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.
In this research article, we propose a class of models for positive and zero responses by means of a zero-augmented mixed regression model. Under this class, we are particularly interested in studying positive responses whose distribution accommodates skewness. At the same time, responses can be zero, and therefore, we justify the use of a zero-augmented mixture model. We model the mean of the positive response in a logarithmic scale and the mixture probability in a logit scale, both as a function of fixed and random effects. Moreover, the random effects link the two random components through their joint distribution and incorporate within-subject correlation because of the repeated measurements and between-subject heterogeneity. A Markov chain Monte Carlo algorithm is tailored to obtain Bayesian posterior distributions of the unknown quantities of interest, and Bayesian case-deletion influence diagnostics based on the q-divergence measure is performed. We apply the proposed method to a dataset from a 24 hour dietary recall study conducted in the city of São Paulo and present a simulation study to evaluate the performance of the proposed methods.
-Dark spots in the fleece area are often associated with dark fibres in wool, which limits its competitiveness with other textile fibres. Field data from a sheep experiment in Uruguay revealed an excess number of zeros for dark spots. We compared the performance of four Poisson and zero-inflated Poisson (ZIP) models under four simulation scenarios. All models performed reasonably well under the same scenario for which the data were simulated. The deviance information criterion favoured a Poisson model with residual, while the ZIP model with a residual gave estimates closer to their true values under all simulation scenarios. Both Poisson and ZIP models with an error term at the regression level performed better than their counterparts without such an error. Field data from Corriedale sheep were analysed with Poisson and ZIP models with residuals. Parameter estimates were similar for both models. Although the posterior distribution of the sire variance was skewed due to a small number of rams in the dataset, the median of this variance suggested a scope for genetic selection. The main environmental factor was the age of the sheep at shearing. In summary, age related processes seem to drive the number of dark spots in this breed of sheep.zero-inflated Poisson / sheep / spot / posterior predictive ability / Bayesian hierarchical model
Dark spots in the fleece area are often associated with dark fibres in wool, which limits its competitiveness with other textile fibres. Field data from a sheep experiment in Uruguay revealed an excess number of zeros for dark spots. We compared the performance of four Poisson and zero-inflated Poisson (ZIP) models under four simulation scenarios. All models performed reasonably well under the same scenario for which the data were simulated. The deviance information criterion favoured a Poisson model with residual, while the ZIP model with a residual gave estimates closer to their true values under all simulation scenarios. Both Poisson and ZIP models with an error term at the regression level performed better than their counterparts without such an error. Field data from Corriedale sheep were analysed with Poisson and ZIP models with residuals. Parameter estimates were similar for both models. Although the posterior distribution of the sire variance was skewed due to a small number of rams in the dataset, the median of this variance suggested a scope for genetic selection. The main environmental factor was the age of the sheep at shearing. In summary, age related processes seem to drive the number of dark spots in this breed of sheep.
Mastitis in cows can be defined as a binary trait, reflecting presence or absence of clinical mastitis (CM), or as a count variable, number of mastitis cases (NCM), within a defined time interval. Many different models have been proposed for genetic analyses of mastitis, and the objective of this study was to evaluate the predictive ability and sire predictions of a set of models for genetic evaluation of CM or NCM. Linear- and threshold liability models for CM, and linear, censored ordinal threshold, and zero-inflated Poisson (ZIP) models for NCM were compared in a cross-validation study. To assess the ability of these models to predict future data, records from 620492 first-lactation Norwegian Red cows, which were daughters of 3064 sires, were evaluated in a fourfold cross-validation scheme. The mean squared error of prediction was used for model comparison. All models but ordinal threshold model equally performed when comparing the overall predictive ability. This result was on average, across sick and healthy cows; however, the models behaved differently for each category of animals. For example, healthy cows were predicted better by the threshold and linear models for binary data and ZIP model, whereas for mastitic cows, the ordinal threshold model was by far the best model. Predicted sire effects and rankings of sires were highly correlated across all models. For practical purposes, the linear models are very competitive with the nonlinear models.
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