Introductory Biostatistics

bis anunknownregression coe‰cient. The estimation ofband subsequent
analyses are performed as follows. Denote the ordered distinct death times by

t 1 <t 2 <<tm

and letRibe the risk set just before timeti,nithe number of subjects inRi,Di
the death set at timeti,dithe number of subjects (i.e., deaths) inDi, andCithe
collection of all possible combinations of subjects fromRi. Each combination,
or subset ofRi, hasdimembers, andDiis itself one of these combinations. For
example, if three subjects (A, B, and C) are at risk just before timetiand two of
them (A and B) die atti, then

Ri¼fA;B;Cg ni¼ 3

Di¼fA;Bg di¼ 2

Ci¼ffA;Bg¼Di;fA;Cg;fB;Cgg

The product of the probabilities,

L¼

Ym

i¼ 1

PrðdijRi;diÞ

¼

Ym

i¼ 1

expðbsiÞ

P

CiexpðbsuÞ

will serve as a likelihood function, called thepartial likelihood function,in
which

si¼

X

Di

xj

su¼

X

Du

xj DuACi

11.3.2 Measures of Association

We first consider the case of a binary covariate with the conventional coding

Xi¼

0 if the patient is not exposed

1 if the patient is exposed



Here, the term exposedmay refer to a risk factor such as smoking, or a
patient’s characteristic such as race (white/nonwhite) or gender (male/female).
It can be seen that from the proportional hazards model,

392 ANALYSIS OF SURVIVAL DATA