signedrank

The signedrank test, often called the Wilcoxon signed-rank test, is a nonparametric statistical test used to compare a pair of related samples or to test a single sample against a hypothesized median. It assesses whether the population median of the differences between paired measurements is zero (or equal to a specified value). It does not assume normality and is appropriate for ordinal data or non-normal interval data.

Procedure overview: For each pair, compute difference d_i = x_i - y_i. Exclude observations with d_i = 0. Rank the absolute differences |d_i| from smallest to largest; assign each rank the sign of d_i. Let W+ be the sum of ranks corresponding to positive d_i and W- the sum of ranks for negative d_i. The test statistic is W = min(W+, W-), or one may use W+ (or W-); under H0, the distribution of W is known.

Inference: For small n, exact critical values are used. For larger samples, a normal approximation can be used with mean E(W) = n(n+1)/4 and variance Var(W) = n(n+1)(2n+1)/24, where n is the number of nonzero differences. A continuity correction may be applied. The p-value is computed from the appropriate distribution.

Assumptions and interpretation: Assumes symmetric distribution of differences around the median, independent observations, and that the data are at least ordinal. It is more powerful than the sign test but less so than the paired t-test when normality holds.

Variants and software: The test exists in one-sample and paired forms; it is a standard nonparametric alternative to the paired t-test. Implementations are widely available in statistics packages (e.g., R's wilcox.test, Python's scipy.stats.wilcoxon). The name "signedrank" appears in some software documentation.