Wilcoxon Signed-Rank Test

The Wilcoxon Signed-Rank Test examines whether independent paired differences have a distribution symmetric about zero. Symmetry is an essential assumption; the statistic is not a general test of a median alone. Under the null hypothesis that the paired differences follow a symmetric distribution around zero, the signs of the differences are equally likely to be positive or negative. The exact conditional sign-randomization distribution enumerates all possible sign assignments, calculating the resulting test statistics, rather than using an asymptotic approximation.

Experiment

You can explore different tied and signed configurations in the Wilcoxon Signed-Rank Test module.

1. Set Difference preset to "Mostly positive". Observe that there are 5 nonzero differences with W+ = 13, W- = 2, and T = 2, giving an exact p-value of 3/16.
2. Set Difference preset to "Tied positive". There are exactly two nonzero absolute differences: 1 and 1. They are tied for ranks 1 and 2, so they both get rank 1.5. Both are positive, making W+ = 3 and W- = 0, giving T = 0. Out of 2^2 = 4 combinations, 2 yield T=0, so p = 0.5.
3. Set Difference preset to "All zero". All differences are zero. They are discarded, leaving 0 nonzero differences. The module reports T = 0 and a p-value of 1.