Introductory Biostatistics

tory variables who survive tot. In other words, survival condition and reason
of loss are independent; under this assumption, there is no need to distinguish
the four forms of censoring described above.
We assume that observations available on the failure time ofnsubjects are
usually taken to be independent. At the end of the study, our sample consists of
npairs of numbersðti;diÞ. Herediis an indicator variable for survival status
(di¼0 if theith individual is censored;di¼1 if theith individual failed) and
tiis the time to failure/event (ifdi¼1) or the censoring time (ifdi¼0);tiis
also called theduration time. We may also consider, in addition toti and
di;ðx 1 i;x 2 i;...;xkiÞ, a set ofkcovariates associated with theith individual rep-
resenting such cofactors as age, gender, and treatment.

11.2 INTRODUCTORY SURVIVAL ANALYSES

In this section we introduce a popular method for estimation of the survival
function and a family of statistical tests for comparison of survival distribu-
tions.

11.2.1 Kaplan–Meier Curve

In this section we introduce theproduct-limit(PL)methodof estimating sur-
vival rates, also called theKaplan–Meier method. Let

t 1 <t 2 <<tk

be the distinct observed death times in a sample of sizenfrom a homogeneous
population with survival functionSðtÞto be estimated (kan;kcould be less
thannbecause some subjects may be censored and some subjects may have
events at the same time). Letnibe the number of subjects at risk at a time just
prior toti(1aiak; these are cases whose duration time is at leastti), anddi
the number of deaths atti. The survival functionSðtÞis estimated by

SS^ðtÞ¼

Y

tiat

1 

di

ni



which is called theproduct-limit estimatororKaplan–Meier estimatorwith a
95% confidence given by

SS^ðtÞexp½G 1 : 96 ^ssðtފ

where

^ss^2 ðtÞ¼

X

tiat

di

niðnidiÞ

384 ANALYSIS OF SURVIVAL DATA