HGLMs

Hierarchical generalized linear models (HGLMs) are a class of statistical models that extend generalized linear models to accommodate multilevel or clustered data and non-Gaussian response distributions by incorporating random effects. HGLMs allow observations within the same group to be correlated and to share information through group-specific random effects, while still modeling the mean structure with a link function.

Formally, for observation j in group i, the conditional mean mu_ij = E[y_ij | b_i] is related to covariates via g(mu_ij) = X_ij beta + Z_ij b_i, where b_i is a vector of random effects for group i, usually assumed to be multivariate normal with mean zero and covariance D. The conditional distribution of y_ij given b_i belongs to an exponential family (for example, normal, binomial, or Poisson distributions).

Estimation typically proceeds by maximizing an approximate marginal likelihood, obtained by integrating over the random effects. Common approaches include penalized quasi-likelihood, Laplace approximation, and Gauss-Hermite or adaptive quadrature; Bayesian inference via MCMC is also used. Software implementations include the R package hglm, which fits hierarchical generalized linear models, as well as broader GLMM frameworks such as gllamm in Stata or lme4 for simpler cases; however, HGLMs emphasize explicit hierarchical structure and random effects at one or more levels.

Applications span education, epidemiology, ecology, and social sciences, wherever data are nested or clustered and responses are non-normal or overdispersed. Compared with standard GLMs, HGLMs provide a way to partition variance, model intra-group correlation, and obtain subject/group-specific predictions.

Limitations include computational complexity, potential bias with small clusters when using some approximation methods, and sensitivity to distributional assumptions for random effects. Model selection and diagnostics often require careful consideration of random-effects structure and checking of residuals.