MARGINAL INFERENCES ABOUT VARIANCE-COMPONENTS IN A MIXED LINEAR-MODEL USING GIBBS SAMPLING

Arguing from a Bayesian viewpoint, Gianola and Foulley (1990) derived a new method for estimation of variance components in a mixed linear model: variance estimation from integrated likelihoods (VEIL). Inference is based on the marginal posterior distribution of each of the variance components. Exact analysis requires numerical integration. In this paper, the Gibbs sampler, a numerical procedure for generating marginal distributions from conditional distributions, is employed to obtain marginal inferences about variance components in a general univariate mixed linear model. All needed conditional posterior distributions are derived. Examples based on simulated data sets containing varying amounts of information are presented for a one-way sire model. Estimates of the marginal densities of the variance components and of functions thereof are obtained, and the corresponding distributions are plotted. Numerical results with a balanced sire model suggest that convergence to the marginal posterior distributions is achieved with a Gibbs sequence length of 20, and that Gibbs sample sizes ranging from 300 - 3 000 may be needed to appropriately characterize the marginal distributions.