Hierarchical Bayesian Methods to Model Heterogeneity in Cow- and Herd-level Relationships between Milk Production and Reproduction in Dairy Cows

Two of the most important broad classifications of phenotypes for successful dairy production are milk yield and fertility. The nature of the relationship between milk production and reproductive performance of dairy cows is uncertain due to conflicting results reported in many studies. A common deficiency in many such studies is an underappreciation of the dual dimension of the production-reproduction relationship, as defined by herd (random or u) level and cow (residual or e) level sources of (co)variation. Our overall hypothesis is that the eand ulevel relationships between milk production and reproduction in dairy cows are heterogeneous and depend upon various herd-related and management factors. Our objective is to develop hierarchical Bayesian extensions that capture heterogeneity in the relationships between traits by mixed effects modeling of u level and e level covariances between traits of interest. We specify a bivariate Bayesian model to jointly model two continuous traits and we apply a square-root free Cholesky decomposition to the variance-covariance matrices of the residuals (cow-level) and random effects (herd-level). As a result, the eand u-level covariances among the traits are reparameterized into unconstrained and easily interpretable eand uregression parameters, respectively. These regression parameters specify the cowand herd-level relationships, respectively, between the traits and can be easily modeled as functions of relevant fixed and random effects, thereby providing a mixed model extension of Pourahmadi’s method. We validate our method using a simulation study and apply it to data on 305-day milk yield and calving interval of Michigan dairy cows.