Exact Gradient Evaluation for Adaptive Quadrature Approximate Marginal Likelihood in Mixed Models for Grouped Data

TL;DR

This paper presents an efficient algorithm for exact gradient computation in adaptive quadrature for mixed models, improving maximum likelihood estimation speed and accuracy for grouped data with small groups.

Contribution

It introduces an exact gradient algorithm for adaptive Gaussian quadrature in mixed models, enhancing computational efficiency and accuracy over existing finite-difference methods.

Findings

Exact gradient computation accelerates maximum likelihood optimization.

Adaptive quadrature improves variance parameter coverage in small-group data.

Method is specialized for Bernoulli mixed models with correlated Gaussian effects.

Abstract

A method is introduced for approximate marginal likelihood inference via adaptive Gaussian quadrature in mixed models with a single grouping factor. The core technical contribution is an algorithm for computing the exact gradient of the approximate log-marginal likelihood. This leads to efficient maximum likelihood via quasi-Newton optimization that is demonstrated to be faster than existing approaches based on finite-differenced gradients or derivative-free optimization. The method is specialized to Bernoulli mixed models with multivariate, correlated Gaussian random effects; here computations are performed using an inverse log-Cholesky parameterization of the Gaussian density that involves no matrix decomposition during model fitting, while Wald confidence intervals are provided for variance parameters on the original scale. Simulations give evidence of these intervals attaining nominal coverage if enough quadrature points are used, for data comprised of a large number of very small groups exhibiting large between-group heterogeneity. The Laplace approximation is well-known to give especially poor coverage and high bias for data comprised of a large number of small groups. Adaptive quadrature mitigates this, and the methods in this paper improve the computational feasibility of this more accurate method. All results may be reproduced using code available at \url{https://github.com/awstringer1/aghmm-paper-code}.