## The Wilcoxon Test

This test is the non-parametric equivalent of the paired samples t-test. The null and alternative hypotheses are the same as those for the paired samples t-test although they are often stated in terms of the median, thus:

H0: In the population the median difference is zero
H1: In the population the median difference is not zero

However, an assumption behind this test is that the population distributions are symmetrical, in which case the mean and medians are identical.

Worked Example

Blood pressures (mmHg)
Subject Standing Supine Difference
1 132 136 4
2 146 145 1
3 135 140 5
4 141 147 6
5 139 142 3
6 162 160 -2
7 128 137 9
8 137 136 -1
9 145 149 4
10 151 158 7
11 131 120 -11
12 143 150 7
Mean 140.83 143.33 2.50
SD 9.49 10.83 5.50

The frugal Minitab output simply shows the estimated median difference and the 94.5% confidence interval (with non-parametric tests it is not always possible to achieve exact probability levels). Since this confidence interval contains zero, we fail to reject the null hypothesis that the population median difference is zero.

The SPSS output provides, among other things, the p-value for a two-tailed test (0.099).

The conclusion is that there is insufficient evidence to suggest a difference in standing and supine systolic blood pressures

Computer output from Minitab and SPSS

The Wilcoxon Test using Minitab

To perform this test use
Stat > Nonparametrics > 1-sample Wilcoxon...
after creating a column of differences between the pairs in the sample.

Wilcoxon Signed Rank Confidence Interval

Estimated Achieved
N Median Confidence Confidence Interval
Difference 12 3.00 94.5 (-1.00, 6.00)

The Wilcoxon Test using SPSS

Use
Analyze > Nonparametric Tests > 2 Related Samples...
to obtain the following output

Wilcoxon Signed Ranks Test  