A Zero-Inflated Poisson Model for Genetic Analysis of the Number of Mastitis Cases in Norwegian Red Cows

Abstract

The objective was to extend a zero-inflated Poisson (ZIP) model to account for correlated genetic effects, and to use this model to analyze the number of clinical mastitis cases in Norwegian Red cows. The ZIP model is suitable for analysis of count data containing an excess of zeros relative to what is expected from Poisson sampling. A ZIP model was developed and compared with a corresponding Poisson model. The Poisson parameter followed a hierarchical structure, and a residual term accounting for overdispersion was included. In both models, the Poisson parameter was regressed 1) on the year, month, and age at first calving; 2) on the logarithm of the number of days elapsed from calving to the end of first lactation; and c) on herd and sire effects. Herd and sire effects were assigned normal prior distributions in a Bayesian analysis, corresponding to a random effects treatment in a frequentist analysis. An analysis of residuals favored the Poisson model when there were 2 or more cases of mastitis during first lactation, with very small differences between the ZIP and Poisson models at 0 and 1 cases. However, the residual assessment was not satisfactory for either of the models. The ZIP model, on the other hand, had a better predictive ability than the corresponding Poisson model. Posterior means of the sire, herd, and residual variances in the ZIP model (log scale) were 0.09, 0.37, and 0.36, respectively, highlighting the importance of herds as a source of variation in clinical mastitis. The correlation between sire rankings from the ZIP and Poisson models was 0.98. A weaker correlation would be expected in a population with more severe inflation at zero than the present one. The estimate of the perfect state probability p was 0.32, indicating that 32% of the animals would be in the perfect state, either because they are resistant or because they were not exposed to mastitis.

A comparison of nonlinear mixed models and response to selection of tick-infestation on lambs

“…In this study, we discarded its implementation given the clear discrepancy with the Gaussian distribution and the wide range of categories of data (Fig 1). Thus, we restricted our comparison to Poisson [51] and negative binomial [52] models and its zero-inflated versions [41,46] using a Bayesian approach. The interpretation of zero-inflated models in the tick count data implies that the records are generated from a mixture of two distributions.…”

“…Data with a Poisson distribution have equal mean and variance whereas negative binomial allows the data to be overdispersed. Moreover, the nature of tick-count data expresses a higher-than-expected incidence of zero scores, but this can be accommodated with zero-inflated versions [39–41] of the above described distributions.…”

Inferences from Mixed Models in Quantitative Genetics

“…The two random effects might be correlated, as in a maternal effects model. Rodrigues-Motta (2006) and Rodrigues-Motta et al (2006) have developed a fully Bayesian analysis via MCMC of a ZIP model, with application to number of mastitis episodes in Norwegian dairy cattle. If the model fits, it would be possible to select animals on the basis of the chance of being truly resistant to the disease, but one cannot rule out lack of exposure.…”

A Genetic Analysis of Mortality in Pigs

“…In the present work the excess of zeros is studied using zero-inflated models that are described in Johnson and Kotz (1969) and extended by Lambert (1992). Ridout et al (1998) provide a review of various zero-inflated models; recent applications of zero-inflated Poisson models in animal breeding are in Rodriguez-Motta et al (2007) and in Naya et al (2008). Zero-inflated models assume that the population consists of two sets of observations.…”