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We describe how to approximate the intractable marginal likelihood thatarises when fitting generalized linear mixed models. We prove that non-adaptivequadrature approximations yield high error asymptotically in every statisticalmodel satisfying weak regularity conditions. We derive the rate of errorincurred when using adaptive quadrature to approximate the marginal likelihoodin a broad class of generalized linear mixed models, which includesnon-exponential family response and non-Gaussian random effects distributions.We provide an explicit recommendation for how many quadrature points to use,and show that this recommendation recovers and explains many empirical resultsfrom published simulation studies and data analyses. Particular attention ispaid to models for dependent binary and survival/time-to-event observations.Code to reproduce results in the manuscript is found athttps://github.com/awstringer1/glmm-aq-paper-code.

Fitting Generalized Linear Mixed Models using Adaptive Quadrature