Bayesian inference strategies for the prediction of genetic merit using threshold models with an application to calving ease scores in Italian Piemontese cattle

Abstract

First parity calving difficulty scores from Italian Piemontese cattle were analysed using a threshold mixed effects model. The model included the fixed effects of age of dam and sex of calf and their interaction and the random effects of sire, maternal grandsire, and herd-year-season. Covariances between sire and maternal grandsire effects were modelled using a numerator relationship matrix based on male ancestors. Field data consisted of 23 953 records collected between 1989 and 1998 from 4741 herd-year-seasons. Variance and covariance components were estimated using two alternative approximate marginal maximum likelihood (MML) methods, one based on expectation-maximization (EM) and the other based on Laplacian integration. Inferences were compared to those based on three separate runs or sequences of Markov Chain Monte Carlo (MCMC) sampling in order to assess the validity of approximate MML estimates derived from data with similar size and design structure. Point estimates of direct heritability were 0.24, 0.25 and 0.26 for EM, Laplacian and MCMC (posterior mean), respectively, whereas corresponding maternal heritability estimates were 0.10, 0.11 and 0.12, respectively. The covariance between additive direct and maternal effects was found to be not different from zero based on MCMC-derived confidence sets. The conventional joint modal estimates of sire effects and associated standard errors based on MML estimates of variance and covariance components differed little from the respective posterior means and standard deviations derived from MCMC. Therefore, there may be little need to pursue computation-intensive MCMC methods for inference on genetic parameters and genetic merits using conventional threshold sire and maternal grandsire models for large datasets on calving ease

Cumulative t-link threshold models for the genetic analysis of calving ease scores

“…We, however, prefered the Metropolis-Hastings and method of composition joint update of L and τ presented by Cowles [11]. She demonstrated and we have further noted in our previous applications [23] that the resulting MCMC mixing properties using this joint update are vastly superior to using separate Gibbs updates on individual elements of L and τ as outlined by Sorensen et al [39]. A lucid exposition on Cowles' update is also provided by Johnson and Albert [22].…”

“…As in Kizilkaya et al [23], the CP and CT models used for the analysis of calving ease data included the fixed effects of age of dam classifications, sex of calf and their interaction in β, the random effects of independent herd-yearseason effects in h, random sire effects in s and random maternal grandsire effects in m. We assume: …”

“…For each chain in the CP model, a total of 5000 cycles of the burnin period followed by saving samples from each of 100 000 additional cycles was executed based on the experiences of Kizilkaya et al [23]. Because of initially anticipated slower mixing, the corresponding burn-in period for each chain in the CT model was 10 000 cycles followed by saving each of 200 000 additional cycles.…”

“…Because of initially anticipated slower mixing, the corresponding burn-in period for each chain in the CT model was 10 000 cycles followed by saving each of 200 000 additional cycles. To facilitate diagnosis of sufficient MCMC convergence, the starting values on variance components for each chain within a model were widely discrepant, with one chain starting at the posterior mean of all (co)variance components based on the analysis of Kizilkaya et al [23], another chain starting at the posterior mean minus 3 posterior standard deviations for each (co)variance component and the final chain starting at the posterior mean plus 3 posterior standard deviations for each (co) variance component.…”

“…Furthermore, key genetic parameters, specifically direct heritability (h 2 d ), maternal heritability (h 2 m ) and the direct-maternal genetic correlation (r dm ) were inferred upon in the calving ease data using the functions of G o as presented by Kizilkaya et al [23] and Luo et al [25], for example. The only difference in the computation of heritabilities between the CP and the CT model was that the marginal residual variance for the underlying liabilities was not σ 2 e in CT, as it is in CP, but is equal to [42].…”