Full conjugate analysis of normal multiple traits with missing records using a generalized inverted Wishart distribution

Abstract

A Markov chain Monte Carlo (MCMC) algorithm to sample an exchangeable covariance matrix, such as the one of the error terms (R0) in a multiple trait animal model withmissing records under normal-inverted Wishart priors is presented. The algorithm (FCG) isbased on a conjugate form of the inverted Wishart density that avoids sampling the missingerror terms. Normal prior densities are assumed for the ‘fixed’ effects and breeding values,whereas the covariance matrices are assumed to follow inverted Wishart distributions. The inverted Wishart prior for the environmental covariance matrix is a product density of all patternsof missing data. The resulting MCMC scheme eliminates the correlation between the sampledmissing residuals and the sampled R0, which in turn has the effect of decreasing the total amountof samples needed to reach convergence. The use of the FCG algorithm in a multiple trait dataset with an extreme pattern of missing records produced a dramatic reduction in the size of theautocorrelations among samples for all lags from 1 to 50, and this increased the effective samplesize from 2.5 to 7 times and reduced the number of samples needed to attain convergence, whencompared with the ‘data augmentation’ algorithm.