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 iÞ
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 ðyÞ^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:
- The choicewi¼nigives thegeneralized Wilcoxon test; it is reduced to the
Wilcoxon test in the absence of censoring. - 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:
- 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