The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

9.2 Wilcoxon Rank-Sum Test 147

As previously mentioned, the test statistic is obtained by ranking the
pooled observations, and then summing the ranks of, say, the fi rst
population (could equally well have chosen the second population).
We will illustrate the test fi rst scores of a leg - lifting test for elderly
men Table 9.1.
For this problem, the sum of all the ranks is ( N 1 + N 2 )( N 1 + N 2 + 1 )
/2 = ( 5 + 5) (5 + 5 + 1)/2 = 10(11)/2 = 55, since N 1 = N 2 = 5. Now,
since N 1 /( N 1 + N 2 ) = probability of randomly selecting a patient from
group 1 and N 2 /( N 1 + N 2 ) = probability of randomly selecting a patient
from group 2, if we multiply N 1 /( N 1 + N 2 ) by ( N 1 + N 2 )( N 1 + N 2 + 1)/2,
it gives the expect rank - sum for group 1. This is N 1 ( N 1 + N 2 + 1)/2,
which in our example is 5(11)/2 = 27.5. From the tables for the Wilcoxon
test, we see that a rank - sum less than 18 or greater than 37 will be
signifi cant at the 0.05 level for a two - side test. Since T 1 = 25, we
cannot reject the null hypothesis.
As a second example, we take another look at the pig blood loss
data. Table 9.2 shows the data the pooled rankings.
The p - value for this test is greater than 0.20, since the 80% confi -
dence interval for T1 is [88, 122], which contains 112. Now, although

Table 9.1

Left Leg Lifting Among Elderly Males Getting Physical Therapy:

Comparing Treatment and Control Groups

Unsorted scores Scores sorted by rank

Control group Treatment group Control group (rank) Treatment group (rank)

25 26 16 (1)

66 85 18 (2)

34 48 25 (3)

18 68 26 (4)

57 16 34 (5)

N 1 = 5 N 2 = 5 48 (6)

57 (7)

66 (8)

68 (9)

85 (10)

T 1 = Σ R = 2 5 T 2 = Σ R = 30