GlmM

GLMM stands for generalized linear mixed model. It extends generalized linear models to handle non-Gaussian responses and to account for correlation among observations by including random effects. It is commonly used for hierarchical data and repeated measures.

Formally, Yi follows a distribution from the exponential family with mean μi. A link function g relates μi to ηi = X_iβ + Z_i b, where β are fixed effects and b are random effects with b ~ N(0, D). Z_i and X_i specify the design; random effects may be intercepts, slopes, or nested/crossed structures.

Estimation integrates over the random effects to obtain a marginal likelihood, typically using Laplace or adaptive Gaussian quadrature. Pseudo-likelihood methods exist but can bias results in some settings. Bayesian approaches using MCMC are also common.

GLMMs are used in medicine, ecology, psychology, and social sciences to analyze binary, count, and other non-normal outcomes, handle overdispersion, and model clustered data. Assumptions include correct distribution and link, and normality of random effects.

Software options include lme4 (R) with glmer, glmer.nb; glmmTMB; SAS PROC GLIMMIX; and Stata commands, all supporting multiple distributions and random-effects structures.