Page 257 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

therefore reflect their true measure of survival. This phenomenon is referred to as
censoring and such records are regarded as censored. There are several types of cen-
soring. When records are based on current values that are less than the unknown end
point, this is called right censoring. Left censoring can occur when, for instance, an
animal has been alive for a certain time before entering the study or the start of data
collection. Interval censoring can occur when there is a break in data collection and
the cow fails somewhere in that interval. However, the most common is right censor-
ing and this is the only type of censoring considered in this chapter.

14.4 Models for Analysis of Survival

14.4.1 Linear models
The linear models described in Chapters 3 or 5 have been used by various researchers
for the analysis of survival traits, including those defined as a binary trait (Everett
et al., 1976; Madgwick and Goddard, 1989; Jairath et al., 1998).
One of the major limitations with analysis of survival traits using a linear model
is the inability or the difficulty of accounting for censoring. Various authors have
attempted to address this problem. Brotherstone et al. (1997) introduced the concept
of lifespan, which is the number of lactations a cow has survived or is expected to
survive. Thus if p is the probability of survival to lactation n + 1 of an animal that has
n
survived to complete lactation n, the expected lifespan (LS) of a cow that has com-
pleted n lactations but has not had time to complete n + 1 is:
*
* *
LS = n + p + p p + p p p +
n n n+1 n n+1 n+2
Thus if all p values above are constant and cows have completed their first lactation
and have had no time restriction in the opportunity to express LS, then:
Prob(Ls = x) = (1 − p)P x−1 with x = (1, 2, 3 ...)
indicating that LS has a geometric distribution with mean = 1 + p/(1 – p) and
2
variance = p/(1 − p) .
Similarly, VanRaden and Klaaskate (1993) evaluated survival using length of pro-
ductive life, and censored records were predicted using phenotypic multiple regression.
Madgwick and Goddard (1989) proposed a multi-trait model for the analysis of survival
in each lactation, with observations in individual lactations treated as a different trait.
Information on the current lactation of living cows can then be included as observed
while their later (future) lactations are treated as missing records, hence accounting for
all information.
While some of these linear models have included methods to predict expected
survival for censored animals, these models are generally inadequate to handle time-
dependent effects. Thus HYS effects, for instance, might be based on information
from first calving, even for cows that have survived several lactations.

14.4.2 Random regression models for survival

Veerkamp et al. (1999) introduced the concept of fitting a random regression model
(RRM) for the analysis of survival defined in terms of survival to the fourth lactation

Survival Analysis 241