Page 263 - Linear Models for the Prediction of Animal Breeding Values

Accelerated failure time model

The accelerated failure time is another procedure to associate the hazard functions
and the risk factors. Here it is assumed that the risk factors not only act multipli-
catively but also accelerate or decelerate the failure time. Let S (t) and S (t) denote
c b
the survival functions of cows housed on concrete floors and floors with straw bed-
ding, respectively. If it is assumed that the survival of cows housed on concrete
floors is lower by a factor g than those housed on floors with bedding, then the
accelerated failure time model assumes that S (t) = S (gt) and g > 0 is the so-called
b c
accelerating factor.
Thus Eqn 14.2 can be written as:
h(t; x) = h (exp[x′ b]t)exp(x′ b)
o
= h (t*)exp(x′ b)
o
where this change in timescale from t to t* denotes an acceleration or a deceleration
depending on whether exp(x′ b) is smaller or greater than unity (Ducrocq, 1997).

Time-dependent risk factors

In the analysis of survival data that span a good length of time, it is possible that
some of the risk factors may change with time. In livestock situations, the effect of
such factors such as year–season of calving or herd management effects are likely to
change over time. Such factors are termed time-dependent variables or risk factors.
The proportional hazard model can be extended to incorporate time-dependent
variables and Eqn 14.2 can then be written as:

h(t; x(t)) = h (t)exp(x(t)′b)
o
where as usual x(t) represents a vector of risk factors, but some of them will be time-
dependent variables. Ducrocq (2000) showed that it is possible to define time-dependent
variables such as HYS as a sequence of indicator variables x(t) = (0 ... 1 ... 0) with x(t)= 1
i
if the observation is affected by the ith HYS at time t, or x (t) = 0 otherwise.
i

14.4.6 Mixed survival models

Mixed survival models, usually called frailty models, refer to the extension of the
proportional hazard function to include random effects such as genetic effects. The
random or frailty term u is defined as an unobserved random quantity that acts
m
multiplicatively on the hazard of individuals or a group of animals (Ducrocq, 1997).
The random vector u can be defined for individual animals or daughters of a sire m
m
as in a sire model. With the simple transformation a = log(u ), the frailty term can
m m
be included in the exponential part of the proportional hazards. Thus the mixed survival
model can be written as:
h(t; x,z) = h (t)exp(x′b + z′a) and
o
S(t; x,z) = exp{−t exp(x′b + z′a)} (14.4)
r

Survival Analysis 247