The Wilcoxon test is a good alternative to the t-test when the normal distribution of the differences between paired individuals cannot be assumed.
[3] The test was popularized by Sidney Siegel (1956) in his influential textbook on non-parametric statistics.
[6] The data for a one-sample test is a sample in which each observation is a real number:
Then the Wilcoxon signed-rank sum test can also be used for the following null and alternative hypotheses:[15][16] The hypothesis that the data are IID can be weakened.
The data points are not required to be independent as long as the conditional distribution of each observation given the others is symmetric about
In this case, the null and alternative hypotheses are:[18][19] These can also be expressed more directly in terms of the original pairs:[20] The null hypothesis of exchangeability can arise from a matched pair experiment with a treatment group and a control group.
Randomizing the treatment and control within each pair makes the observations exchangeable.
When this happens, the test procedure defined above is usually undefined because there is no way to uniquely rank the data.
Wilcoxon's original paper did not address the question of observations (or, in the paired sample case, differences) that equal zero.
[21] Then the standard signed-rank test could be applied to the resulting data, as long as there were no ties.
Pratt[22] observed that the reduced sample procedure can lead to paradoxical behavior.
Suppose that we are in the one-sample situation and have the following thirteen observations: The reduced sample procedure removes the zero.
To the remaining data, it assigns the signed ranks: This has a one-sided p-value of
Pratt argues that one would expect that decreasing the observations should certainly not make the data appear more positive.
Conover found examples of null and alternative hypotheses that show that neither Wilcoxon's and Pratt's methods are uniformly better than the other.
When testing a binomial distribution centered at zero to see whether the parameter of each Bernoulli trial is
Once the ranks are assigned, the test statistic is computed in the same way as usual.
Under the average rank procedure, the null distribution is different in the presence of ties.
It is possible that a sample can be judged significantly positive by the average rank procedure; but increasing some of the values so as to break the ties, or breaking the ties in any way whatsoever, results in a sample that the test judges to be not significant.
The rank assigned to an observation depends on its absolute value and the tiebreaking rule.
The tiebreaking rule is used to assign ranks to observations with the same absolute value.
[36] Random tiebreaking has the advantage that the probability that a sample is judged significantly positive does not decrease when some observations are increased.
[37] Conservative tiebreaking breaks the ties in favor of the null hypothesis.
significant, ties are broken the other way, and when large absolute values of
On the other hand, any tiebreaking rule will assign the ranks At the same one-sided level
Two other options for handling ties are based around averaging the results of tiebreaking.
under the null hypothesis is equal to the number of sign combinations that yield
[53] When consideration is restricted to continuous distributions, this is a minimum variance unbiased estimator of
To compute an effect size for the signed-rank test, one can use the rank-biserial correlation.
[55] To continue with the current example, the sample size is 9, so the total rank sum is 45.