Introduction to Categorical Data Analysis

300 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
(yi1 = 1,yi2 = 1), whereas with a large negative ui it is common to see outcomes
(yi1 = 0,yi2 = 0). When a high proportion of cases have these outcomes, the association
between the repeated responses is positive. Greater association results from
greater heterogeneity (i.e., larger σ ).
10.1.3 Example: Sacrifices for the Environment Revisited
Table 10.1 shows a 2 × 2 table from the General Social Survey, analyzed originally in
Chapter 8. Subjects were asked whether, to help the environment, they were willing
to (1) raise taxes, (2) accept a cut in living standards. The ML fit of model (10.3),
treating {ui} as normal, yields ˆβ = 0.210 (SE = 0.130), with ˆσ = 2.85. For a given
subject, the estimated odds of a “yes” response on accepting higher taxes equal
exp(0.210) = 1.23 times the odds of a “yes” response on accepting a lower standard
of living.
Table 10.1. Opinions Relating to Environment
Cut Living Standards
Pay Higher Taxes Yes No Total
Yes 227 132 359
No 107 678 785
Total 334 810 1144
The relatively large ˆσ value of 2.85 reflects a strong association between the two
responses. In fact, Table 10.1 has a sample odds ratio of 10.9. Whenever the sample
log odds ratio in such a table is nonnegative, as it usually is, the ML estimate of β with
this random effects approach is identical to the conditional ML estimate from treating
{αi} in model (10.2) as fixed effects. Section 8.2.3 presented this conditional ML
approach. For these data the conditional ML estimate is ˆβ = log(132/107) = 0.210,
with SE =[(1/107) + (1/132)] 1/2 = 0.130.
10.1.4 Differing Effects in Conditional Models and Marginal Models
As Sections 9.1.3 and 8.2.2 discussed, parameters in GLMMs and marginal models
have different interpretations. The parameters in GLMMs have conditional (clusterspecific)
intepretations, given the random effect. By contrast, effects in marginal
models are averaged over all clusters (i.e., population-averaged), and so those effects
do not refer to a comparison at a fixed value of a random effect.
Section 8.2.2 noted that the cluster-specific model (10.2) applies naturally to the
data as displayed in a separate partial table for each cluster, displaying the two matched
responses. For the survey data on the environmental issues, each subject is a cluster
and has their own table. The first row shows the response on taxes (a 1 in the first
column for “yes” or in the second column for “no”), and the second row shows the