Marginal maximum likelihood estimation of variance components in poisson mixed models using laplacian integration

Abstract

An algorithm for computing marginal maximum likelihood (MML) estimates of variance components in Poisson mixed models is presented. A Laplacian approximation was used to integrate fixed and random effects out of the joint posterior density of all parameters. This approximation was found to be identical to that invoked in the more commonly used expectation-maximization type algorithm for MML. Numerically, however, a different sequence of iterates was obtained, although the same variance component estimates should result. The Laplacian algorithm was precisely DFREML (derivative-free maximum likelihood) optimization when applied to normally distributed data, and could then be termed DFMML (derivative-free marginal maximum likelihood). Because DFMML is based on an approximation to the marginal likelihood of the variance components, it provides a mechanism for testing hypotheses about such components via posterior odds ratios or marginal likelihood ratio tests. Also, asymptotic posterior standard errors of the variance components can be computed with DFMML. A Tierney-Kadane procedure for computing the posterior mean of a variance component was also developed; however, it required 2 joint maximizations, and consequently might not be expected to perform well in many linear and non-linear mixed models. An example of a Poisson model is presented in which the null estimate commonly found when jointly estimating variance components with fixed and random effects was observed; thus, the Tierney-Kadane procedure for computing the posterior mean failed. On the other hand, the Laplacian method succeeded in locating the mode of the marginal distribution of the variance components in a Bayesian model with flat priors for fixed effects and variance components; i.e. the MML estimate.