Maximum Likelihood Algorithm for Spatial Generalized Linear Mixed Models without Numerical Evaluations of Intractable Integrals

Abstract

Spatial generalized linear mixed effects models are popular in spatial or spatiotemporal data analysis when the responses are counts and the random effects are modeled by multivariate normal distributions. Direct computation of the MLEs of model parameters is impossible because the likelihood functions contain high-dimensional intractable integrals. To overcome the difficulty, a new method called the prediction-maximization algorithm is proposed. The method has a maximization step for the MLEs of spatial linear mixed effects models for normal responses and a prediction step for the prediction of the random effects. None of them involves high-dimensional intractable integrals. Because only algorithms for the normal responses are needed, the derivation of the MLEs of a spatial generalized linear mixed effects model for count responses by the proposed method is not computationally harder than a model for normal responses. The simulation study shows that the performance of the proposed method is comparable to that of the previous maximum likelihood algorithms formulated by high-order Laplace approximations and is better than that of Bayesian methods formulated by MCMC algorithms. Supplementary materials for this article are available online.

Keywords:

• Exponential family distribution
• High-dimensional intractable integrals
• Laplace approximation
• Penalized quasi-likelihood
• Prediction-maximization algorithm
• Profile likelihood

Supplementary Materials

The online supplementary materials contain the standard MLEs for spatial linear mixed models (SLMMs) for normal data, and the prediction of random effects based on the SLMMs, the formulations of the Fisher information matrix for the MLEs of the SLMMs, and the proofs of all of the lemmas, theorems, and corollaries.

Acknowledgments

The author appreciates great comments from the Associate Editor and two anonymous referees which significantly improve the quality of the article.