A Zero-Inflated Poisson Model with Correlated Parameters and Application to Animal Breeding

Lambert (1992, \textit{Technometrics} \textbf{34}, 1--14) described the zero-inflated Poisson (ZIP) regression, a class of models for count data with an excess of zeros. In this paper, Lambert's methodology is extended to accommodate correlated genetic effects in the regression structure of the Poisson and mixture parameters. In addition, an inter correlation structure between these random genetic effects is introduced, and used to infer pleiotropy, an expression of the extent to which the mixture and Poisson parameters are influenced by common genes. The methods described here are implemented and illustrated with data on number of mastitis cases from Norwegian Red cows. Bayesian analysis yields posterior distributions useful for studying environmental and genetic variability, as well as genetic correlation. The model is assessed using posterior predictive checks.