
For model (GLS, GI, IL, IS), each pair of variables is conditionally dependent,
and at each level of I the association between G and L or between G and S or
between L and S varies across the levels of the remaining variable. For this model, it
is inappropriate to interpret the GL, GS, and LS two-factor terms on their own. For
example, the presence of the GLS term implies that the GS odds ratio varies across
the levels of L. Because I does not occur in a three-factor term, the conditional odds
ratio between I and each variable is the same at each combination of levels of the
other two variables. The first three lines of Table 7.11 report the fitted odds ratios for
the GI, IL, and IS associations.
When a model has a three-factor term, to study the interaction, calculate fitted
odds ratios between two variables at each level of the third. Do this at any levels of
remaining variables not involved in the interaction. The bottom six lines of Table 7.11
illustrates this for model (GLS, GI, IL, IS). For example, the fitted GS odds ratio of
0.66 for (L = urban) refers to four fitted values for urban accidents, both the four
with (injury = no) and the four with (injury = yes); that is,
0.66 = (7273.2)(10, 959.2)/(11, 632.6)(10, 358.9)
= (1009.8)(389.8)/(713.4)(834.1)
7.2.8 Large Samples and Statistical Versus Practical Significance
The sample size can strongly influence results of any inferential procedure. We are
more likely to detect an effect as the sample size increases. This suggests a cautionary
remark. For small sample sizes, reality may be more complex than indicated by the
simplest model that passes a goodness-of-fit test. By contrast, for large sample sizes,
statistically significant effects can be weak and unimportant.
We saw above that model (GLS, GI, IL, IS) seems to fit much better than
(GI, GL, GS, IL, IS, LS): The difference in G 2 values is 23.4 − 7.5 = 15.9, based
on df = 5 − 4 = 1(P = 0.0001). The fitted odds ratios in Table 7.11, however,
show that the three-factor interaction is weak. The fitted odds ratio between any two