Marginal inferences about variance components in a mixed linear model using Gibbs sampling

Abstract

Arguing from a Bayesian viewpoint, Gianola and Foulley (1990) derived anew method for estimation of variance components in a mixed linearmodel: variance estimation from integrated likelihoods (VEIL).Inference is based on the marginal posterior distribution of each ofthe variance components. Exact analysis requires numerical integration.In this paper, the Gibbs sampler, a numerical procedure for generatingmarginal distributions from conditional distributions, is employed toobtain marginal inferences about variance components in a generalunivariate mixed linear model. All needed conditional posteriordistributions are derived. Examples based on simulated data setscontaining varying amounts of information are presented for a one-waysire model. Estimates of the marginal densities of the variancecomponents and of functions thereof are obtained, and the correspondingdistributions are plotted. Numerical results with a balanced sire modelsuggest that convergence to the marginal posterior distributions isachieved with a Gibbs sequence length of 20, and that Gibbs samplesizes ranging from 300 - 3 000 may be needed to appropriatelycharacterize the marginal distributions.