mixedeffectsmGLMM

Mixed-effects generalized linear mixed models (mGLMM) are a class of statistical models that extend generalized linear models by incorporating both fixed effects and random effects to account for dependence among observations within clusters or hierarchical structures. They are appropriate for non-Gaussian response variables and can model various distributions from the exponential family through appropriate link functions, such as the logit for binary data or the log for counts.

Typically, the linear predictor is η_i = X_iβ + Z_i b, where β are fixed-effect coefficients and b are random effects assumed to follow a multivariate normal distribution with covariance D. Random effects capture clustering by units such as subjects, groups, sites, or repeated measures, enabling partial pooling and more accurate standard errors.

Estimation is generally done by maximum likelihood or restricted maximum likelihood, using approximations such as Laplace, adaptive Gaussian quadrature, or Bayesian methods via Markov chain Monte Carlo or INLA. Software packages include lme4, glmmTMB, brms, MCMCglmm, and INLA. Model selection and inference rely on likelihood ratio tests, Wald tests, and information criteria (AIC/BIC). Diagnostics address convergence, overdispersion, residual patterns, and the fit of the random-effects distribution.

Applications span ecology, psychology, medicine, and social sciences, where hierarchical data or repeated measurements are common. Limitations include computational complexity, potential convergence issues, sensitivity to distributional assumptions for random effects, and interpretational challenges for complex random-effects structures. Extensions cover crossed or multiple membership designs, zero-inflated or overdispersed responses, and alternative estimation frameworks.