L¼
Ym
i¼ 1
PrðdijRi;diÞ
¼
Ym
i¼ 1
expð
Pk
j¼ 1 bjsjiÞ
P
Ciexpð
Pk
j¼ 1 bjsjuÞ
where
sji¼
X
lADi
xjl
sju¼
X
lADu
xjl DuACi
andRiis the risk set just before timeti,nithe number of subjects inRi,Dithe
death set at timeti,dithe number of subjects (i.e., deaths) inDi, andCithe
collection of all possible combinations of subjects fromRi. In this approach we
try to explain whyeventðsÞoccurred tosubjectðsÞinDiwhereas all subjects in
Riare equally at risk.This explanation, through the use of sjiand sju, is based on
the covariate values measured at time ti. Therefore, this needs some modifica-
tion in the presence of time-dependent covariates because events at timeti
should be explained byvalues of covariates measured at that particular moment.
Blood pressure measured years before, for example, may become irrelevant.
First, notations are expanded to handle time-dependent covariates. Letxjil
be the value of factorxjmeasured from subject l at timeti; then the likelihood
function above becomes
L¼
Ym
i¼ 1
PrðdijRi;diÞ
¼
Ym
i¼ 1
expð
Pk
j¼ 1 bjsjiiÞ
P
Ciexpð
Pk
j¼ 1 bjsjiuÞ
where
sjii¼
X
lADi
xjil
sjiu¼
X
lADu
xjil DuACi
From this new likelihood function, applications of subsequent steps (esti-
mation ofb’s, formation of test statistics, and estimation of the baseline sur-
vival function) are straightforward. In practical implementation, most standard
computer programs have somewhat di¤erent procedures for two categories of
402 ANALYSIS OF SURVIVAL DATA