Computation and application of generalized linear mixed model derivatives using lme4

TL;DR

This paper presents a method for efficiently computing derivatives of generalized linear mixed models using lme4, enabling improved statistical inference and model testing with practical R implementations.

Contribution

It introduces a quadrature-based approach for derivative computation in GLMMs with a single clustering variable, leveraging psychometric insights for accuracy verification.

Findings

Efficient derivative computation methods implemented in R packages.

Enhanced tools for robust standard errors and hypothesis testing.

Validation of derivatives using psychometric principles.

Abstract

Maximum likelihood estimation of generalized linear mixed models(GLMMs) is difficult due to marginalization of the random effects. Computing derivatives of a fitted GLMM's likelihood (with respect to model parameters) is also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we describe GLMM derivatives along with a quadrature method to efficiently compute them, focusing on lme4 models with a single clustering variable. We describe how psychometric results related to IRT are helpful for obtaining these derivatives, as well as for verifying the derivatives' accuracies. After describing the derivative computation methods, we illustrate the many possible uses of these derivatives, including robust standard errors, score tests of fixed effect parameters, and likelihood ratio tests of non-nested models. The derivative computation methods and applications described in the paper are all available in easily-obtained R packages.