Introduction to Categorical Data Analysis

308 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS
(1 = new, 0 = standard), and time of observation t, we used the model
logit[P(Yt = 1)] =α + β1s + β2d + β3t + β4(d × t)
to evaluate effects on the marginal distributions.
Now let yit denote observation t for subject i. The model
logit[P(Yit = 1)] =ui + α + β1s + β2d + β3t + β4(d × t)
has subject-specific rather than population-averaged effects. Table 10.6 shows the
ML estimates. The time trend estimates are ˆβ3 = 0.48 for the standard drug and
ˆβ3 + ˆβ4 = 1.50 for the new one. These are nearly identical to the GEE estimates
for the corresponding marginal model, which Table 10.6 also shows. (Sections 9.1.2
and 9.2.3 discussed these.) The reason is that the repeated observations are only
weakly correlated, as the GEE analysis observed. Here, this is reflected by
ˆσ = 0.07, which suggests little heterogeneity among subjects in their response
probabilities.
Table 10.6. Model Parameter Estimates for Marginal and Conditional
Models Fitted to Table 9.1 on Depression Longitudinal Study
GEE Marginal Random Effects
Parameter Estimate SE ML Estimate SE
Diagnosis −1.31 0.15 −1.32 0.15
Treatment −0.06 0.23 −0.06 0.22
Time 0.48 0.12 0.48 0.12
Treat × time 1.02 0.19 1.02 0.19
When we assume σ = 0 in this model, the log-likelihood decreases by less than
0.001. For this special case of the model, the ML estimates and SE values are the
same as if we used ordinary logistic regression without the random effect and ignored
the clustering (e.g., acting as if each observation comes from a different subject).
10.2.7 Choosing Marginal or Conditional Models
Some statisticians prefer conditional models (usually with random effects) over
marginal models, because they more fully describe the structure of the data. However,
many statisticians believe that both model types are useful, depending on the
application. We finish the section by considering issues in choosing one type over the
other.
With the marginal model approach, ML is sometimes possible but the GEE
approach is computationally simpler and more readily available with standard software.
A drawback of the GEE approach is that likelihood-based inferences are not