A class of generalized linear mixed models adjusted for marginal interpretability

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 (GLMM) 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 individual-level predictions for future observations. The second approach overcomes these deficiencies, but leads to parameter estimates that must be interpreted conditional on the latent variables. To obtain marginal summaries, one needs to evaluate an analytically intractable integral or use attenuation factors as an approximation. Further, we note an unpalatable implication of the standard GLMM. To resolve these issues, we turn to a class of marginally interpretable GLMMs that lead to parameter estimates with a marginal interpretation while maintaining the desirable statistical properties of a conditionally specified model and avoiding problematic implications. We establish the form of these models under the most commonly used link functions and address computational issues. For logistic mixed effects models, we introduce an accurate and efficient method for evaluating the logistic-normal integral.