Introductory Biostatistics

fore, by default, is more sensitive to exposures with a constant relative

risk (the proportional hazards e¤ect; in fact, we have derived the log-rank

test as a score test using the proportional hazards model). Because of

these characteristics, applications of both tests may reveal not only

whether or not an exposure has any e¤ect, but also the nature of the

e¤ect, short term or long term.

2. Because of the way the tests are formulated (terms in the summation are
   not squared),
   X

alli

wi½d 1 iE 0 ðd 1 iފ

they are powerful only when one risk is greater than the other at all times.

Otherwise, some terms in this sum are positive, other terms are negative,

and they cancel each other out. For example, the tests are virtually pow-

erless for the case of crossing survival curves; in this case the assumption

of proportional hazards is severely violated.

3. Some cancer treatments (e.g., bone marrow transplantation) are thought
   to have cured patients within a short time following initiation. Then,
   instead of all patients having the same hazard, a biologically more
   appropriate model, the cure model, assumes that an unknown propor-
   tionð 1 pÞare still at risk, whereas the remaining proportionðpÞhave
   essentially no risk. If the aim of the study is to compare the cure pro-
   portionspvalues, neither the generalized Wilcoxon nor log-rank test is
   appropriate (low power). One may simply choose a time point tfar
   enough out for the curves to level o¤, then compare the estimated sur-
   vival rates by referring to percentiles of the standard normal distribution:

z¼

SS^ 1 ðtÞSS^ 2 ðtÞ

fVar½SS^ 1 ðtފþVar½SS^ 2 ðtފg^1 =^2

:

Estimated survival rates,SS^iðtÞ, and their variances are obtained as dis-

cussed in Section 11.2.1 (the Kaplan–Meier procedure).

Example 11.2 Refer back to the clinical trial (Example 11.1) to evaluate the
e¤ect of 6-mercaptopurine (6-MP) to maintain remission from acute leukemia.
The results of the tests indicate a highly significant di¤erence between survival
patterns of the two groups (Figure 11.5). The generalized Wilcoxon test shows
a slightly larger statistic, indicating that the di¤erence is slightly larger at earlier
times; however, the log-rank test is almost equally significant, indicating that
the use of 6-MP has a long-term e¤ect (i.e., the e¤ect does not wear o¤).

Generalized Wilcoxon: w^2 ¼ 13 : 46 ð1dfÞ;p< 0 : 0002

Log-rank: w^2 ¼ 16 : 79 ð1dfÞ;p¼ 0 : 0001

INTRODUCTORY SURVIVAL ANALYSES 389