Introduction to Categorical Data Analysis

298 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
treatments. By contrast, random effects apply to a sample. For a study with repeated
measurement of subjects, for example, a cluster is a set of observations for a particular
subject, and the model contains a random effect term for each subject. The random
effects refer to a sample of clusters from all the possible clusters.
10.1.1 The Generalized Linear Mixed Model
Generalized linear models (GLMs) extend ordinary regression by allowing nonnormal
responses and a link function of the mean (recall Chapter 3). The generalized linear
mixed model, denoted by GLMM, is a further extension that permits random effects
as well as fixed effects in the linear predictor. Denote the random effect for cluster i
by ui. We begin with the most common case, in which ui is an intercept term in the
model.
Let yit denote observation t in cluster i. Let xit be the value of the explanatory
variable for that observation. (The model extends in an obvious way for multiple
predictors.) Conditional on ui, a GLMM resembles an ordinary GLM. Let μit =
E(Yit|ui), the mean of the response variable for a given value of the random effect.
With the link function g(·), the GLMM has the form
g(μit) = ui + βxit, i = 1,...,n, t = 1,...,T
A GLMM with random effect as an intercept term is called a random intercept model.
In practice, the random effect ui is unknown, like the usual intercept parameter. It is
treated as a random variable and is assumed to have a normal N(α, σ ) distribution,
with unknown parameters. The variance σ 2 is referred to as a variance component.
When xit = 0 in this model, the expected value of the linear predictor is α, the
mean of the probability distribution of ui. An equivalent model enters α explicitly in
the linear predictor,
g(μit) = ui + α + βxit
(10.1)
compensating by taking ui to have an expected value of 0. The ML estimate of α
is identical either way. However, it would be redundant to have both the α term in
the model and permit an unspecified mean for the distribution of ui. We will specify
models this second way, taking ui to have a N(0,σ) distribution. The separate α
parameter in the model then is the value of the linear predictor when xit = 0 and ui
takes value at its mean of 0.
Why not treat the cluster-specific {ui} terms as fixed effects (parameters)? Usually
a study has a large number of clusters, and so the model would then contain a
large number of parameters. Treating {ui} as random effects, we have only a single
additional parameter (σ ) in the model, describing their dispersion.
Section 10.5 outlines the model-fitting process for GLMMs. As in ordinary models
for a univariate response, for given predictor values ML fitting treats the observations
as independent. For the GLMM, this independence is assumed conditional on the