BRUNNERMUNZEL

Overview

The BRUNNERMUNZEL function performs the Brunner-Munzel nonparametric test for comparing two independent samples, which is robust to differences in variance and distribution between groups. Unlike the Mann-Whitney U test, the Brunner-Munzel test does not assume equal variances and is suitable for heteroscedastic data. The test statistic is computed as:

BM = \frac{\hat{p} - 0.5}{\sqrt{\hat{V}}}

where $\hat{p}$ is the estimated probability that a randomly selected value from sample $x$ is less than a randomly selected value from sample $y$ , and $\hat{V}$ is the estimated variance of $\hat{p}$ .

For more details, see the scipy.stats.brunnermunzel documentation .

This wrapper exposes only the most commonly used parameters: x, y, alternative, and distribution. Input arrays are always flattened to 1D, and options like nan_policy, axis, and keepdims are not supported. This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=BRUNNERMUNZEL(x, y, [alternative], [distribution])

x (2D list, required): First sample data. Must be a rectangular range with at least two rows.
y (2D list, required): Second sample data. Must be a rectangular range with at least two rows.
alternative (string, optional, default=‘two-sided’): Defines the alternative hypothesis. Allowed values: two-sided, less, greater.
distribution (string, optional, default=‘t’): Method to compute p-value. Allowed values: t, normal.

The function returns a single-row, two-column array: [statistic, pvalue] (both floats), or an error message (string) if the input is invalid.

Examples

Example 1: Basic Case

Inputs:

x		y		alternative	distribution
1.2	2.3	2.1	3.2	two-sided	t
3.4	4.5	4.3	5.4

Excel formula:


=BRUNNERMUNZEL({1.2,2.3;3.4,4.5}, {2.1,3.2;4.3,5.4})

Expected output:

statistic	pvalue
0.548	0.604

Example 2: Alternative Hypothesis ‘less’

Inputs:

x		y		alternative	distribution
1.2	2.3	2.1	3.2	less	t
3.4	4.5	4.3	5.4

Excel formula:


=BRUNNERMUNZEL({1.2,2.3;3.4,4.5}, {2.1,3.2;4.3,5.4}, "less")

Expected output:

statistic	pvalue
0.548	0.302

Example 3: Distribution ‘normal’

Inputs:

x		y		alternative	distribution
1.2	2.3	2.1	3.2	two-sided	normal
3.4	4.5	4.3	5.4

Excel formula:


=BRUNNERMUNZEL({1.2,2.3;3.4,4.5}, {2.1,3.2;4.3,5.4}, "two-sided", "normal")

Expected output:

statistic	pvalue
0.548	0.584

Example 4: All Arguments Specified

Inputs:

x		y		alternative	distribution
1.2	2.3	2.1	3.2	greater	t
3.4	4.5	4.3	5.4

Excel formula:


=BRUNNERMUNZEL({1.2,2.3;3.4,4.5}, {2.1,3.2;4.3,5.4}, "greater", "t")

Expected output:

statistic	pvalue
0.548	0.698

Python Code


from scipy.stats import brunnermunzel as scipy_brunnermunzel
from typing import List, Union
 
def brunnermunzel(x: List[List[float]], y: List[List[float]], alternative: str = 'two-sided', distribution: str = 't') -> Union[List[List[float]], str]:
    """
    Computes the Brunner-Munzel nonparametric test for two independent samples.
 
    Args:
        x: 2D list of float values. First sample data.
        y: 2D list of float values. Second sample data.
        alternative: Defines the alternative hypothesis ('two-sided', 'less', 'greater'). Default is 'two-sided'.
        distribution: Method to compute p-value ('t' or 'normal'). Default is 't'.
 
    Returns:
        2D list with a single row: [statistic, pvalue], or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate x and y are 2D lists with at least two rows
    if not (isinstance(x, list) and all(isinstance(row, list) for row in x) and len(x) >= 2):
        return "Invalid input: x must be a 2D list with at least two rows."
    if not (isinstance(y, list) and all(isinstance(row, list) for row in y) and len(y) >= 2):
        return "Invalid input: y must be a 2D list with at least two rows."
    # Flatten x and y
    try:
        x_flat = [float(item) for row in x for item in row]
        y_flat = [float(item) for row in y for item in row]
    except Exception:
        return "Invalid input: x and y must contain only numeric values."
    if len(x_flat) < 2 or len(y_flat) < 2:
        return "Invalid input: x and y must each contain at least two values."
    # Validate alternative
    if alternative not in ('two-sided', 'less', 'greater'):
        return "Invalid input: alternative must be 'two-sided', 'less', or 'greater'."
    # Validate distribution
    if distribution not in ('t', 'normal'):
        return "Invalid input: distribution must be 't' or 'normal'."
    import math
    try:
        result = scipy_brunnermunzel(x_flat, y_flat, alternative=alternative, distribution=distribution)
        stat, pval = float(result.statistic), float(result.pvalue)
        # If result is nan/inf and distribution is 't', retry with 'normal'
        if distribution == 't' and (
            (math.isnan(stat) or math.isinf(stat)) or
            (math.isnan(pval) or math.isinf(pval))
        ):
            try:
                result2 = scipy_brunnermunzel(x_flat, y_flat, alternative=alternative, distribution='normal')
                stat2, pval2 = float(result2.statistic), float(result2.pvalue)
                if any([
                    math.isnan(stat2) or math.isinf(stat2),
                    math.isnan(pval2) or math.isinf(pval2)
                ]):
                    return "Invalid result: statistic or pvalue is NaN or infinite."
                return [[stat2, pval2]]
            except Exception as e2:
                return f"scipy.stats.brunnermunzel error (normal fallback): {e2}"
        # Disallow nan/inf
        if any([
            math.isnan(stat) or math.isinf(stat),
            math.isnan(pval) or math.isinf(pval)
        ]):
            return "Invalid result: statistic or pvalue is NaN or infinite."
        return [[stat, pval]]
    except Exception as e:
        return f"scipy.stats.brunnermunzel error: {e}"

Example Workbook

Link to Workbook