(Statistics)
The Wilcoxon Rank-Sum test (hereafter, WRS) is a test for determining which of two sets of data ranks higher than the other. Typically statistic tests need to know something about the underlying distribution from which data is collected; non-parametric statistics concerns the study of statistics without knowing this. WRS is the non-parametric counterpart to the t-test.
Assumptions
- We are given two data sets (call them A and B)
- These data sets were sampled from two sources thought to have the same distribution, save for a possible phase shift.
- Null hypothesis: μA = μB (i.e., both A and B come from the same distribution).
- Alternative hypothesis: μA ≠ μB (i.e., they don't).
Take the union of A and B, and sort it. Assign to each element a rank from 1 to |A ∪ B| (the total sample size for both A and B). The test statistic (called U for various historical reasons) for A is the sum of the ranks of all the elements in A, and similiarly for B. Both test statistics give the same p-value, so it doesn't matter which you calculate. Let's say we find the U of data set A.
The p-value associated with U is the probability of a sample of |A| ranks having a more extreme sum than the observed sum U. Thankfully, the distribution of the sum of n ranks sampled from a population of m ranks is known: it's the null distribution, a discrete probability distribution that for large n (say ≥ 10) approximates the normal distribution (see Central Limit Theorem).
There's a few problems with this test. One, it's annoying to write tables for the null distribution because it depends on two parameters, but for large sample sizes it doesn't matter; one can just approximate it with the normal distribution. Two, figuring out what to do in the case of ties is annoying. Three, since it doesn't use any information about the underlying distribution, its usefulness is bounded above by the parametric t-test.
Frank Wilcoxon published the test in a biometrics journal around 1945; in 1947 (the same year that saw the development of the simplex method) two blokes by the names of Mann and Whitney published a similar test in the Annals of Mathematical Statistics. So of course Mann and Whitney get a huge chunk of credit when Wilcoxon was really first to the punch. They're the ones who called the test statistic U, by the way.