Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear mixed models

Mixed-effects models have become a popular approach for the analysis of grouped data that arise in many areas as diverse as clinical trials, epidemiology, and sociology. Examples of grouped data include longitudinal data, repeated measures, and multilevel data. In the case of linear mixed-effects (LME) models, the likelihood function can be expressed in closed form, with efficient computational algorithms having been proposed for maximum likelihood and restricted maximum likelihood estimation. For nonlinear mixed-effects (NLME) models and generalized linear mixed models (GLMMs), however, the likelihood function does not have a closed form. Different likelihood approximations, with varying degrees of accuracy and computational complexity, have been proposed for these models. This article describes algorithms for one such approximation, the adaptive Gaussian quadrature (AGQ), for GLMMs which scale up efficiently to multilevel models with arbitrary number of levels. The proposed algorithms greatly reduce the computational complexity and the memory usage for approximating the multilevel GLMM likelihood, when compared to a direct application of a single-level AGQ approximation algorithm to the multilevel case. The accuracy of the associated estimates is evaluated and compared to that of estimates obtained from other approximations via simulation studies.