Introductory Biostatistics

clinically important itself, but adjustments are still needed in e¤orts to investi-
gate other covariates. We define the hazard function for a person in the jth
stratum (or level) of this factor as

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

for j¼ 1 ; 2 ;...;q, wherel 0 ðtÞis anunspecifiedbaseline hazard for the jth
stratum andX represents otherkcovariates under investigation (excluding
the stratification itself). The baseline hazard functions are allowed to be arbi-
trary and are completely unrelated (and, of course,notproportional). The basic
additional assumption here, which is the same as that inanalysis of covariance,
requires the theb’s are the same across strata (called theparallel lines assump-
tion, which is testable).
In the analysis we identify distinct times of events for thejth stratum and
form the partial likelihoodLjðbÞas in earlier sections. Theoverallpartial like-
lihood ofbis then the product of thoseqstratum-specific likelihoods:

LðbÞ¼

Yq

j¼ 1

LjðbÞ

Subsequent analyses, finding maximum likelihood estimates as well as using
score statistics, are straightforward. For example, if the null hypothesisb¼ 0
for a given covariate is of interest, the score approach would produce astrati-
fiedlog-rank test. An important application of stratification, the analysis of
epidemiologic matched studies resulting in the conditional logistic regression
model, is presented in Section 11.5.

Example 11.5 Refer to the myelogenous leukemia data of Example 11.3
(Table 11.3). Patients were classified into the two groups according to the
presence or absence of a morphologic characteristic of white cells, and the pri-
mary covariate is the white blood count (WBC). Using

X 1 ¼lnðWBCÞ

X 2 ¼AG groupð0 if negative and 1 if positiveÞ

we fit the following model with one interaction term:

l½tjX¼ðx 1 ;x 2 ފ¼l 0 ðtÞeb^1 x^1 þb^2 x^2 þb^3 x^1 x^2

From the results shown in Table 11.4, it can be seen that the interaction e¤ect

400 ANALYSIS OF SURVIVAL DATA