Introduction to Categorical Data Analysis

PROBLEMS 163
PROBLEMS
5.1 For the horseshoe crab data (available at www.stat.ufl.edu/∼aa/
intro-cda/appendix.html), fit a model using weight and width as predictors.
a. Report the prediction equation.
b. Conduct a likelihood-ratio test of H0: β1 = β2 = 0. Interpret.
c. Conduct separate likelihood-ratio tests for the partial effects of each variable.
Why does neither test show evidence of an effect when the test in (b)
shows very strong evidence?
5.2 For the horseshoe crab data, use a stepwise procedure to select a model for
the probability of a satellite when weight, spine condition, and color (nominal
scale) are the predictors. Explain each step of the process.
5.3 For the horseshoe crab data with width, color, and spine as predictors, suppose
you start a backward elimination process with the most complex model possible.
Denoted by C ∗ S ∗ W , it uses main effects for each term as well as the
three two-factor interactions and the three-factor interaction. Table 5.9 shows
the fit for this model and various simpler models.
a. Conduct a likelihood-ratio test comparing this model to the simpler model
that removes the three-factor interaction term but has all the two-factor
interactions. Does this suggest that the three-factor term can be removed
from the model?
b. At the next stage, if we were to drop one term, explain why we would select
model C ∗ S + C ∗ W .
c. For the model at this stage, comparing to the model S + C ∗ W results in
an increased deviance of 8.0 on df = 6(P = 0.24); comparing to the model
W + C ∗ S has an increased deviance of 3.9 on df = 3(P = 0.27). Which
term would you take out?
Table 5.9. Logistic Regression Models for Horseshoe
Crab Data
Model Predictors Deviance df AIC
1 C ∗ S ∗ W 170.44 152 212.4
2 C ∗ S + C ∗ W + S ∗ W 173.68 155 209.7
3a C ∗ S + S ∗ W 177.34 158 207.3
3b C ∗ W + S ∗ W 181.56 161 205.6
3c C ∗ S + C ∗ W 173.69 157 205.7
4a S + C ∗ W 181.64 163 201.6
4b W + C ∗ S 177.61 160 203.6
5 C + S + W 186.61 166 200.6