Wilcoxon

Wilcoxon refers to a pair of nonparametric statistical tests developed by Frank Wilcoxon in 1945: the Wilcoxon signed-rank test for paired or matched samples, and the Wilcoxon rank-sum test for two independent samples. These tests provide nonparametric alternatives to the paired t-test and the two-sample t-test, respectively. They rely on the ranks of data rather than their raw values, making them robust to outliers and suitable for data that do not follow a normal distribution.

Wilcoxon signed-rank test: This test analyzes paired data by computing differences d_i = x_i − y_i for each pair, discarding zero differences, ranking the absolute differences |d_i|, and assigning the signs of d_i to those ranks. The test statistic is the sum of signed ranks. Under the null hypothesis of zero median difference, the distribution of the statistic is known exactly for small samples and can be approximated by a normal distribution for larger samples. A p-value indicates whether there is a systematic difference between the paired observations. Assumptions include paired data and at least ordinal measurement.

Wilcoxon rank-sum test (Mann–Whitney U test): For two independent samples, all observations are pooled and ranked. The test uses the sum of ranks for one group to derive the U statistic, with exact p-values for small samples and normal approximations for large samples. The test assesses whether one population tends to yield larger values than the other and is robust to outliers and non-normal distributions. Assumptions are independence of samples and at least ordinal data; interpretation in terms of medians holds when group distributions are similarly shaped.

Wilcoxon tests are widely used in biostatistics, psychology, and social sciences and are commonly referred to as the Wilcoxon signed-rank test and the Wilcoxon rank-sum test. The rank-sum test is also known as the Mann–Whitney U test.