Bayesian Analysis via Gibbs Sampling of Susceptibility to Intramammary Infection in Holstein Cattle

Abstract

A Bayesian analysis was undertaken to assess the susceptibility of Holsteins to mastitis from 120 to 305 d in milk. Data included 595 lactations from 267 cows. The response variable was presence or absence of intramammary infection; explanatory variables were period and season of calving, somatic cell score, and cow. The logistic model adopted had period and season of calving and the regression on somatic cell score with vague prior distributions, and cow effects had a normal prior with unknown variance σu2, which, in turn, had a gamma prior. Implementation was by Gibbs sampling. Posterior densities of location parameters were unimodal and symmetric. The probability of intramammary infection of a sample cow was skewed. The posterior distribution of σu2 was skewed also. Gibbs samples of σu2 had high lag correlations, which gave an effective sample ranging between 47 and 117 from a chain of size 3000. There were differences between estimates of σu2 found using Gibbs sampling and those obtained using approximations. The low information content arising from the small size of the data and the binary nature of the response are reasons for such differences. A sensitivity analysis revealed influences of hyperparameters of the prior distribution of σu2 on inferences about this parameter.