Statistical Methods in Medical Research 4ed

588 Survival analysis
McGilchrist and Aisbett (1991) considered recurrence times to infection in
patients on kidney dialysis. Following an infection a patient is treated and, when
the infection is cleared, put back on dialysis. Thus a patient may have more than
one infection so the events are not independent; some patients may be more
likely to have an infection than others and, in general, it is useful to consider
that, in addition to the covariates that may influence the hazard rate, each
individual has an unknown tendency to become infected, referred to as the
frailty. The concept of frailty may be extended to any situation where observations
on survival may not be independent. For example, individuals in families
may share a tendency for long or short survival because of their common genes,
or household members because of a common environment. Subjects in the same
family or the same environment would have a common value for their frailty.
The proportional hazards model (17.25) is modified to
or, equivalently, to
l…t, xik† ˆl0…t†exp…b T xik†exp…sfi†
l…t, xik† ˆl0…t†exp…b T xik ‡ sfi†, …17:28†
where i represents a group sharing a common value of the frailty, fi, and k a
subject within the group. The parameter s expresses the strength of the frailty
effect on the hazard function. Of course, the frailties, fi, are unobservable and
there will usually be insufficient data within each group to estimate the frailties
for each group separately. The situation is akin to that discussed in §12.5 and the
approach is to model the frailties as a set of random effects, in terms of a
distributional form. The whole data set can then be used to estimate the parameters
of this distribution as well as the regression coefficients for the covariates.
McGilchrist and Aisbett (1991) fitted a log-normal distribution to the frailties
but other distributional forms may be used. For a fuller discussion, see Klein and
Moeschberger (1997, Chapter 13). The situation is similar to those where empirical
Bayesian methods may be employed (§6.5) and the frailty estimates are
shrunk towards the mean. This approach is similar to that given by Clayton
and Cuzick (1985), and Clayton (1991) discusses the problem in terms of
Bayesian inference.
17.9 Diagnostic methods
Plots of the survival against time, usually with some transformation of one or
both of these items, are useful for checking on the distribution of the hazard. The
integrated or cumulative hazard, defined as
H…t† ˆ
… t
0
l…u† du ˆ ln S…t†, …17:29†