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A method is introduced for approximate marginal likelihood inference viaadaptive Gaussian quadrature in mixed models with a single grouping factor. Thecore technical contribution is an algorithm for computing the exact gradient ofthe approximate log marginal likelihood. This leads to efficient maximumlikelihood via quasi-Newton optimization that is demonstrated to be faster thanexisting approaches based on finite-differenced gradients or derivative-freeoptimization. The method is specialized to Bernoulli mixed models withmultivariate, correlated Gaussian random effects; here computations areperformed using an inverse log-Cholesky parameterization of the Gaussiandensity that involves no matrix decomposition during model fitting, while Waldconfidence intervals are provided for variance parameters on the originalscale. Simulations give evidence of these intervals attaining nominal coverageif enough quadrature points are used, for data comprised of a large number ofvery small groups exhibiting large between-group heterogeneity. In contrast,the Laplace approximation is shown to give especially poor coverage and highbias for data comprised of a large number of small groups. Adaptive quadraturemitigates this, and the methods in this paper improve the computationalfeasibility of this more accurate method. All results may be reproduced usingcode available at \url{https://github.com/awstringer1/aghmm-paper-code}.

Approximate Marginal Likelihood Inference in Mixed Models for Grouped Data