Marginal Inference for Hierarchical Generalized Linear Mixed Models with Patterned Covariance Matrices Using the Laplace Approximation

TL;DR

This paper introduces a flexible hierarchical generalized linear mixed model framework using Laplace approximation for marginal inference of covariance parameters, fixed effects, and predictions across various data types and covariance structures.

Contribution

It develops a comprehensive methodology for marginal inference in hierarchical GLMMs with patterned covariance matrices, extending to six distributions and various real-world applications.

Findings

Accurate covariance parameter estimation demonstrated in simulations.

Effective predictions for unobserved data shown in case studies.

Framework adaptable to diverse data types and covariance structures.

Abstract

Using a hierarchical construction, we develop methods for a wide and flexible class of models by taking a fully parametric approach to generalized linear mixed models with complex covariance dependence. The Laplace approximation is used to marginally estimate covariance parameters while integrating out all fixed and latent random effects. The Laplace approximation relies on Newton-Raphson updates, which also leads to predictions for the latent random effects. We develop methodology for complete marginal inference, from estimating covariance parameters and fixed effects to making predictions for unobserved data, for any patterned covariance matrix in the hierarchical generalized linear mixed models framework. The marginal likelihood is developed for six distributions that are often used for binary, count, and positive continuous data, and our framework is easily extended to other distributions. The methods are illustrated with simulations from stochastic processes with known parameters, and their efficacy in terms of bias and interval coverage is shown through simulation experiments. Examples with binary and proportional data on election results, count data for marine mammals, and positive-continuous data on heavy metal concentration in the environment are used to illustrate all six distributions with a variety of patterned covariance structures that include spatial models (e.g., geostatistical and areal models), time series models (e.g., first-order autoregressive models), and mixtures with typical random intercepts based on grouping.