statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, the Brunner Munzel test (also called the generalized Wilcoxon test) is a

nonparametric Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric sta ...

test Test(s), testing, or TEST may refer to: * Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities Arts and entertainment * ''Test'' (2013 film), an American film * ''Test'' (2014 film) ...

of the

null hypothesis The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...

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 Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW/MWU), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test) is a nonparametric statistical test of the null hypothesis that randomly selected values ''X'' and ''Y'' f ...

. 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.

Assumptions and formal statement of hypotheses

# All the observations from both groups are

independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...

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 ''H''₀, 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., or . # The alternative hypothesis ''H''₁ is that or Under these assumptions, the test is consistent and approximately exact. The crucial difference compared to the

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

The Brunner Munzel test is available in the following packages * R

(function ) *

Python (programming language) Python is a high-level programming language, high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is type system#DYNAMIC, dynamically type-checked a ...

* jamovi

References

Nonparametric statistics Statistical tests {{Statistics-stub