Using Spherical-Radial Quadrature to Fit Generalized Linear Mixed Effects Models

Although generalized linear mixed effects models have received much attention in the statistical literature, there is still no computationally efficient algorithm for computing maximum likelihood estimates for such models when there are a moderate number of random effects. Existing algorithms are either computationally intensive or they compute estimates from an approximate likelihood. Here we propose an algorithm—the spherical–radial algorithm—that is computationally efficient and computes maximum likelihood estimates. Although we concentrate on two-level, generalized linear mixed effects models, the same algorithm can be applied to many other models as well, including nonlinear mixed effects models and frailty models. The computational difficulty for estimation in these models is in integrating the joint distribution of the data and the random effects to obtain the marginal distribution of the data. Our algorithm uses a multidimensional quadrature rule developed in earlier literature to integrate the jo...