Perinatal piglet mortality poses a substantial challenge to the economic viability and ethical standards of pig farming. The categorical nature of stillbirth data, frequently including a high proportion of zeros, complicates traditional statistical analysis which often assumes Gaussian distributions. Previous research suggested a binomial model with a logit link function as suitable for analyzing piglet survival. However, these models struggle to accommodate non-genetic relationships between traits, a significant limitation given the established link between litter size (Total Number Born, TNB) and piglet mortality. This relationship is often non-linear, influenced by factors like ovulation rate, embryo loss, and farrowing duration. Recursive models, which incorporate the influence of one trait on another, offer a potential solution. Existing recursive models, however, assume a uniform relationship across all data, failing to account for the potential variation in this relationship across different genetic groups or environmental conditions. Furthermore, the assumption of a strict binomial distribution might be violated due to overdispersion in the data. The multiplicative binomial (MBN) distribution, a generalization of the binomial distribution, offers a more flexible framework that incorporates an overdispersion parameter. This study aims to address these limitations by developing a recursive model that allows for cross-specific recursive functions and employs the MBN distribution to capture overdispersion. The model is applied to a comprehensive dataset from a diallel cross among three Iberian pig varieties (Entrepelado, Retinto, and Torbiscal) to estimate Dickerson crossbreeding parameters for TNB and stillbirths (SB).

Several studies have highlighted the strong, often non-linear relationship between litter size and piglet mortality. Varona and Sorensen (2010, 2014) demonstrated the effectiveness of binomial models with logit link functions and recursive models for analyzing pig mortality and litter size. Other researchers (Roehe & Kalm, 2000; Lund et al., 2002; Ibáñez-Escriche et al., 2009; Bidanel, 2011) explored various statistical approaches and risk factors related to piglet mortality. Ibáñez-Escriche et al. (2010) employed change-point recursive models to analyze the relationship between litter size and stillbirths. Mulder et al. (2015) investigated the impact of heritable environmental variance on this relationship. The use of recursive models, as proposed by Gianola and Sorensen (2004) and further developed by Varona et al. (2007), provides a suitable framework for handling the inherent dependency between traits like litter size and mortality. The multiplicative binomial distribution, first introduced by Altham (1978) and further explored by Lovison (1998) and Lawal (2017), provides a statistically robust approach to address overdispersion which frequently affects count data like stillbirths.

This study utilized a dataset of 10,255 records from 2110 sows, encompassing three Iberian pig varieties and their reciprocal crosses in a full diallel cross. Stillbirths (SB) were recorded, and the Total Number Born (TNB) was also recorded for each litter. The pedigree included 4609 individuals extending across three generations. A hierarchical Bayesian framework was employed, with the multiplicative binomial (MBN) distribution modeling the conditional distribution of SB given TNB. Five models were compared, varying in their handling of the recursive relationship between TNB and the logit of the MBN distribution's φ parameter. Model I assumed no relationship, while Models II–V incorporated linear and quadratic recursive relationships, some allowing for cross-specific parameters. Litter size (TNB) was modeled linearly, incorporating systematic effects (cross type, parity order, year-season), additive genetic effects, and dam permanent environmental effects. Prior distributions for the model parameters were specified, and the models were implemented using a Gibbs sampler with a Metropolis-Hasting step in custom FORTRAN90 software. Convergence was assessed using the CODA package in R. Model comparison was based on the logarithm of the conditional predictive ordinate (LogCPO). Finally, Dickerson model parameters (direct, maternal, and heterosis effects) were estimated by transforming the posterior distributions of cross effects.

Model comparison using LogCPO indicated that Model II, which included a linear recursive relationship between TNB and the logit of φ, provided the best fit to the data. This model revealed a positive relationship between litter size and the probability of stillbirths, confirming previous findings. All models indicated significant overdispersion, supporting the use of the MBN distribution over the standard binomial distribution. Variance component estimates revealed substantial permanent environmental variance for both TNB and the logit of φ, indicating significant non-genetic influences on both traits. The genetic correlation between TNB and the logit of φ was near zero after accounting for the recursive relationship, suggesting that modeling the relationship directly improves the accuracy of variance component estimates. Analysis of cross effects and their re-parametrization into Dickerson model parameters provided insights into the genetic merit of different crosses. The RE and TE crosses showed superior TNB performance, while TT was the lowest. Direct, maternal, and heterosis effects were estimated for both TNB and the logit of φ. Heterosis consistently reduced the probability of stillbirths.

The findings demonstrate the utility of the MBN distribution for analyzing overdispersed count data in animal breeding. The significant overdispersion highlights the limitations of assuming independent Bernoulli trials in modeling piglet survival. The strong positive relationship between litter size and stillbirth probability underscores the challenges in breeding for increased litter size without concurrently impacting piglet survival. The relatively high permanent environmental variance highlights the importance of environmental management factors in influencing both litter size and survival. The significant maternal heterosis effects suggest that crossbreeding strategies can effectively improve piglet survival. The results provide valuable information for Iberian pig breeding programs, indicating the potential for improving reproductive efficiency through strategies such as optimizing crossbreeding schemes and improving sow management practices.

This study successfully demonstrated the effectiveness of a recursive model incorporating the multiplicative binomial distribution for analyzing perinatal mortality in a diallel cross of Iberian pigs. The model accounted for overdispersion in the data and the non-genetic relationship between litter size and stillbirths. The results showed that piglet mortality increased with litter size, but maternal heterosis significantly improved piglet survival. These findings have important implications for breeding programs aimed at improving reproductive efficiency in Iberian pig populations. Future research could explore alternative recursive relationships, such as those using change-point techniques, and investigate the genetic architecture of overdispersion in more detail.

The study assumed a priori independence of the systematic effects associated with the diallel crosses, potentially confounding the additive genetic effects and cross effects. However, the low additive variance for TNB and SB suggests that this assumption did not significantly affect the results. The model only considered linear and quadratic relationships between TNB and the logit of φ, neglecting potential non-linear relationships that could be captured using change-point techniques in future studies. The data came from a specific commercial farm and may not fully represent the diversity of Iberian pig populations.