Rangsumtest

Rangsumtest, commonly referred to as the rank-sum test, is a nonparametric statistical method used to compare two independent samples to assess whether their population distributions differ in central tendency. It is particularly suited when the data are ordinal or when the assumptions of normality required for a t-test are not met.

In practice, data from both samples are pooled and ranked together from smallest to largest. Let n1 and n2 be the sample sizes, and let R1 be the sum of the ranks for the first sample (and R2 for the second). The test computes a statistic U, which can be defined as U1 = n1 n2 + n1(n1+1)/2 − R1 and U2 = n1 n2 − U1. Often the smaller of U1 and U2 is used as the test statistic U. Under the null hypothesis that the two populations have identical distributions, the expected value is E[U] = n1 n2 / 2.

For small samples, the exact distribution of U is used to obtain a p-value. For larger samples, a normal approximation is employed with mean μU = n1 n2 / 2 and variance σU^2 = n1 n2 (n1 + n2 + 1) / 12, with a tie correction if tied values are present. The test can be two-sided or one-sided, depending on whether the alternative hypothesis specifies a direction.

Assumptions include independence within and between samples and measurement on at least an ordinal scale. The rangsumtest is related to the Wilcoxon rank-sum test and the Mann–Whitney U test; they are different formulations of the same underlying nonparametric comparison.

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