Computation and application of generalized linear mixed model derivatives using lme4

Maximum likelihood estimation of generalized linear mixed models (GLMMs) is difficult due to marginalization of the random effects. Derivative computations of a fitted GLMM's likelihood are also difficult, especially because the derivatives are not by-products of popular estimation algorithms. In this paper, we first describe theoretical results related to GLMM derivatives along with a quadrature method to efficiently compute the derivatives, focusing on fitted lme4 models with a single clustering variable. We describe how psychometric results related to item response models are helpful for obtaining the derivatives, as well as for verifying the derivatives' accuracies. We then provide a tutorial on 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.