Page 266 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

indicate the relative risk of daughters of these animals being culled. Usually these
estimates are transformed to relative breeding values, say, with mean 100 and stand-
ard deviation of 12, so they are comparable with breeding values of other traits.
The results can also be presented in several other forms. The interest may be to
predict the percentage of live daughters for sires at 40 months of productive life; for
instance, (i) in herd 1 and in the 4th YSP or (ii) across all herds and YSPs. If (i), then
for sire 1:
ˆ
ˆ
d = b + b + aˆ = 0.00 + −2.982 + −0.779 = −3.761
1 1 6 1
Then using Eqn 14.4, S(40, x,z) = exp{−40 exp(d )} = 0.394.
r
1
For (ii), a weighted mean for fixed effect solutions might be computed based on the
number of daughters the sire has in each fixed effect subclass. Thus for sire 1:
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
d = (2b + 2b )/4 + (1b + 0b + 1b + 2b )/4 + aˆ
1 1 2 3 4 5 6 1
= (2*0.0 + 2*0.196)/4 + (0.0 + 0 + −3.149 + 2*−2.982)/4 + −0.779 = −2.959
and S(40, x,z) = exp{−40 exp(d )} = 0.126
r
1
Equation 14.6 and its application in Example 14.1 was mainly to illustrate the
basic principles of survival analysis using proportional hazard models with a frailty
term. The parameter r has been assumed known and in practice this has to be esti-
mated simultaneously, and usually more terms including time-dependent variables are
included in the models. The ‘Survival kit’ (Ducrocq and Solkner, 1998; Mészáros
et al., 2013) is currently used for the genetic evaluation of survival traits at the
national level by a number of countries. A summary of methods utilized for the evalu-
ation of survival at the national level for the Holstein breed on the Interbull website
(http://www-interbull.slu.se/national_ges_info2/framesida-ges.htm) indicates that eight
countries (France, Germany, Italy, the Netherlands, Hungary, Slovenia, Spain and
Switzerland) use proportional hazard models in their genetic evaluation systems.
Similarly, nine countries (Canada, Denmark, Finland, Japan, New Zealand, Sweden and
the UK) currently use a multi-trait animal model, while the USA, Israel and Australia
employ a single-trait animal model. The only country that uses a random regression ani-
mal model is Belgium (Walloon region).

14.4.7 Group data survival model

When survival is defined as a discrete trait such as number of lactations completed
or number of years completed, the Cox and Weibull models may not be suitable for
the analysis of such traits. This is because these models assume continuity of the
baseline hazard distribution and/or absence of ties between ordered failure times.
Thus, with discrete survival traits, the grouped data version of the proportional haz-
ards model introduced by Prentice and Gloeckler (1978) can be used. The group data
proportional hazard model involves grouping failure time into intervals Q = (q , q ),
i i−1 i
i = 1, ... ,r with q = 0, q = +infinite and failure times in Q are recorded as t . Thus the
0 r i i
regression vector is assumed to be time-dependent but fixed within each time interval.
Grouped data models have been used in beef cattle (Phocas and Ducrocq, 2006) and
rabbits (Piles et al., 2006). Mészáros et al. (2010) demonstrated this grouped data
model was more appropriate in length of productive life in pigs.

250 Chapter 14