Introduction to Categorical Data Analysis

94 GENERALIZED LINEAR MODELS
Table 3.7. Table for Problem 3.10 on Cancer Remission
Standard Likelihood Ratio 95% Chi-
Parameter Estimate Error Confidence Limits Square Pr > ChiSq
Intercept −2.3178 0.7795 −4.0114 −0.9084 8.84 0.0029
LI 0.0878 0.0328 0.0275 0.1575 7.19 0.0073
as independent Poisson variates having means μA and μB. Consider the model
log μ = α + βx, where x = 1 for treatment B and x = 0 for treatment A.
a. Show that β = log μB − log μA = log(μB/μA) and e β = μb/μA.
b. Fit the model. Report the prediction equation and interpret ˆβ.
c. Test H0: μA = μB by conducting the Wald or likelihood-ratio test of
H0: β = 0. Interpret.
d. Construct a 95% confidence interval for μB/μA. [Hint: Construct one for
β = log(μB/μA) and then exponentiate.]
3.12 Refer to Problem 3.11. The wafers are also classified by thickness of silicon
coating (z = 0, low; z = 1, high). The first five imperfection counts reported
for each treatment refer to z = 0 and the last five refer to z = 1. Analyze these
data, making inferences about the effects of treatment type and of thickness of
coating.
3.13 Access the horseshoe crab data of Table 3.2 at www.stat.ufl.edu/∼aa/introcda/appendix.html.
a. Using x = weight and Y = number of satellites, fit a Poisson loglinear
model. Report the prediction equation.
b. Estimate the mean of Y for female crabs of average weight 2.44 kg.
c. Use ˆβ to describe the weight effect. Construct a 95% confidence interval
for β and for the multiplicative effect of a 1 kg increase.
d. Conduct a Wald test of the hypothesis that the mean of Y is independent of
weight. Interpret.
e. Conduct a likelihood-ratio test about the weight effect. Interpret.
3.14 Refer to the previous exercise. Allow overdispersion by fitting the negative
binomial loglinear model.
a. Report the prediction equation and the estimate of the dispersion parameter
and its SE. Is there evidence that this model gives a better fit than the
Poisson model?
b. Construct a 95% confidence interval for β. Compare it with the one in (c)
in the previous exercise. Interpret, and explain why the interval is wider
with the negative binomial model.