Marginal Inference for Hierarchical Generalized Linear Mixed Models with Patterned Covariance Matrices Using the Laplace Approximation
Using a hierarchical construction, we develop methods for a wide and flexible class of modelsby taking a fully parametric approach to generalized linear mixed models with complex covariancedependence. The Laplace approximation is used to marginally estimate covarianceparameters while integrating out all fixed and latent random effects. The Laplace approximationrelies on Newton-Raphson updates, which also leads to predictions for the latentrandom effects. We develop methodology for complete marginal inference, from estimatingcovariance parameters and fixed effects to making predictions for unobserved data, for anypatterned 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 knownparameters, and their efficacy in terms of bias and interval coverage is shown through simulationexperiments. Examples with binary and proportional data on election results, countdata for marine mammals, and positive-continuous data on heavy metal concentration inthe environment are used to illustrate all six distributions with a variety of patterned covariancestructures that include spatial models (e.g., geostatistical and areal models), time seriesmodels (e.g., first-order autoregressive models), and mixtures with typical random interceptsbased on grouping.