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We develop hierarchical models and methods in a fully parametric approach to generalized linear mixed models for any patterned covariance matrix. The Laplace approximation is used to marginally estimate covariance parameters by integrating over 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. 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. We compare our methods to fully Bayesian methods, automatic differentiation, and integrated nested Laplace approximations (INLA) for bias, mean‐squared (prediction) error, and interval coverage, and all methods yield very similar results. However, our methods are much faster than Bayesian methods, and more general than INLA. Examples with binary and proportional data, count data, and positive‐continuous data are used to illustrate all six distributions with a variety of patterned covariance structures that include spatial models (both geostatistical and areal models), time series models, and mixtures with typical random intercepts based on grouping. [ABSTRACT FROM AUTHOR]

Quasi-Poisson and negative binomial regression models have equal numbers of parameters, and either could be used for overdispersed count data. While they often give similar results, there can be striking differences in estimating the effects of covariates. We explain when and why such differences occur. The variance of a quasi-Poisson model is a linear function of the mean while the variance of a negative binomial model is a quadratic function of the mean. These variance relationships affect the weights in the iteratively weighted least-squares algorithm of fitting models to data. Because the variance is a function of the mean, large and small counts get weighted differently in quasi-Poisson and negative binomial regression. We provide an example using harbor seal counts from aerial surveys. These counts are affected by date, time & day, and time relative to low tide. We present results on a data set that showed a dramatic difference on estimating abundance of harbor seals when using quasi-Poisson vs. negative binomial regression. This difference is described and explained in light of the different weighting used in each regression method. A general understanding of weighting can help ecologists choose between these two methods. Key words: covariates; generalized linear models; harbor seals; iteratively weighted least squares; overdispersion; quasi models.

Models for spatial autocorrelation and cross-correlation depend on the distance and direction separating two locations, and are constrained so that for all possible sets of locations, the covariance matrices implied [...]

Spatial confounding, that is, collinearity between fixed effects and random effects in a spatial generalized linear mixed model, can adversely affect estimates of the fixed effects. Restricted spatial regression methods have been proposed as a remedy for spatial confounding. Such methods replace inference for the fixed effects of the original model with inference for those effects under a model in which the random effects are restricted to a subspace orthogonal to the column space of the fixed effects model matrix; thus, they "deconfound" the two types of effects. We prove, however, that frequentist inference for the fixed effects of a deconfounded linear model is generally inferior to that for the fixed effects of the original spatial linear model; in fact, it is even inferior to inference for the corresponding nonspatial model. We show further that deconfounding also leads to inferior predictive inferences, though its impact on prediction appears to be relatively small in practice. Based on these results, we argue that deconfounding a spatial linear model is bad statistical practice and should be avoided. [ABSTRACT FROM AUTHOR]

The population of harbor seals (Phoca vitulina vitulina) along the coast of Maine, U.S.A., has experienced rapid growth in abundance following passage of the Marine Mammal Protection Act in 1972 but current information on trends in abundance is lacking. In this study, we apply a Bayesian hierarchical model to aerial survey data of nonpups and pups. Prior to 2001, estimates of growth rates from 8‐year moving averages reached a high of 2.1% and 9.4% per year with posterior probabilities of positive growth of.97 and >.99 for nonpups and pups, respectively. Between 2001 and 2012, estimated growth rates for nonpups decreased to a low of −1.9% per year with a posterior probability of negative growth of.95. Between 2012 and 2018, posterior estimates of growth were close to zero suggesting little change in abundance of nonpups. Estimates of growth for pups were close to zero between 2001 and 2012 but reached a low of −2.5% per year with a posterior probability of negative growth of 0.94 at the end of the time series suggesting a decrease in pup abundance between 2012 and 2018. The total abundance estimate for 2018 is 61,336 (CV = 0.08). [ABSTRACT FROM AUTHOR]