Rank tests and distribution-free inference

Let $X_1,\ldots,X_m$ and $Y_1,\ldots,Y_n$ be independent samples, iid within each group. Under the null that both groups have the same continuous distribution, pooled observations have no ties almost surely and every subset of m pooled ranks is equally likely to belong to X. The rank sum R for X therefore has a combinatorial null distribution depending only on the sample sizes. This exact statement assumes the common-distribution null, not merely equality of two numerical summaries.

Subtracting the smallest possible X rank sum gives a Mann–Whitney count U, here defined as the number of cross-group pairs with X greater than Y. Some references reverse the inequality or report a normalized count, so the convention must be declared. Under the stated no-tie null, its expectation is mn/2 and its variance is mn(m+n+1)/12. Rank invariance under a common strictly increasing transformation follows because such a transformation preserves all pooled order relations.

Under alternatives, the normalized pair count estimates the probability that a new X exceeds an independent Y. Different distributions can have the same median while producing different rank behavior, and equal pairwise probability alone does not generally restore the common-distribution permutation law. Interpreting the test as a location-shift comparison is justified within an additional common-shape shift model. Without that model, a rejection need not mean that only the medians differ.

Ties require their own treatment. Midranks are natural for representing tied observations, but the no-tie variance and exact tables cannot simply be reused unchanged. A conditional permutation distribution given the observed pooled data can still be exact under exchangeable labels, including ties; the attainable probabilities and correction formulas change. Randomized experiments can provide exchangeability through assignment under an appropriate sharp null, while observational sampling requires an argument about the group distributions and independence.

For independent identically distributed continuous differences, a null median of zero with equal probability on either side gives a binomial positive-sign count. It uses only signs, so symmetry of the full distribution is unnecessary. If zeros occur, removing them changes the conditioning and the positive-sign probability must still be justified among nonzero values. Merely knowing that zero is one possible median of a distribution with atoms does not automatically supply a fair conditional sign experiment.

The Wilcoxon signed-rank statistic instead weights signs by ranks of absolute differences. Its usual exact null relies on a continuous distribution symmetric about zero, making signs fair and independent of magnitudes. A skew distribution can have median zero without satisfying that condition. Paired measurements should first be converted into within-pair differences, with independent pairs as sampling units. Predetermined one-sided or two-sided rejection rules also matter: switching direction after seeing the result can turn a valid exact calculation into an incorrectly calibrated procedure.