Introductory Biostatistics

Var 0 ðd 1 iÞ¼

n 1 in 2 iaidi

ni^2 ðni 1 Þ

the formula we used in the Mantel–Haenszel method in Chapter 6.
After constructing a 22 table for each uncensored observation, the evi-
dence against the null hypothesis can be summarized in the following statistic:

y¼

Xm

i¼ 1

wi½d 1 iE 0 ðd 1 iފ

wherewiis the weight associated with the 22 table atti. We have under the
null hypothesis:

E 0 ðyÞ¼ 0

Var 0 ðyÞ¼

Xm

i¼ 1

w^2 iVar 0 ðd 1 iÞ

¼

Xm

i¼ 1

w^2 in 1 in 2 iaidi

ni^2 ðni 1 Þ

The evidence against the null hypothesis is summarized in the standardized
statistic

z¼

y

½Var 0 ðyފ^1 =^2

which is referred to the standard normal percentilez 1 afor a specified sizea
of the test. We may also referz^2 to a chi-square distribution at 1 degree of
freedom.
There are two important special cases:

1. The choicewi¼nigives thegeneralized Wilcoxon test; it is reduced to the
   Wilcoxon test in the absence of censoring.


2. The choicewi¼1 gives thelog-rank test(also called theCox–Mantel
   test; it is similar to the Mantel–Haenszel procedure of Chapter 6 for the
   combination of several 22 tables in the analysis of categorical data).

There are a few other interesting issues:

1. Which test should we use? The generalized Wilcoxon statistic puts more
   weight on the beginning observations, and because of that its use is more
   powerful in detecting the e¤ects of short-term risks. On the other hand,
   the log-rank statistic puts equal weight on each observation and there-

388 ANALYSIS OF SURVIVAL DATA