Basic Statistics

THE WILCOXON-MANN-WHITNEY TEST 197

With a tie the formula for the variance becomes more complicated (see van Belle et
al. [2004]). Let c be the number of ties. In this example c = 1.

Var (w) = mn(m + n + 11/12 - 'j-"ti(ti - l)(ti + I)]

c

i=l

x[mn/(l2(m + n)(m + n - l)]

= 146.67 - (2 x 1 x 3)[88/(12 x 19 x (19 - l)] = 146.67 - 528/4104

= 146.47 - .13 = 146.34

The correction for ties reduces the numerical value of the Var(W). In this case the
effect on the size of Var(W) is very small, but if ordinal data are used that can have
numerous ties, the correction for ties has a larger effect. When the size of Var(W) is
reduced, the size of the computed z increases. Bergmann et al. [2000] note that the
correction for ties is needed only when the ties are across the two groups, not within
one group. The test statistic is

z = (W-E(W))/&pq

= (101.5 - 80)/12.10 = 1.78

so the p value for a two-sided test would be p = .08, and for a one-sided test p = .04.
A continuity correction of plus or minus 1/2 can also be included in the numerator of
the test statistic. The correction is made to reduce the value of the numerator.
Van Belle et al. [2004] gives the relationship between the Wilcoxon W and the
Mann-Whitney U, so the numerical value of W can be compared with the values in
tables made for U for large samples without ties. The Mann-Whitney U statistic can
be computed from
U = m(m + 2n + 1)/2 - W
or for the example given above, 8(8 + 22 + 1)/2 - 80 = 136 - 80 = 56.
The software programs vary in how the tests are done by programs and in some
cases within programs. Bergmann et al. [2000] reported that the SPSS large-sample
approximation was corrected for ties only and that SPSS has an exact permutation,
as does SAS and StatXact. The SAS large-sample permutation was corrected for
ties and continuity. Stata was corrected for ties. Minitab has a correction for ties
and continuity. This nonparametric test is often used in medical and public health
analyses since it does not require a normal distribution, and even if it is used on
normally distributed data, the results are likely to be fairly similar.

13.3.2 Wilcoxon Rank Sum Test for Small Samples

Tables that present p-values when the sample sizes are small do not include a correction
for ties. Note that the null hypothesis will be rejected when there is a lot of either
high or low ranks in the two samples. Tables for the Mann-Whitney test are included
in van Belle et al. [2004], Conover [1999], and Daniel [1978], as well as numerous
other texts on nonparametric statistics. The results can be converted to results for the
Wilcoxon rank sum test by using the formula given at the end of Section 13.3.1.