In statistics, the Brunner Munzel test[1][2][3] (also called the generalized Wilcoxon test) is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.
It is thus highly similar to the well-known Mann–Whitney U test. The core difference is that the Mann-Whitney U test assumes equal variances and a location shift model, while the Brunner Munzel test does not require these assumptions, making it more robust and applicable to a wider range of conditions. As a result, multiple authors recommend using the Brunner Munzel instead of the Mann-Whitney U test by default.[4][5]
Assumptions and formal statement of hypotheses[edit]
- All the observations from both groups are independent of each other,
- The responses are at least ordinal (i.e., one can at least say, of any two observations, which is the greater),
- Under the null hypothesis H0, is that the probability of an observation from population X exceeding an observation from population Y is the same than the probability of an observation from Y exceeding an observation from X; i.e., P(X > Y) = P(Y > X) or P(X > Y) + 0.5 · P(X = Y) = 0.5.
- The alternative hypothesis H1 is that P(X > Y) ≠ P(Y > X) or P(X > Y) + 0.5 · P(X = Y) ≠ 0.5
Under these assumptions, the test is consistent and approximately exact test.[1] The crucial difference compared to the Mann–Whitney U test is that the latter is not approximately exact under these assumptions. Both tests are exact when additionally assuming equal distributions under the null hypothesis.
Software implementations[edit]
The Brunner Munzel test is available in the following packages
References[edit]
- ^ a b Brunner, Edgar; Bathke, Arne; Konietschke, Frank (2019). Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer. p. 137. doi:10.1007/978-3-030-02914-2. ISBN 978-3-030-02912-8.
- ^ Brunner, E.; Munzel, U. (2000). "The nonparametric Behrens-Fisher problem: Asymptotic theory and a small-sample approximation". Biometrical Journal. 42 (1): 17–25. doi:10.1002/(SICI)1521-4036(200001)42:1<17::AID-BIMJ17>3.0.CO;2-U.
- ^ Neubert, K.; Brunner, E. (2007). "A studentized permutation test for the non-parametric Behrens-Fisher problem". Computational Statistics & Data Analysis. 51 (10): 5192–5204. doi:10.1016/j.csda.2006.05.024.
- ^ Karch, J. D. (2021). "Psychologists should use Brunner-Munzel's instead of Mann-Whitney's U test as the default nonparametric procedure". Advances in Methods and Practices in Psychological Science. 4 (2). doi:10.1177/2515245921999602. hdl:1887/3209569.
- ^ Noguchi, K.; Konietschke, F.; Marmolejo-Ramos, F.; Pauly, M. (2021). "Permutation tests are robust and powerful at 0.5% and 5% significance levels". Behavior Research Methods. 53 (6): 2712–2724. doi:10.3758/s13428-021-01595-5. PMID 34050436.