Computational Methods for Single Level Linear Mixed-effects Models

Linear mixed-effects models are an important class of statistical models that are used directly in many fields of applications and are also used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood (ML) or restricted maximum likelihood (REML). In general there is no closed form solution for these estimates and they must be determined by iterative algorithms such as the EM algorithm or Fisher scoring or by general nonlinear optimizers. We recommend using a moderate number of EM iterations followed by general nonlinear optimization of a profiled log-likelihood. In this paper we present a method of calculating analytic gradients of the profiled loglikelihood. This gradient calculation can be implemented very efficiently using matrix decompositions as is done in the nlme packages for R and S-plus. Furthermore, the same type of calculation as is used to evaluate the gradient of the profiled log-likelihood can be used to implment an ECME algorithm.