Brunner Munzel Test: Formula, Interpretation, R, Python, SPSS, SAS and Excel Guide

Brunner Munzel Test Quick Answer

For the worked analysis, the Brunner Munzel test compares final grade G3 between female students (F, n = 383) and male students (M, n = 266). The groups are independent, G3 is ordered numeric data, the sample sizes are unequal, and identical distribution shapes are not assumed.

Brunner Munzel Test Contents

What Is the Brunner Munzel Test?

The Brunner Munzel test is a rank-based nonparametric procedure for comparing two independent populations without requiring equal variances or identically shaped distributions. It was developed for the nonparametric Behrens-Fisher problem: the same broad situation handled by Welch’s t test for means, but expressed through a probability-based relative effect rather than a difference in population means.

The central estimand is p = P(X < Y) + 0.5P(X = Y). It describes the probability that a randomly selected observation from population X is smaller than a randomly selected observation from population Y, with ties split equally. Under stochastic equality, p = 0.5. A value below 0.5 favors larger observations in X; a value above 0.5 favors larger observations in Y. Because the direction depends on group order, every Brunner Munzel test report should name X and Y explicitly.

The method is sometimes called the generalized Wilcoxon test. That name can be useful, but it should not obscure the difference: the Brunner Munzel test studentizes the relative-effect estimator with group-specific rank variability. This is why it remains appropriate when the populations have unequal spreads and different shapes, conditions that complicate the usual interpretation of the Wilcoxon-Mann-Whitney procedure.

When Should You Use the Brunner Munzel Test?

The Brunner Munzel test is especially attractive when a box plot or ECDF shows unequal spread. However, test selection should follow the research estimand and design, not a preliminary significance test for equal variances. Read the site’s guide to parametric versus nonparametric tests and the overview of the independent-samples t test before treating any rank test as an automatic fallback.

Brunner Munzel Test Null and Alternative Hypotheses

Write the hypotheses in terms of the relative effect, not generic scale parameters and not an unsupported statement about medians. For the worked Brunner Munzel test, define X as female G3 and Y as male G3.

In the observed data, p = 0.421696. Equivalently, P(F > M) + 0.5P(F = M) = 1 – 0.421696 = 0.578304. The Brunner Munzel test therefore estimates a 57.83% tie-adjusted probability that a randomly selected female student has the higher G3 score.

Brunner Munzel Test Assumptions

The Brunner Munzel test is flexible, but it is not assumption-free. The important assumptions concern design and measurement, not normality or equal variances.

Brunner Munzel Test Formula and Calculation

The Brunner Munzel test begins with pooled midranks and within-group midranks. Let n₁ and n₂ be the two group sizes. For observation j in group i, let R_ij be its midrank in the pooled sample and R^*_ij its midrank within its own group.

The name “nonparametric Behrens-Fisher” becomes clear here: the Brunner Munzel test uses separate rank-variance contributions and a Welch-type degrees-of-freedom approximation. The p-value is commonly obtained from a t distribution. SciPy defaults to the t reference distribution, while a normal option is available for large-sample approximation.

Variables Used in the Brunner Munzel Test

All 649 records had usable values for the selected variables in the supplied reports. The Brunner Munzel test therefore used 383 F observations and 266 M observations. For more on summaries such as means, quartiles and spread, see descriptive statistics, standard deviation and interquartile range.

Worked Brunner Munzel Test Result

The worked Brunner-Munzel method uses 101,878 cross-group pairs (383 × 266). Of these, 37,750 pairs have F < M, 53,705 have F > M, and 10,423 are tied. Splitting ties equally gives:

The complementary F-superiority effect is 1 – 0.4216956 = 0.5783044. The pooled mean rank is 345.829 for F and 295.009 for M. The rank-variance components are 5,761.750 for F and 11,587.195 for M. Studentization produces W = -3.468824 and df = 575.612839.

Brunner Munzel Test Charts and Statistical Meaning

Visuals do not replace the Brunner-Munzel method. They show why a probability-based rank comparison is informative and help readers detect features that a single p-value cannot communicate.

Brunner Munzel Test in Python

The most direct Brunner-Munzel method in Python uses scipy.stats.brunnermunzel. SciPy accepts two one-dimensional samples, supports two-sided and one-sided alternatives, and offers t or normal reference distributions. The t distribution is the default and matches the supplied Python and R reports.

import pandas as pd
from scipy import stats

# Read the data and preserve the intended group order.
df = pd.read_csv("dataset.csv")
x = df.loc[df["sex"].eq("F"), "G3"].dropna().to_numpy()
y = df.loc[df["sex"].eq("M"), "G3"].dropna().to_numpy()

result = stats.brunnermunzel(
    x,
    y,
    alternative="two-sided",
    distribution="t"
)

less = sum(a < b for a in x for b in y)
ties = sum(a == b for a in x for b in y)
relative_effect = (less + 0.5 * ties) / (len(x) * len(y))

print(f"W = {result.statistic:.6f}")
print(f"p = {result.pvalue:.9f}")
print(f"P(X<Y)+0.5P(X=Y) = {relative_effect:.6f}")

W = -3.468824
p = 0.000561828
P(X<Y)+0.5P(X=Y) = 0.421696

For this Brunner-Munzel method, using distribution="normal" produces p = 0.000522741 instead of the t-based p = 0.000561828. Both lead to the same decision, but the report must identify the approximation. Do not switch between them silently. The site’s t test in Python guide provides related guidance on reproducible data preparation and software reporting.

