Introductory Biostatistics

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