  Two popular approaches for relating correlated measurements of a non-Gaussian
response variable to a set of predictors are to fit a marginal model using
generalized estimating equations and to fit a generalized linear mixed model by
introducing latent random variables. The first approach is effective for
parameter estimation, but leaves one without a formal model for the data with
which to assess quality of fit or make predictions for future observations. The
second approach overcomes the deficiencies of the first, but leads to parameter
estimates that must be interpreted conditional on the latent variables. Further
complicating matters, obtaining marginal summaries from a generalized linear
mixed model often requires evaluation of an analytically intractable integral
or use of attenuation factors that are not exact. We define a class of
marginally interpretable generalized linear mixed models that lead to parameter
estimates with a marginal interpretation while maintaining the desirable
statistical properties of a conditionally-specified model. We discuss the form
of these models under various common link functions and also address
computational issues associated with these models. For logistic mixed effects
models, we introduce an accurate and efficient method for evaluating the
logistic-normal integral.
