Introduction to Categorical Data Analysis

PROBLEMS 321
10.10 For the previous exercise, compare estimates of βB − βA and βC − βA
and their SE values to those using the corresponding marginal model of
Problem 9.6.
10.11 Refer to Table 5.5 on admissions decisions for Florida graduate school applicants.
For a subject in department i of gender g (1 = females, 0 = males),
let yig = 1 denote being admitted.
a. For the fixed effects model, logit[P(Yig = 1)] =α + βg + β D i , the
estimated gender effect is ˆβ = 0.173 (SE = 0.112). Interpret.
b. The corresponding model (10.7) that treats departments as a normal
random effect has ˆβ = 0.163 (SE = 0.111). Interpret.
c. The model of form (10.8) that allows the gender effect to vary randomly
by department has ˆβ = 0.176 (SE = 0.132), with ˆσv = 0.20. Interpret.
Explain why the standard error of ˆβ is slightly larger than with the other
analyses.
d. The marginal sample log odds ratio between gender and whether admitted
equals −0.07. How could this take different sign from ˆβ in these models?
10.12 Consider Table 8.14 on premarital and extramarital sex. Table 10.11 shows
the results of fitting a cumulative logit model with a random intercept.
a. Interpret ˆβ.
b. What does the relatively large ˆσ value suggest?
Table 10.11. Computer Output for Problem 10.12
Subjects 475 Parameter Estimate
Std
Error t Value
Max Obs Per Subject 2 inter1 −1.5422 0.1826 −8.45
Parameters 5 inter2 −0.6682 0.1578 −4.24
Quadrature Points 100 inter3 0.9273 0.1673 5.54
Log Likelihood −890.1 beta 4.1342 0.3296 12.54
sigma 2.0757 0.2487 8.35
10.13 Refer to the previous exercise. Analyze these data with a corresponding
cumulative logit marginal model.
a. Interpret ˆβ.
b. Compare ˆβ to its value in the GLMM. Why are they so different?
10.14 Refer to Problem 9.11 for Table 7.25 on government spending.
a. Analyze these data using a cumulative logit model with random effects.
Interpret.
b. Compare the results with those with a marginal model (Problem 9.11).