Heritability, reliability of genetic evaluations and response to selection in proportional hazard models

The purposes of this study were 1) to investigate the heritability, reliability, and selection response for survival traits following a Weibull frailty proportional hazard model; and 2) to examine the relationship between genetic parameters from a Weibull model, a discrete proportional hazard model, and a binary data analysis using a linear model. Both analytical methods and Monte Carlo simulations were used to achieve these aims. Data were simulated using the Weibull frailty model with two different shapes of the Weibull distribution. Breeding values of 100 unrelated sires with 50 to 100 progeny (with different levels of censoring) were generated from a normal distribution and two different sire variances. For analysis of longevity data on the discrete scale, simulated data were transformed to a discrete scale using arbitrary ends of discrete intervals of 400, 800, or 1200 d. For binary data analysis, an individual's longevity was either 0 (when longevity was less than the end of interval) or 1 (when longevity was equal or greater than the end of interval).

Three different statistical models were investigated in this study: a Weibull model, a discrete-time model (a proportional hazard model assuming that the survival data are measured on a discrete scale with few classes), and a linear model based upon binary data. An alternative derivation using basic expressions of reliabilities in sire models suggests a simple equation for the heritability on the original scale (effective heritability) that is not dependent on the Weibull parameters.

The predictions of reliabilities using the proposed formulae in this study are in very good agreement with reliabilities observed from simulations. In general, the estimates of reliability from either the discrete model or the binary data analysis were close to estimates from the Weibull model for a given number of uncensored records in this simplified case of a balanced design. Although selection response from the binary data analysis depends on the end of interval point, there is a relatively good agreement between selection responses in the Weibull model and the binary data analysis. In general, when the underlying survival data is from a Weibull distribution, it appears that the method of analyzing data does not greatly affect the results in terms of sire ranking or response to selection, at least for the simplified context considered in this study.