Estimation of variance components of threshold characters by marginal posterior modes and means via Gibbs sampling

A Gibbs sampling scheme for Bayesian analysis of binary threshold data was derived. A simulation study was conducted to evaluate the accuracy of 3 variance component estimators, deterministic approximate marginal maximum likelihood (AMML), Monte-Carlo marginal posterior mode (MCMML), and Monte-Carlo marginal posterior mean (MCMPM). Several designs with different numbers of genetic groups, herd-year-seasons (HYS), sires and progeny per sire were simulated. HYS were generated as fixed, normally distributed or drawn from a proper uniform distribution. The downward bias of the AMML estimator for small family sizes (50 sires, average of 40 progeny) was eliminated with the MCMML estimator. For designs with many HYS, 0.9 incidence, 50 sires and 40 progeny on average, the marginal posterior distribution of heritability was non-normal; MCMML and MCMPM significantly overestimated heritability under the sire mode, while under the animal model the Gibbs sampler did not converge. For designs with 100 sires and 200 progeny per sire, the marginal posterior distribution of heritability became more normal and the discrepancy among MCMML and MCMPM estimates vanished. Heritability estimates under the animal model were less accurate than those under the sire model. For the smaller designs, the MCMML estimates were very close to the true value when using a normal prior for HYS effects, irrespective of the true state of nature of the HYS effects. For extreme incidence, small data sets and many HYS, observations within an HYS will frequently fall into the same category of response. With flat priors for the HYS effects, the posterior density is likely improper, supported by an analytical proof for a simplified model and analyses from Gibbs output. In analyses of limited binary data with extreme incidence, effects of a factor with many levels should be given a normal prior. Assigning a proper uniform prior or fixing values of such levels was not useful. Most accurate estimation of genetic parameters requires very large data sets. Further work is needed on diagnosis of improperness and on alternative proper priors.