Genet. Sel. Evol. 35 (2003) 159-183
DOI: 10.1051/gse:2003002

Multivariate Bayesian analysis of Gaussian, right censored Gaussian, ordered categorical and binary traits using Gibbs sampling

Inge Riis Korsgaard^a, Mogens Sandø Lund^a, Daniel Sorensen^a, Daniel Gianola^b, Per Madsen^a and Just Jensen<sup>a

a</sup>  Department of Animal Breeding and Genetics, Danish Institute of Agricultural Sciences, PO Box 50, 8830 Tjele, Denmark
^b  Department of Meat and Animal Sciences, University of Wisconsin-Madison, WI 53706-1284, USA

(Received 5 October 2001; accepted 3 September 2002)

Abstract
A fully Bayesian analysis using Gibbs sampling and data augmentation in a multivariate model of Gaussian, right censored, and grouped Gaussian traits is described. The grouped Gaussian traits are either ordered categorical traits (with more than two categories) or binary traits, where the grouping is determined via thresholds on the underlying Gaussian scale, the liability scale. Allowances are made for unequal models, unknown covariance matrices and missing data. Having outlined the theory, strategies for implementation are reviewed. These include joint sampling of location parameters; efficient sampling from the fully conditional posterior distribution of augmented data, a multivariate truncated normal distribution; and sampling from the conditional inverse Wishart distribution, the fully conditional posterior distribution of the residual covariance matrix. Finally, a simulated dataset was analysed to illustrate the methodology. This paper concentrates on a model where residuals associated with liabilities of the binary traits are assumed to be independent. A Bayesian analysis using Gibbs sampling is outlined for the model where this assumption is relaxed.

Key words: categorical / Gaussian / multivariate Bayesian analysis / right censored Gaussian

Correspondence and reprints: Inge Riis Korsgaard
    e-mail: IngeR.Korsgaard@agrsci.dk

© INRA, EDP Sciences 2003