Relationship between (co)variance components on the liability and the observed scale for binary traits

Abstract

In livestock breeding programs, many economically important traits such as diseases, calving ease, and conception or survivability at certain age, are binary outcomes. Since these are considered quantitative traits, their genetic analyses are based on an underlying continuous distributed liability mapping to the observed responses via a fixed threshold. Statistically, this is modelled using mixed threshold probit or logit models (Gianola and Foulley, 1983), yielding solutions on the liability scale (model scale) that are easily translated to probabilities of expressing the trait. The theoretical bases for threshold models are solid; however, the resulting mixed model equations should be solved iteratively. In the case of the threshold model, it implies dealing with integrals that involves the normal distribution, which could be a time- consuming task when working with large datasets.