Introduction to Categorical Data Analysis

CHAPTER 10
Random Effects: Generalized
Linear Mixed Models
Chapter 9 focused on modeling the marginal distributions of clustered responses. This
chapter presents an alternative model type that has a term in the model for each cluster.
The cluster-specific term takes the same value for each observation in a cluster. This
term is treated as varying randomly among clusters. It is called a random effect.
Section 8.2.3 introduced cluster-specific terms in a model for matched pairs. Such
models have conditional interpretations, the effects being conditional on the cluster.
The effects are called cluster-specific, orsubject-specific when each cluster is
a subject. This contrasts with marginal models, which have population-averaged
interpretations in which effects are averaged over the clusters.
The generalized linear mixed model, introduced in Section 10.1, extends generalized
linear models to include random effects. Section 10.2 shows examples of logistic
regression models with random effects and discusses connections and comparisons
with marginal models. Section 10.3 shows examples of multinomial models and
models with multiple random effect terms. Section 10.4 introduces multilevel models
having random effects at different levels of a hierarchy. For example, an educational
study could include a random effect for each student as well as for each school that
the students attend. Section 10.5 summarizes model fitting methods. The appendix
shows how to use software to do the analyses in this chapter.
10.1 RANDOM EFFECTS MODELING OF CLUSTERED
CATEGORICAL DATA
Parameters that describe a factor’s effects in ordinary linear models are called fixed
effects. They apply to all categories of interest, such as genders, age groupings, or
An Introduction to Categorical Data Analysis, Second Edition. By Alan Agresti
Copyright © 2007 John Wiley & Sons, Inc.
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