Introduction to Categorical Data Analysis

10.2 EXAMPLES OF RANDOM EFFECTS MODELS FOR BINARY DATA 309
possible because the joint distribution of the responses is not specified. In addition,
this approach does not explicitly include random effects and therefore does not allow
these effects to be estimated.
The conditional modeling approach is preferable if one wants to fully model the
joint distribution. The marginal modeling approach focuses only on the marginal
distribution. The conditional modeling approach is also preferable if one wants to
estimate cluster-specific effects or estimate their variability, or if one wants to specify
a mechanism that could generate positive association among clustered observations.
For example, some methodologists use conditional models whenever the main focus
is on within-cluster effects. In the depression study (Section 10.2.6), the conditional
model is appropriate if we want the estimate of the time effect to be “within-subject,”
describing the time trend at the subject level.
By contrast, if the main focus is on comparing groups that are independent samples,
effects of interest are “between-cluster” rather than “within-cluster.” It may then be
adequate to estimate effects with a marginal model. For example, if after a period of
time we mainly want to compare the rates of depression for those taking the new drug
and for those taking the standard drug, a marginal model is adequate. In many surveys
or epidemiological studies, a goal is to compare the relative frequency of occurrence
of some outcome for different groups in a population. Then, quantities of primary
interest include between-group odds ratios comparing marginal probabilities for the
different groups.
When marginal effects are the main focus, it is simpler to model the margins
directly. One can then parameterize the model so regression parameters have a direct
marginal interpretation. Developing a more detailed model of the joint distribution
that generates those margins, as a random effects model does, provides greater opportunity
for misspecification. For instance, with longitudinal data the assumption that
observations are independent, given the random effect, need not be realistic.
Latent variable constructions used to motivate model forms (such as the probit and
cumulative logit) usually apply more naturally at the cluster level than the marginal
level. Given a conditional model, one can in principle recover information about
marginal distributions, although this may require extra work not readily done by
standard software. That is, a conditional model implies a marginal model, but a
marginal model does not itself imply a conditional model. In this sense, a conditional
model has more information.
We have seen that parameters describing effects are usually larger in conditional
models than marginal models, moreso as variance components increase. Usually,
though, the significance of an effect (e.g., as determined by the ratio of estimate
to SE) is similar for the two model types. If one effect seems more important than
another in a conditional model, the same is usually true with a marginal model. The
choice of the model is usually not crucial to inferential conclusions.
10.2.8 Conditional Models: Random Effects Versus Conditional ML
For the fixed effects approach with cluster-specific terms, a difficulty is that the
model has a large number of parameters. To estimate the other effects in the model,