MARGINAL MAXIMUM-LIKELIHOOD-ESTIMATION OF VARIANCE-COMPONENTS IN POISSON MIXED MODELS USING LAPLACIAN INTEGRATION

An algorithm for computing marginal maximum likelihood (MML) estimates of variance components in Poisson mixed models is presented. A Laplacian approximation is used to integrate fixed and random effects out of the joint posterior density of all parameters. This approximation is 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 is obtained, although the same variance component estimates should result. The Laplacian algorithm is precisely DFREML (derivative free REML) 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 is also developed; however, it requires 2 joint maximizations, and consequently may 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 is 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 component in a Bayesian model with flat priors for fixed effects and variance components; that is, the MML estimate.