Abstract
Response variables that are scored as counts, for example, number of mastitis cases in dairy cattle, often arise in quantitative genetic analysis. When the number of zeros exceeds the amount expected such as under the Poisson density, the zero-inflated Poisson (ZIP) model is more appropriate. In using the ZIP model in animal breeding studies, it is necessary to accommodate genetic and environmental covariances. For that, this study proposes to model the mixture and Poisson parameters hierarchically, each as a function of two random effects, representing the genetic and environmental sources of variability, respectively. The genetic random effects are allowed to be correlated, leading to a correlation within and between clusters. The environmental effects are introduced by independent residual terms, accounting for overdispersion above that caused by extra-zeros. In addition, an inter correlation structure between random genetic effects affecting mixture and Poisson parameters is used to infer pleiotropy, an expression of the extent to which these parameters are influenced by common genes. The methods described here are 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.
Highlights
Some traits in animal breeding are scored as counts, for example, litter size in pigs, embryo yield produced after superovulation, and number of mastitis cases in dairy cattle
R2n with mean zero is an identity matrix and covariance matrix of order n, and σε2λ and σε2p are residual variances reflecting overdispersion over and above that caused by extra zeros
Count data models have been developed for animal breeding applications, which pose either a Poisson mixed effects model (Foulley et al, 1987) or accommodate “extra-Poisson” residual variation explicitly (Tempelman and Gianola, 1996)
Summary
Some traits in animal breeding are scored as counts, for example, litter size in pigs, embryo yield produced after superovulation, and number of mastitis cases in dairy cattle. In using the ZIP model in animal breeding studies, it is necessary to accommodate genetic and environmental covariances. In this article, modeling genetic effects of zero-inflated count data presents special challenges; in addition of the problem of extra zeros, the correlation within and between clusters, e.g., half-sibs families, needs to be taken into account. The ZIP model accomodates overdispersion (variance > mean) caused by extra zeros, independent environmental effects (residual effects) accommodate overdispersion above that. Both genetic and environmental random effects are independent, and considered at the level of the mixture and Poisson parameters. A model checking was conducted via an analysis of residuals and predictive ability
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