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

D Gianola,CS Wang,JJ Rutledge

doi:10.1186/1297-9686-25-1-41

Abstract

Summary - 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. variance components / linear models / Bayesian methods / marginalization / Gibbs sampler

Highlights

Variance components and functions thereof are important in quantitative genetics and other areas of statistical inquiry
From a Bayesian viewpoint, REML estimates are the elements of the mode of the joint posterior density of all variance components when flat priors are employed for all parameters in the model (Harville, 1974)
In REML, fixed effects are viewed as nuisance parameters and are integrated out from the posterior density of fixed effects and variance components, which is proportional to the full likelihood in this case

Summary

Introduction

Variance components and functions thereof are important in quantitative genetics and other areas of statistical inquiry. Favored has been restricted maximum likelihood under normality, known as REML (Thompson, 1962; Patterson and Thompson, 1971). This method accounts for the degrees of freedom used in estimating fixed effects, which full maximum likelihood (ML) does not do. ML estimates are obtained by maximizing the full likelihood, including its location variant part, while REML estimation is based on maximizing the &dquo;restricted&dquo; likelihood, ie, that part of the likelihood function independent of fixed effects. From a Bayesian viewpoint, REML estimates are the elements of the mode of the joint posterior density of all variance components when flat priors are employed for all parameters in the model (Harville, 1974). In REML, fixed effects are viewed as nuisance parameters and are integrated out from the posterior density of fixed effects and variance components, which is proportional to the full likelihood in this case

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