Perform Mann-Whitney U Test for two samples and required significance. For additional information check documentation of mannWhitneyU and mannWhitneyUSignificant. This is just a helper function.

One-tailed test checks whether first sample is significantly larger than second. Two-tailed whether they are significantly different.

The Mann-Whitney U Test.

This is sometimes known as the Mann-Whitney-Wilcoxon U test, and confusingly many sources state that the Mann-Whitney U test is the same as the Wilcoxon's rank sum test (which is provided as wilcoxonRankSums). The Mann-Whitney U is a simple transform of Wilcoxon's rank sum test.

Again confusingly, different sources state reversed definitions for U₁ and U₂, so it is worth being explicit about what this function returns. Given two samples, the first, xs₁, of size n₁ and the second, xs₂, of size n₂, this function returns (U₁, U₂) where U₁ = W₁ - (n₁(n₁+1))/2 and U₂ = W₂ - (n₂(n₂+1))/2, where (W₁, W₂) is the return value of wilcoxonRankSums xs1 xs2.

Some sources instead state that U₁ and U₂ should be the other way round, often expressing this using U₁' = n₁n₂ - U₁ (since U₁ + U₂ = n₁n₂).

All of which you probably don't care about if you just feed this into mannWhitneyUSignificant.

Calculates the critical value of Mann-Whitney U for the given sample sizes and significance level.

This function returns the exact calculated value of U for all sample sizes; it does not use the normal approximation at all. Above sample size 20 it is generally recommended to use the normal approximation instead, but this function will calculate the higher critical values if you need them.

The algorithm to generate these values is a faster, memoised version of the simple unoptimised generating function given in section 2 of "The Mann Whitney Wilcoxon Distribution Using Linked Lists"

Calculates whether the Mann Whitney U test is significant.

If both sample sizes are less than or equal to 20, the exact U critical value (as calculated by mannWhitneyUCriticalValue) is used. If either sample is larger than 20, the normal approximation is used instead.

If you use a one-tailed test, the test indicates whether the first sample is significantly larger than the second. If you want the opposite, simply reverse the order in both the sample size and the (U₁, U₂) pairs.

The Wilcoxon Rank Sums Test.

This test calculates the sum of ranks for the given two samples. The samples are ordered, and assigned ranks (ties are given their average rank), then these ranks are summed for each sample.

The return value is (W₁, W₂) where W₁ is the sum of ranks of the first sample and W₂ is the sum of ranks of the second sample. This test is trivially transformed into the Mann-Whitney U test. You will probably want to use mannWhitneyU and the related functions for testing significance, but this function is exposed for completeness.