Introduction to Categorical Data Analysis

140 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS
With either approach, for categorical predictors with more than two categories,
the process should consider the entire variable at any stage rather than just individual
indicator variables. Otherwise, the result depends on how you choose the baseline
category for the indicator variables. Add or drop the entire variable rather than just
one of its indicators.
Variable selection methods need not yield a meaningful model. Use them with
caution! When you evaluate many terms, one or two that are not truly important may
look impressive merely due to chance.
In any case, statistical significance should not be the sole criterion for whether to
include a term in a model. It is sensible to include a variable that is important for
the purposes of the study and report its estimated effect even if it is not statistically
significant. Keeping it in the model may help reduce bias in estimating effects of other
predictors and may make it possible to compare results with other studies where the
effect is significant (perhaps because of a larger sample size). Likewise, with a very
large sample size sometimes a term might be statistically significant but not practically
significant. You might then exclude it from the model because the simpler model is
easier to interpret – for example, when the term is a complex interaction.
5.1.4 Example: Backward Elimination for Horseshoe Crabs
When one model is a special case of another, we can test the null hypothesis that the
simpler model is adequate against the alternative hypothesis that the more complex
model fits better. According to the alternative, at least one of the extra parameters
in the more complex model is nonzero. Recall that the deviance of a GLM is the
likelihood-ratio statistic for comparing the model to the saturated model, which has
a separate parameter for each observation (Section 3.4.3). As Section 3.4.4 showed,
the likelihood-ratio test statistic −2(L0 − L1) for comparing the models is the difference
between the deviances for the models. This test statistic has an approximate
chi-squared null distribution.
Table 5.2 summarizes results of fitting and comparing several logistic regression
models. To select a model, we use a modified backward elimination procedure. We
start with a complex model, check whether the interaction terms are needed, and then
successively take out terms.
We begin with model (1) in Table 5.2, symbolized by C ∗ S + C ∗ W + S ∗ W.It
contains all the two-factor interactions and main effects. We test all the interactions
simultaneously by comparing it to model (2) containing only the main effects. The
likelihood-ratio statistic equals the difference in deviances, which is 186.6 − 173.7 =
12.9, with df = 166 − 155 = 11. This does not suggest that the interactions terms
are needed (P = 0.30). If they were, we could check individual interactions to see
whether they could be eliminated (see Problem 5.3).
The next stage considers dropping a term from the main effects model. Table 5.2
shows little consequence from removing spine condition S (model 3c). Both remaining
variables (C and W ) then have nonnegligible effects. For instance, removing C
increases the deviance (comparing models 4b and 3c) by 7.0 on df = 3(P = 0.07).
The analysis in Section 4.4.3 revealed a noticeable difference between dark crabs