  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.
