Introduction to Categorical Data Analysis

9.2 MARGINAL MODELING: THE GEE APPROACH 279
This model assumes that the linear time effect β3 is the same for each group. The
sample proportions in Table 9.2, however, show a higher rate of improvement for the
new drug. A more realistic model permits the time effect to differ by drug. We do this
by including a drug-by-time interaction term,
logit[P(Yt = 1)] =α + β1s + β2d + β3t + β4(d × t)
Here, β3 describes the time effect for the standard drug (d = 0) and β3 + β4 describes
the time effect for the new drug (d = 1).
We will fit this model, interpret the estimates, and make inferences in Section 9.2.
We will see that an estimated slope (on the logit scale) for the standard drug is
ˆβ3 = 0.48. For the new drug the estimated slope increases by ˆβ4 = 1.02, yielding an
estimated slope of ˆβ3 + ˆβ4 = 1.50.
9.1.3 Conditional Models for a Repeated Response
The models just considered describe how P(Yt = 1), the probability of a normal
response at time t, depends on severity, drug, and time, for a randomly selected subject.
By contrast, for matched pairs Section 8.2.3 presented a different type of model
that describes probabilities at the subject level. That model permits heterogeneity
among subjects, even at fixed levels of the explanatory variables.
Let Yit denote the response for subject i at time t. For the depression data, a
subject-specific analog of the model just considered is
logit[P(Yit = 1)] =αi + β1s + β2d + β3t + β4(d × t)
Each subject has their own intercept (αi), reflecting variability in the probability
among subjects at a particular setting (s,d,t)for the explanatory variables.
This is called a conditional model, because the effects are defined conditional on
the subject. For example, the model identifies β3 as the time effect for a given subject
using the standard drug. The effect is subject-specific, because it is defined at the
subject level. By contrast, the effects in the marginal models specified in the previous
subsection are population-averaged, because they refer to averaging over the entire
population rather than to individual subjects.
The remainder of this chapter focuses only on marginal models. The following
chapter presents conditional models and also discusses issues relating to the choice
of model.
9.2 MARGINAL MODELING: THE GENERALIZED
ESTIMATING EQUATIONS (GEE) APPROACH
ML fitting of marginal logit models is difficult. We will not explore the technical
reasons here, but basically, it is because the models refer to marginal probabilities
whereas the likelihood function refers to the joint distribution of the clustered