Product versus additive threshold models for analysis of reproduction outcomes in animal genetics

The phenotypic observation of some reproduction traits (e.g., insemination success, interval from lambing to insemination) is the result of environmental and genetic factors acting on 2 individuals: the male and female involved in a mating couple. In animal genetics, the main approach (called additive model) proposed for studying such traits assumes that the phenotype is linked to a purely additive combination, either on the observed scale for continuous traits or on some underlying scale for discrete traits, of environmental and genetic effects affecting the 2 individuals. Statistical models proposed for studying human fecundability generally consider reproduction outcomes as the product of hypothetical unobservable variables. Taking inspiration from these works, we propose a model (product threshold model) for studying a binary reproduction trait that supposes that the observed phenotype is the product of 2 unobserved phenotypes, 1 for each individual. We developed a Gibbs sampling algorithm for fitting a Bayesian product threshold model including additive genetic effects and showed by simulation that it is feasible and that it provides good estimates of the parameters. We showed that fitting an additive threshold model to data that are simulated under a product threshold model provides biased estimates, especially for individuals with high breeding values. A main advantage of the product threshold model is that, in contrast to the additive model, it provides distinct estimates of fixed effects affecting each of the 2 unobserved phenotypes.