Page 262 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

14.4.5 Regression survival models

Initially, a fixed effects survival model is considered to introduce the concept. Assume
that x is a vector of risk fixed effect factors or variables that influence failure time and
b is the vector of corresponding solutions. One of the most popular procedures used to
associate the hazard function h(t) and x is the proportional hazard model (Cox, 1972;
Ducrocq, 1997). The hazard function with vector of risk factors can be written as:
h(t; x) = h (t)exp(x′b) (14.2)
o
where h (t) is the baseline hazard function, representing the ageing process of the
o
whole population. Thus the hazard function has been factored into two parts.
First, the baseline hazard function (h (t)), which is independent of the risk factors,
o
and hence the ratio of the hazard functions of two animals, is equal to a constant
at any time, i.e. their hazard functions are proportional (Ducrocq, 1997). Second,
the remaining part of the equation, exp(x′b), can be regarded as the scalar that
does not depend on time and denotes the specific risk associated with animals with
the factors x and acts multiplicatively on the baseline hazard function.
When h (t) = l = a constant, then the baseline hazard is exponential. When the
o
baseline hazard function is left completely arbitrary, then the proportion model is
termed a Cox model (Cox, 1972).
With the Weibull model, the baseline hazard function can be derived as:
h(t; x) = rl(lt) r−1 exp(x′b)
= rt r−1 exp(rlog(l) + x′b)

= h (t)exp(x′b) (14.3)
o
where h (t) = rt r−1 models the baseline hazard function and exp(x′b), the scalar, mod-
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els the relative risk above or below the baseline risk. Note that the x′b in Eqn 14.3
includes the intercept term such that x = (1, x′) and b = (rlog(l), b).
The corresponding survival function (Kachman, 1999) is:

S(t; x) = exp{−t exp(x′b)}
r

Stratified proportional hazard model

At times, the assumption of a single baseline hazard function for the whole population
in proportional hazard models may be inappropriate. Therefore, data may be divided
into subclasses on the basis of factors such as year or season of birth, treatment or
region. Then for individuals in a subclass c, a baseline hazard function can be fitted as:
h(t; x, c) = h (t)exp(x′b)
o,c
Therefore, the hazards of two animals A1 and A2 in the same subclass with covariates
x and x , respectively, are proportional:
A1 A2
ht(;x A1 , c) = exp[(x′ − x′
b
ht(;x c) A1 A2 ) ] = constant
A2 ,
and the baseline can have a known parametric form or be left arbitrary.

246 Chapter 14