Fitting Generalized Linear Mixed-Effects Models using lme4

2026-07-13

AI Summary

This work proposes an efficient fitting approach for generalized linear mixed-effects models in which the response variable follows a non-Gaussian distribution and its mean is nonlinearly related to the linear predictor through a link function. Built within the lme4 framework, the method jointly estimates fixed effects, conditional modes of random effects, and their variance–covariance structure via penalized iteratively reweighted least squares. High-accuracy approximations to the likelihood are achieved using either Laplace approximation or adaptive Gauss–Hermite quadrature. The approach accommodates user-specified exponential family distributions and link functions, supporting common non-Gaussian responses such as binomial and Poisson outcomes. By balancing computational efficiency with modeling flexibility, the proposed method substantially extends the applicability of generalized linear mixed models while maintaining robust performance.

Abstract

The lme4 R package can be used to fit generalized linear mixed models (GLMMs), which extend the class of linear mixed models (LMMs). The two main extensions provided by GLMMs are (1) allowing for the conditional distribution of the response given the random effects to be non-Gaussian (e.g. binomial, Poisson) and (2) allowing the conditional mean to be a nonlinear function of a linear combination of the fixed and random effect coefficients, via an inverse link function. The conditional mode of the random effects given the observed data, the variance-covariance matrix of the random effects, and the fixed effect parameters are determined using penalized iteratively reweighted least squares. We compute an approximation of the integral over the distributions of the conditional modes to compute the maximum likelihood estimate for a given set of parameters (by default we use the Laplace approximation or, alternatively, the more computationally expensive adaptive Gauss-Hermite quadrature). The package provides all the standard features available for GLMs in base R, including the standard set of accessor functions as well as the possibility of user-specified distributions (within the exponential dispersion family) and link functions.

Problem

• Generalized Linear Mixed Models
• Non-Gaussian responses
• Link functions
• Random effects
• Maximum likelihood estimation

Innovation

• Generalized Linear Mixed Models
• Penalized Iteratively Reweighted Least Squares
• Laplace Approximation
• Adaptive Gauss-Hermite Quadrature
• Exponential Dispersion Family