Information Technology Reference
In-Depth Information
Survived
Died
Total
Men
9
1
10
Women
4
10
14
Total
13
11
24
The 9 denotes the number of males who survived, the 1 denotes the
number of males who died, and so forth. The four marginal totals or
marginals are 10, 14, 13, and 11. The total number of men in the study
is 10, while 14 denotes the total number of women, and so forth.
The marginals in this table are fixed because, indisputably, there are 11
dead bodies among the 24 persons in the study and 14 women. Suppose
that before completing the table, we lost the subject IDs so that we could
no longer identify which subject belonged in which category. Imagine you
are given two sets of 24 labels. The first set has 14 labels with the word
“woman” and 10 labels with the word “man.” The second set of labels
has 11 labels with the word “dead” and 13 labels with the word “alive.”
Under the null hypothesis, you are allowed to distribute the labels to sub-
jects independently of one another. One label from each of the two sets
per subject, please.
24
10
14
10
10
1
Ê
Á
ˆ
˜
Ê
Á
ˆ
˜
Ê
ˆ
˜
There are a total of
ways you could hand out the labels.
Á
of the assignments result in tables that are as extreme as our original table
14
11
10
0
Ê
Á
ˆ
˜
Ê
ˆ
˜
(that is, in which 90% of the men survive) and
in tables that are
Á
more extreme (100% of the men survive). This is a very small fraction of
the total, so we conclude that a difference in survival rates of the two
sexes as extreme as the difference we observed in our original table is very
unlikely to have occurred by chance alone. We reject the hypothesis that
the survival rates for the two sexes are the same and accept the alternative
hypothesis that, in this instance at least, males are more likely to profit
from treatment (Table 2.1).
In the preceding example, we tested the hypothesis that survival rates
do not depend on sex against the alternative that men diagnosed with
cancer are likely to live longer than women similarly diagnosed. We
rejected the null hypothesis because only a small fraction of the possible
tables were as extreme as the one we observed initially. This is an example
of a one-tailed test. But is it the correct test? Is this really the alternative
hypothesis we would have proposed if we had not already seen the data?
Wouldn't we have been just as likely to reject the null hypothesis that men
