SignedRanktoets

SignedRanktoets, commonly referred to in English as the Wilcoxon signed-rank test, is a nonparametric method for comparing two related samples or matched pairs. It tests whether the population median difference between paired observations is zero, without assuming that the differences are normally distributed.

In practice, data consist of paired measurements (X1, Y1), (X2, Y2), …. For each pair, compute the difference di = Xi − Yi. Zeros are discarded. The absolute differences |di| are ranked from smallest to largest, with ties given average ranks. Each rank is given the sign of its corresponding di. The test statistic can be formed as the sum of the positive signed ranks (W+) and the sum of the negative signed ranks (W−); often the smaller of these sums, W = min(W+, W−), is used. Some presentations use the sum of signed ranks as the statistic T.

For small samples, exact p-values are available. For larger samples, a normal approximation is used, with established mean and variance for the signed-rank distribution, sometimes with a continuity correction. The null hypothesis states that the median difference is zero (no systematic change between paired measurements). A two-sided test assesses any difference, while one-sided tests examine a consistent increase or decrease.

Assumptions are relatively mild: paired observations should be independent, data should be at least ordinal, and the distribution of differences should be symmetric around zero. The SignedRanktoets is a robust alternative to the paired t-test when normality cannot be assumed, but it has less power when the symmetry assumption is violated or when many ties or zero differences occur. It is closely related to the Wilcoxon signed-rank test and is widely used in small-sample and nonparametric analyses.