wherel 0 ðtÞislðt;E^0 Þ—the hazard function of the unexposed subpopulation—
and the indicator (orcovariate)xis defined as
x¼0 if unexposed
1 if exposedThe regression coe‰cientbrepresents the relative risk on the log scale. This
model works with any covariate X, continuous or categorical; the binary
covariate above is only a very special case. Of course, the model can be
extended to include several covariates; it is usually referred to asCox’s regres-
sion model.
A special source of di‰culty in the analysis of survival data is the possibility
that some subjects may not be observed for the full time to failure or event.
Suchrandom censoringarises in medical applications with animal studies, epi-
demiological applications with human studies, or in clinical trials. In these
cases, observation is terminated before the occurrence of the event. In a clinical
trial, for example, patients may enter the study at di¤erent times; then each is
treated with one of several possible therapies after a randomization.
Figure 11.3 shows a description of a typical clinical trial. Of course, in con-
ducting this trial, we want to observe their lifetimes of all subjects from enroll-
ment, but censoring may occur in one of the following forms:
Loss to follow-up (the patient may decide to move elsewhere)
Dropout (a therapy may have such bad e¤ects that it is necessary to dis-
continue treatment)
Termination of the study
Death due to a cause not under investigation (e.g., suicide)To make it possible for statistical analysis we make the crucial assumption
that conditionally on the values of any explanatory variables (or covariates),
the prognosis for any person who has survived to a certain timetshould not be
a¤ected if the person is censored att. That is, a person who is censored att
should be representative of all those subjects with the same values of explana-
Figure 11.3 Clinical trial.SURVIVAL DATA 383