Introductory Biostatistics

11.4.1 Proportional Hazards Model with Several Covariates

Suppose that we want to considerkcovariates simultaneously. The propor-
tional hazards model of Section 11.3 can easily be generalized and expressed as

l½tjX¼ðx 1 ;x 2 ;...;xkފ¼l 0 ðtÞeb^1 x^1 þb^2 x^2 þþbkxk

wherel 0 ðtÞis anunspecifiedbaseline hazard (i.e., hazard atX¼ 0 ),Xis the
vector representing thekcovariates, andbT¼ðb 1 ;b 2 ;...;bkÞarek unknown
regression coe‰cients. To have a meaningful baseline hazard, it may be neces-
sary to standardize continuous covariates about their means:

newx¼xx

so thatl 0 ðtÞis the hazard function associated witha typical patient(i.e., a
hypothetical patient who has all covariates at their average values).
The estimation ofband subsequent analyses are performed similar to the
univariate case using the partial likelihood function:

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

Also similar to the univariate case, expðbiÞrepresents one of the following:

1. The relative risk associated with an exposure ifXiis binary (exposed
   Xi¼1 versus unexposedXi¼0); or


2. The relative risk due to a 1-unit increase ifXiis continuous (Xi¼xþ 1
   versusXi¼x)

Afterbb^iand its standard error have been obtained, a 95% confidence interval
for the relative risk above is given by

exp½bb^iG 1 :96 SEðbb^iފ

396 ANALYSIS OF SURVIVAL DATA