Abstract

Linear mixed effects models arise quite naturally in a number of settings. Two of the more prominent uses are in experimental designs and multilevel models. Furthermore, Bayesian analysis has also been utilized with respect to such models. Here we will consider such an approach with emphasis placed on estimation of the covariance matrix for the random effects. With respect to the covariance structure, however, we depart from the traditional Bayesian prior usage of the Inverse Wishart distribution. The rationale for such a departure is that this distribution is somewhat constraining. Instead, we employ a multivariate Normal approximation procedure for the likelihood of the matrix logarithm of the random effects covariance matrix. Such an approximation allows us to use a multivariate Normal prior for the logarithm of the random effects covariance matrix and still maintain the tractability of conjugacy, at least in an approximate sense. All posterior moments are calculated via Markov Chain Monte Carlo (MCMC) techniques. The Metropolis--Hastings accept reject algorithm is utilized to appropriately account for the approximation procedures. As a particular application we consider a multilevel model where student grade point average relate to a number of standardized test scores.

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