Computationally efficient Monte Carlo EM algorithms for generalized linear mixed models

Abstract

Maximum likelihood estimation in generalized linear mixed models usually involves intractable integrals that may be of high dimension. To reduce the dimensions of the integrals involved in computation, a reduced form of the score equation obtained by exploiting conditional independence and random effects structures can be used. Two Monte Carlo methods, one based on direct Monte Carlo integration and the other based on a stochastic version of the Gauss–Hermite quadrature, are proposed for approximating the integrals involved in the reduced score equation. The proposed Monte Carlo EM algorithms need to simulate random effects only in the first iteration; hence, the computation burden can be reduced and the likelihood that the increasing property of the original EM algorithm can be preserved. A reduced form of the information matrix is proposed for standard error estimation. We illustrate the proposed methods with an application to the salamander data and examine their performance via simulation studies.