Brunner Munzel Test in R

A reliable Brunner-Munzel method in R can be run with the brunnermunzel package. It returns the statistic, degrees of freedom, p-value, confidence interval and relative-effect estimate. The formula interface is convenient, but factor-level order must be verified because the first and second levels determine effect direction.

library(brunnermunzel)

dat <- read.csv("dataset.csv", stringsAsFactors = FALSE)
dat <- subset(dat, sex %in% c("F", "M") & !is.na(G3))
dat$sex <- factor(dat$sex, levels = c("F", "M"))

fit <- brunnermunzel.test(
  G3 ~ sex,
  data = dat,
  alternative = "two.sided",
  alpha = 0.05,
  perm = FALSE,
  est = "original"
)

fit$statistic
fit$parameter
fit$p.value
fit$estimate
fit$conf.int

Brunner-Munzel Test Statistic = -3.468824
df = 575.612839
p-value = 0.0005618282
P(X<Y)+0.5P(X=Y) = 0.4216956

For a small-sample Brunner-Munzel method in R, set perm = TRUE when exact enumeration is computationally feasible, or use the package’s permutation function. Enumeration grows combinatorially, so large samples automatically require asymptotic or Monte Carlo strategies. This dataset has 649 observations, making full enumeration infeasible and the t approximation appropriate.

The supplied R report gives a bootstrap 95% interval of approximately [0.3798, 0.4655] for the reported effect. The supplied Python report gives [0.3785, 0.4681]. Small differences are expected when bootstrap seeds or percentile procedures differ. Both intervals lie below 0.50 and support the same directional interpretation of the Brunner-Munzel method.

Brunner Munzel Test in SPSS

Standard SPSS menus do not provide a native one-click Brunner-Munzel method in SPSS. The supplied SPSS workflow uses embedded Python to calculate pooled and within-group ranks, estimate W and df, create a prepared SAV file, and export SPV/PDF output. SPSS’s ordinary Mann-Whitney command is included only as familiar descriptive output; it is not a substitute for the custom Brunner-Munzel calculation.

Researchers who need a GUI may also use a validated jamovi implementation rather than relabeling SPSS’s Mann-Whitney output. For broader SPSS workflow guidance, see t tests in SPSS and SPSS, R and data-visualization results.

Brunner Munzel Test in Excel

A transparent Brunner-Munzel method in Excel is possible, but it requires more than ranking the pooled outcome. A correct workbook must calculate pooled midranks, within-group midranks, group-specific variance components, W, Welch-type df and the t-distribution p-value. A workbook that stops at a rank sum is performing a Mann-Whitney-style calculation, not the complete Brunner-Munzel method.

The downloadable workbook supplies a worked Brunner-Munzel method in Excel with the full calculation. For related spreadsheet techniques, consult t tests in Excel, percentiles and quartiles and variance.

Brunner Munzel Test in SAS

The keyword Brunner-Munzel method in SAS often leads to a misleading expectation that a standard SAS procedure has a dedicated option. Standard rank-test procedures do not provide the complete studentized Brunner-Munzel calculation as a simple menu choice. A valid implementation generally requires SAS/IML code, a validated macro, or integration with R or Python.

Because a long custom macro is difficult to audit inside a general tutorial, this guide emphasizes the formula, verification targets and reproducible cross-software checks rather than presenting unverified SAS code. That is safer than giving readers a generic rank-sum script labeled as a Brunner-Munzel method.

Brunner Munzel Test vs Mann-Whitney, Welch and Other Tests

The Brunner-Munzel method is not simply “Mann-Whitney with unequal variances.” It targets the same relative-effect family but uses a variance estimator that remains consistent under more general heteroscedastic distributions. The decision between methods should begin with the estimand: mean, median, full distribution, or probability of superiority.

Read more about Welch’s t test, the t test for unequal variances, p-values, effect size and confidence intervals.

Brunner Munzel Test Diagnostics and Common Errors

How to Report the Brunner Munzel Test in APA Style

A complete Brunner-Munzel method report should identify the outcome, both groups, sample sizes, descriptive statistics, W, degrees of freedom, p-value, relative effect, interval method and direction. Avoid converting the result into a claim about means or medians unless a separate method supports that claim.

For a concise manuscript, round W and df to two decimals, report p < .001, and retain three decimals for the relative effect. For reproducibility, preserve full precision in a supplement. See the site’s guides to p-values, significance levels and test statistics and null and alternative hypotheses.

Brunner Munzel Test Downloads

Use the reports to verify software output and the workbook to inspect the full rank calculation. Each file corresponds to the same G3-by-sex Brunner-Munzel method unless otherwise stated in the file itself.

Technical Sources for the Brunner Munzel Test

This article’s method description was checked against the current SciPy brunnermunzel documentation, the R brunnermunzel package documentation, the package’s permutation and direction guide, and Brunner and Munzel’s original nonparametric Behrens-Fisher paper. These sources support the definition, effect estimate, t approximation, software arguments and small-sample discussion used throughout the Brunner-Munzel method guide.