Statistical Methods in Medical Research 4ed

8 7 3 3 2 1 ‡1 ‡1 ‡1 ‡8
8 8 7 5 5 5 5 5 5 4 5 3 5 3 5 3 5 0
7 7 5 5 4 5 4 3 3 3 ‡0 5
3 3 3 2 5 2 1 1 1 ‡2 5
3 3 2 5 2 1 1 1 ‡2 5
2 2 1 5 0.5 0 5 0 5 ‡3
1 1 0 0 0 ‡3 5
‡1 ‡1 ‡1 ‡1 ‡4 5
‡1 ‡1 ‡1 ‡4 5
‡1 ‡1 ‡4 5
‡8 ‡8
Note that the numbers of positive and negative pair means (counting zero values as
contributing 1
2 to each sum) are 17 and 38, respectively, agreeing with the values of T‡
and T obtained earlier. The estimate ^m is the median value of the pair means, namely 1.
For 95% confidence limits, note that the entry in Table A6 for n ˆ 10 and P ˆ 0 05 is 8.
The confidence limits are therefore the pair of means whose ranks are 9 and 47 (ˆ 56 9).
From the display above, these values are 4 5 and ‡1 0. For comparison, the t distribution
used for these data in Example 4.3 gave limits of 4 55 and ‡1 95, not too dissimilar
from the present values.
10.3 Comparison of two independent groups
Suppose we have two groups of observations: a random sample of n1 observations,
xi, from population X and a random sample of n2 observations, yj, from
population Y. The null hypothesis to be tested is that the distribution of x
in population X is exactly the same as that of y in population Y. We should
like the test to be sensitive to situations in which the two distributions differ
primarily in location, so that x tends to be greater (or less) than y.
The normal-theory test is the two-sample (unpaired) t test described in §4.3.
Three distribution-free tests in common usage are all essentially equivalent to
each other. They are described briefly here.
The Mann±Whitney U test
10.3 Comparison of two independent groups 277
The observations are ranked together in order of increasing magnitude. There
are n1n2 pairs (xi, yj); of these
UXY is the number of pairs for which xi < yj,
andUYX is the number of pairs for which xi > yj:
Any pairs for which xi ˆ yj, count 1 2 a unit towards both UXY and UYX .
Either of these statistics may be used for a test, with exactly equivalent
results. Using UYX, for instance, the statistic must lie between 0 and n1n2. On
the null hypothesis its expectation is 1
2 n1n2. High values will suggest a difference