5.4 Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is a nonparametric alternative to the paired-samples t-test for cases in which the paired differences fails the normality condition, but is at least symmetrically distributed.

The test statistic is the sum product of the difference signs (-1, +1) and the rank of the difference absolute values, \(W = \sum_{i=1}^n sign (d_i) \cdot R_i\). The more differences that are of one sign, or of extreme magnitude, the larger \(W\) is likely to be, and the more likely to reject \(H_0\) of equality of medians.

Sign Test

The sign test is an alternative to the Wilcoxon signed-rank test for cases in which the paired differences fails the symmetrical distribution condition.

The test statistic is the count of pairs whose difference is positive, \(W = cnt(d_i > 0)\). \(W \sim b(n, 0.5)\), so the sign test is really just an exact binomial test (exact sign test), or for large n-size, the normal approximation to the binomial (sign test).