Introduction to Categorical Data Analysis

310 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
the conditional ML approach removes the cluster-specific terms from the model.
Section 8.2.3 introduced the conditional ML approach for binary matched pairs.
Compared with the random effects approach, it has the advantage that it does not
assume a parametric distribution for the cluster-specific terms.
However, the conditional ML approach has limitations and disadvantages. It is
restricted to inference about within-cluster fixed effects. The conditioning removes
the source of variability needed for estimating between-cluster effects. This approach
does not provide information about cluster-specific terms, such as predictions of their
values and estimates of their variability or of probabilities they determine. When
the number of observations per cluster is large, it is computationally difficult to
implement. Finally, conditional ML can be less efficient than the random effects
approach for estimating the other fixed effects.
One application in which conditional ML with cluster-specific terms in logistic
regression models has been popular is case–control studies. A case and the matching
control or controls form a cluster. Section 8.2.4 discussed this for the matchedpairs
case. For further details, see Breslow and Day (1980), Fleiss et al. (2003,
Chapter 14), and Hosmer and Lemeshow (2000, Chapter 7).
10.3 EXTENSIONS TO MULTINOMIAL RESPONSES OR
MULTIPLE RANDOM EFFECT TERMS
GLMMs extend directly from binary outcomes to multiple-category outcomes. Modeling
is simpler with ordinal responses, because it is often adequate to use the same
random effect term for each logit. With cumulative logits, this is the proportional
odds structure that Section 6.2.1 used for fixed effects. However, GLMMs can have
more than one random effect term in a model. Most commonly this is done to allow
random slopes as well as random intercepts. We next show examples of these two
cases.
10.3.1 Example: Insomnia Study Revisited
Table 9.6 in Section 9.3.2 showed results of a clinical trial at two occasions comparing
a drug with placebo in treating insomnia patients. The response, time to fall asleep,
fell in one of four ordered categories. We analyzed the data with marginal models in
Section 9.3.2 and with transitional models in Section 9.4.3.
Let yt = time to fall asleep at occasion t (0 = initial, 1 = follow-up), and let
x = treatment (1 = active, 0 = placebo). The marginal model
logit[P(Yt ≤ j)]=αj + β1t + β2x + β3(t × x)
permits interaction. Table 10.7 shows GEE estimates.