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

Jay M. Ver Hoef; Eryn Blagg; Michael Dumelle; Philip M Dixon; Dale L Zimmerman; Paul Conn

doi:10.48550/arxiv.2305.02978

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

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

Jay M. Ver Hoef, Eryn Blagg, Michael Dumelle, Philip M Dixon, Dale L Zimmerman and Paul Conn

ArXiv.org

05/04/2023

DOI: 10.48550/arxiv.2305.02978

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https://doi.org/10.48550/arXiv.2305.02978View

Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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.

Details

Title: Subtitle: Marginal Inference for Hierarchical Generalized Linear Mixed Models with Patterned Covariance Matrices Using the Laplace Approximation
Creators: Jay M. Ver Hoef
Eryn Blagg
Michael Dumelle
Philip M Dixon
Dale L Zimmerman
Paul Conn
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.2305.02978
ISSN: 2331-8422
Language: English
Date posted: 05/04/2023
Academic Unit: Statistics and Actuarial Science; Biostatistics
Record Identifier: 9984406604002771

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