Brunner-Munzel Test

Basic Concepts

The Mann-Whitney test is the commonly used non-parametric substitute for the two independent t-test when the normality assumption is not met. It turns out, however, that it generally requires that the two samples come from populations with the same shape (with possible differences in location), an assumption which is rarely met. In particular, the Mann-Whitney test requires that the two populations have the same variance (or at least similar variances).

We can consider the Mann-Whitney test to be the non-parametric version of the t-test with equal variances. We now describe the Brunner-Munzel test as the non-parametric version of the t-test even with unequal variances (i.e. the nonparametric version of Welch’s t-test). Just as Welch’s t-test can be used even when the variances are equal, the same is true for the Brunner-Munzel test.

Test Overview

Suppose that you have a sample X of size m for the random variable x (group 1) and a sample Y of size n for y (group 2). The BM test is designed to test the (two-sided) null hypothesis:

           H₀: P(x < y) = P(y < x)

This hypothesis is called stochastic equality.

We can also test the one-sided null hypothesis

H₀: P(x < y) > P(y < x)

or

H₀: P(x < y) < P(y < x)

Suppose sample X = {x₁₁, …, x_1m} and sample Y = {x₂₁, …, x_2n} and let N = m+n. We also find it convenient to denote m by n₁, n by n₂, X by X₁ and Y by X₂.

Define R_ij to be the rank of x_ij in X₁ ∪ X₂, R_ij* to be the rank of x_ij in X_i, and R_i to be the average of the pooled rankings R_ij.

Effect Size and Null Hypothesis

We also define the relative treatment effect size as

p = P(x < y) + .5P(x = y)

The two-sided null hypothesis is therefore equivalent to

p = .5

The one-sided null hypotheses are equivalent to p < .5 or p > .5.

The effect size p can be approximated by

We define the studentized version of p as

where

which is estimated from the samples by

where s_i² is the sample variance of the P_ij with

The test statistic for the BM test is

For m and n sufficiently large (each at least 10), t has a standard normal distribution. Thus, we reject the null hypothesis if |t| > t_crit = T.INV.2T(α/2, df) where

as for Welch’s t-test.

As noted previously, the MW test is the nonparametric version of the t-test with equal variances and the BM test is the nonparametric version of the t-test with unequal variances (i.e. Welch’s t-test).

Continuation

We continue the description of the Brunner-Munzel Test with the following topics:

• Brunner-Munzel Test Example
• Permutation Brunner-Munzel Test
• Data Analysis Tool

References

Brunner, E., Munzel, U. (2000) The Nonparametric Behrens-Fisher problem: Asymptotic theory and a small-sample approximation
https://community.jmp.com/t5/JMP-Wish-List/Brunner-Munzel-test/idi-p/631723?attachment-id=17778

Nowak, C. P., Pauly, M., Brunner, E. (2022) The nonparametric Behrens-Fisher problem in small samples
https://arxiv.org/pdf/2208.01231

Karch, J. D. (2021) Psychologists should use Brunner-Munzel’s instead of Mann-Whitney’s U test as the default nonparametric procedure.
https://osf.io/preprints/psyarxiv/kgdwn

Noguchi, K., Konietschke, F., Marmolejo-Ramos, F., Pauly, M. (2021) Permutation tests are robust and powerful at 0.5% and 5% significance levels
https://link.springer.com/article/10.3758/s13428-021-01595-5

Karch, J. D. (2022) Choosing between the two-sample t test and its alternatives: a practical guide
https://osf.io/preprints/psyarxiv/ye2d4

Akinshin, A. (2023) Corner case of the Brunner-Munzel test
https://aakinshin.net/posts/brunner-munzel-corner-case/

Toshiaki A. (2022) Usage of brunnermunzel package
https://cran.r-project.org/web/packages/brunnermunzel/vignettes/usage.html

Pauly, M., Asendorf, T., Konietschke, F. (2014) Permutation tests and confidence intervals for the area under the ROC-curve
https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.105/Publikationen/FrankThomas_8.pdf