Conjugate generalized linear mixed models for clustered data

TL;DR

This paper introduces a class of conjugate generalized linear mixed models for clustered data, providing conditions for closed-form marginal likelihoods, which enhances computational efficiency especially in big data applications.

Contribution

It derives necessary and sufficient conditions for closed-form marginal likelihoods in conjugate GLMMs, unifying various distributions under a single framework and improving computational convenience.

Findings

Closed-form marginal likelihoods derived for Gaussian, Poisson, binomial, and gamma distributions.

Models achieve conjugacy, allowing for efficient computation and inclusion of covariates.

Enhanced applicability in big data contexts due to explicit likelihood expressions.

Abstract

This article concerns a class of generalized linear mixed models for clustered data, where the random effects are mapped uniquely onto the grouping structure and are independent between groups. We derive necessary and sufficient conditions that enable the marginal likelihood of such class of models to be expressed in closed-form. Illustrations are provided using the Gaussian, Poisson, binomial and gamma distributions. These models are unified under a single umbrella of conjugate generalized linear mixed models, where "conjugate" refers to the fact that the marginal likelihood can be expressed in closed-form, rather than implying inference via the Bayesian paradigm. Having an explicit marginal likelihood means that these models are more computationally convenient, which can be important in big data contexts. Except for the binomial distribution, these models are able to achieve simultaneous conjugacy, and thus able to accommodate both unit and group level covariates.