Page 264 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

where z is an incidence matrix for random effects and the baseline hazard function can
assume a parametric or arbitrary form. Ducrocq et al. (1988a, 1988b) and Ducrocq
(1997) discussed the various distributions (gamma or log-gammas or inverse Gaussian)
that have been assumed for the frailty term and various estimation procedures for the
parameters of the frailty model. In the following section, the parametric model pre-
sented by Kachman (1999) is used to illustrate the prediction of a in the frailty model.
The parameters of interest in a survival model with or without the frailty term can
be estimated using non-parametric, semi-parametric or parametric approaches. In this
section, a brief outline of the parametric approach is presented. The basic parametric
approach involves obtaining the joint likelihood of the survival time and the random
effects, getting the marginal likelihood of survival time by integrating over the random
effects or taking a second-order Taylor’s series expansion of the joint log-likelihood.
The joint log-likelihood for the Weibull function can be written (Kachman, 1999) as:
(
L bu, ,r) = ∑ {log h t ( ) + (x i b + z a) − H t ( )exp(x i b + z a)}
o
i
i
o
i
i
i (14.5)
′
/log| |G − 12
− 12 | / a Ga
The posterior mode estimates of the fixed and random effects can then be obtained
by taking the first and second partial derivatives of Eqn 14.5. The resulting equations
for the estimation are:
⎡ X′RX X′RZ⎤ ⎡ ⎤ ˆ b ⎡ X′y ⎤ *
⎢ − ⎥ ⎢ ⎥ = ⎢ ⎥ [14.6]
⎣ Z′RX Z′RZ G ⎦ ⎣ ˆ a⎦ ⎣ Z′y ⎦ *
1
+
where R is a diagonal matrix with elements r = w × exp(x b + z a), with w =
ii i i i i
*
exp(r*log(t )) and t is the survival record for animal i, and y = q − r {1 − (x b + z a)},
i i i i ii i i
with q = 1 for uncensored records or 0 if records are censored.
i
The use of Eqn 14.6 involves an iterative procedure with d = (xb + za) being initially
i
i
i
*
computed for record or individual i, then r and y are calculated assuming that the esti-
ii i
mate r is known for the data. Then Eqn 14.6 can be set up. Once all records have been
ˆ
processed, estimates of b and a are obtained by solving Eqn 14.6. The new estimates of b ˆ
ˆ
and â are then fed into the iterative procedure again until convergence is achieved.
Example 14.1
Presented in Table 14.1 is the length of productive life in months for a group of cows in
two herds. The aim is to undertake a survival analysis using Eqn 14.6, fitting herd and
year–season–parity as fixed risk factors and random animal effects. It is assumed that r
is 1 and the genetic variance is 20. The full pedigree is incorporated into the analysis.
Considering the fixed effects, the design matrix X is:
⎛ 1 1 1 1 1 1 0 0 0 0 0 0 ⎞
⎜ 0 0 000 0 1 1 1 1 1 1 ⎟
⎜ ⎟
X′= ⎜ ⎜ 00 1 0 0 1 0 1 0 0 0 0 ⎟ ⎟
⎜ 00 0 1 0 0 0 0 1 0 0 1 ⎟
⎜ 10 0 0 10 0 0 0 0 1 0 0 ⎟
⎜ ⎟
⎝ 0 1 00 00 1 0 0 0 1 0 ⎠
248 Chapter 14