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.

翻译：利用分层构造，我们通过完全参数化方法处理广义线性混合模型中具有复杂协方差依赖的模型，从而发展出一类广泛且灵活的建模方法。采用拉普拉斯逼近对协方差参数进行边际估计，同时积分掉所有固定效应和潜在随机效应。拉普拉斯逼近依赖牛顿-拉夫森迭代更新，这一过程同时可得到潜在随机效应的预测值。我们针对分层广义线性混合模型框架中的任意模式化协方差矩阵，建立了完整的边际推断方法论，涵盖协方差参数与固定效应的估计以及未观测数据的预测。针对常用于二分类、计数和正连续数据的六种分布，发展了边际似然函数，且该框架易于扩展至其他分布类型。通过已知参数的随机过程模拟验证方法有效性，并利用模拟实验展示其偏差特性和区间覆盖能力。基于选举结果的二分类和比例数据、海洋哺乳动物计数数据、以及环境中重金属浓度的正连续数据等实例，演示了包含空间模型（如地统计模型和区域模型）、时间序列模型（如一阶自回归模型）以及基于分组的典型随机截距混合模型在内的多种模式化协方差结构下的六种分布应用。