Page 260 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

120
100
Percentage alive 60
80

40
20
0
1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000

Days alive
Fig. 14.1. Distribution of length of productive life for a group of Holstein dairy cows in the
United Kingdom.

where f(t) is the density function that equals h(t)S(t). Another way of looking at the
h(t) is that for short periods of time (Dt), the probability that an animal fails is
approximately equal to (h(t)Dt) (Kachman, 1999).

Exponential distribution

Several distributions can be used to define h(t). If h(t) is assumed to be constant over
time then this is an exponential distribution. This implies that the chance of an animal
surviving, for instance, an additional 2 years, is the same independent of how old the
animal is. Assuming the exponential distribution, then h(t) = l and S(t) = exp(−lt),
where l is the parameter of the exponential distribution.

Weibull distribution

The Weibull distribution, which is a two-parameter generalization of the exponen-
tial distribution, has also been used to model the hazard function to account for
increasing or decreasing hazard function. With the Weibull distribution, h(t) and
S(t) are:
h(t) = rl(lt) r−1 and S(t) = exp(−(lt)r)
with r > 0 and l > 0. When r = 1, the Weibull distribution reduces to the exponential
distribution. The Weibull distribution has a decreasing hazard function when r < 1
and an increasing hazard function when r > 1 (Fig. 14.2). Kachman (1999) showed
that at a given l, survival functions based on a Weibull model will all intersect at t =
1/l, and that at t = 1/l, the percentage survival is equal to exp(−1) » 37%. The role of
the l is to adjust the intercept.
Other possible distributions to model the hazard function include the gamma
distribution, log-logistics and the log-normal distribution (Ducrocq, 1997). A sum-
mary of the commonly used distributions and parameters are given in Table 14.2.

244 Chapter 14