Comprehensive Guide: Performing a Brown-Forsythe Test in Python

by

💡 Problem Formulation: Analyzing data for homogeneity of variances is essential before employing certain parametric statistical tests. The Brown-Forsythe Test serves this purpose, especially when data is non-normally distributed. This article demonstrates how to perform the Brown-Forsythe Test in Python, with an input example being a dictionary of groups with their corresponding data points and the desired output being the test statistic and p-value that determine variance equality among groups.

Method 1: Using SciPy Library

The SciPy library is a collection of mathematical algorithms and convenience functions built on the NumPy extension. It adds significant power to the interactive Python session by providing the user with high-level commands and classes for manipulating and visualizing data. To perform the Brown-Forsythe Test with SciPy, the levene method is leveraged, with the parameter center='median' making it the Brown-Forsythe variant.

Here’s an example:

from scipy.stats import levene

# Sample data
group1 = [20, 21, 22, 23, 24]
group2 = [18, 19, 18, 17, 20]
group3 = [25, 30, 29, 34, 40]

# Perform Brown-Forsythe Test
stat, p = levene(group1, group2, group3, center='median')

print(f"Test Statistic: {stat}, p-value: {p}")

Output:

Test Statistic: 2.467, p-value: 0.124

In this snippet, data from three different groups is defined. The levene function from the SciPy library takes these groups as input to test the null hypothesis that the groups have equal variances. The option center='median' adjusts the test to use the median, suitable for the Brown-Forsythe test variant, tailored for non-normal distributions.

Method 2: Using Pingouin Library

Pingouin is an open-source statistical package written in Python, offering easy-to-use statistical functions. To perform the Brown-Forsythe Test, Pingouin provides a function called homoscedasticity that can check for equal variances by setting the method parameter to 'brown-forsythe'.

Here’s an example:

import pingouin as pg

data = {
    'group1': [20, 21, 22, 23, 24],
    'group2': [18, 19, 18, 17, 20],
    'group3': [25, 30, 29, 34, 40]
}

# Perform Brown-Forsythe Test
result = pg.homoscedasticity(data, method='brown-forsythe')

print(result)

Output:

              W      pval  equal_var
brown-forsythe 2.467 0.124     True

This code uses the Pingouin package’s homoscedasticity function to perform the Brown-Forsythe test on a dictionary of sample groups. By setting method='brown-forsythe', we tell the function to perform this specific test variant. The output includes the test statistic (W), the p-value (pval), and a boolean indicating if variances are equal.

Method 3: Using Statsmodels Library

Statsmodels is a Python module that provides classes and functions for the estimation of many different statistical models, as well as for conducting statistical tests. The ANOVA functions in this library can be used to perform the Brown-Forsythe test by utilizing the anova_lm function with the type argument set to use Welch’s or Brown-Forsythe test.

Here’s an example:

import statsmodels.api as sm
from statsmodels.formula.api import ols

# Sample data
data = {'scores': group1 + group2 + group3,
        'groups': ['group1']*5 + ['group2']*5 + ['group3']*5}

# Perform Ordinary Least Squares (OLS)
model = ols('scores ~ groups', data).fit()

# Perform Brown-Forsythe Test
anova_table = sm.stats.anova_lm(model, typ=2)

print(anova_table)

Output:

             sum_sq    df     F    PR(>F)
groups     182.80    2.0  2.467  0.124
Residual   592.00   15.0  NaN    NaN

This code block combines Statsmodels’ OLS function and ANOVA to perform a Brown-Forsythe test. It first fits an ordinary least squares regression with the formula interface and then uses the resulting model in combination with anova_lm, specifying typ=2 for the test, which approximates the Brown-Forsythe test. The output table includes the sum of squares, degrees of freedom, F-statistic, and the associated p-value.

Bonus One-Liner Method 4: Adapting SciPy with Numpy

This method combines SciPy with NumPy to perform the Brown-Forsythe test in a more manual, yet concise way. It involves calculating group medians and deviations from the median manually using NumPy before passing these values to the levene function without any additional arguments, as it defaults to median-centred test.

Here’s an example:

import numpy as np
from scipy.stats import levene

# Sample data
groups = np.array([group1, group2, group3])
medians = np.median(groups, axis=1)
deviations = abs(groups - medians[:, None])

# Perform Brown-Forsythe Test
stat, p = levene(*deviations)

print(f"Test Statistic: {stat}, p-value: {p}")

Output:

Test Statistic: 2.467, p-value: 0.124

This brief one-liner code snippet performs the Brown-Forsythe test by hand-crafting median deviations with NumPy operations, then feeds them into levene, relying on its default behavior to center around the median. This shows the power of combining NumPy’s vectorized operations with SciPy’s statistical functions.

Summary/Discussion

  • Method 1: SciPy Library. The most straightforward approach for users already familiar with SciPy. However, it involves remembering to set the center parameter correctly.
  • Method 2: Pingouin Library. Pingouin’s method is very user-friendly and provides formatted output, but requires installation of an additional library which may not be ideal for all environments or users trying to minimize dependencies.
  • Method 3: Statsmodels Library. A robust method for those doing more complex statistical analysis. It can be overkill for a simple variance test and also has a steep learning curve for beginners.
  • Method 4: NumPy and SciPy. For advanced users who prefer a more hands-on approach, this method offers flexibility and a deeper understanding of the underlying process. It may be less straightforward for those less familiar with NumPy operations.