Marginally Interpretable Generalized Linear Mixed Models

TL;DR

This paper introduces a new class of generalized linear mixed models that provide marginally interpretable parameter estimates, combining the advantages of existing approaches while addressing their limitations, especially for logistic models.

Contribution

It defines marginally interpretable GLMMs, discusses their forms under various links, and presents an efficient method for logistic-normal integral evaluation.

Findings

Models achieve marginal interpretability with desirable statistical properties.

Proposed method improves accuracy and efficiency for logistic mixed effects models.

Addresses computational challenges in marginal inference for GLMMs.

Abstract

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.