Application of survival analysis in breeding for longevity

Genetic evaluation for the length of productive life (LPL) in livestock species requires specific methods because of following reasons: (i) some animals are still alive at the time of evaluation and their complete LPL is not known, which implies that records of such individuals must be treated as censored; (ii) effects influencing LPL do not act linearly and vary with time; and (iii) the distribution of LPL data is often unknown and extremely skewed. Models based on survival analysis provide appropriate statistical tools for the analysis of LPL. These models are based on a hazard function λ(t) which describes the risk of being culled at time t conditional upon survival to time t. The hazard function of an animal is modeled as a product of a baseline hazard function, λ0(t), describing the ageing process, and a vector containing explanatory variables that supposedly influence the culling process and may depend on time. Because the form of λ0(t) is not always known with certainty, λ0(t) can be left unspecified, as in a Cox model. However, with complex models and large amount of data typical for national genetic evaluation, it is computationally advantageous to assume a parametric, e.g., Weibull hazard function. Such a model can also be easily extended to include a random sire or animal effect and enable estimation of genetic parameters for LPL. Routine genetic evaluation for LPL based on a Weibull model has been implemented in many countries worldwide. This paper gives an example of implementation of survival analysis techniques in dairy populations in Switzerland.