How to Perform a Brown ñ Forsythe Test in Python

The Brown-Forsythe test is a statistical test used to determine whether the variances of two or more groups are equal. While Levene's test uses the absolute deviations from the mean, the Brown-Forsythe test uses the deviations from the median.

The null hypothesis used in the test is as follows −

H0: The variance of the groups (populations) is equal

The alternative hypothesis is that the variances are not equal −

H1: The variance of the groups (populations) is not equal

To perform the test, we calculate each group's median and the absolute deviations from the median. Then we calculate the F-statistic based on the variances of these deviations. Suppose the calculated F-statistic is greater than the critical value from the F-distribution table. In that case, we reject the null hypothesis and conclude that the variances of the groups are not equal.

In Python, the scipy and statsmodels library provides a method to perform the Brown-Forsythe test.

It's important to note that The Brown-Forsythe test is sensitive to outliers but is more robust to non-normality than Levene's test. It is generally recommended to use the Brown-Forsythe test if the data is not normal.

Brown ñ Forsythe Test in Python

Syntax

levene(sample1, sample2, …sampleN, center=’median’, proportiontocut=0.05)

Parameters

• sample1, sample2, …sampleN − the sample data, which may come in various lengths. Samples must only be one dimension to be accepted.

• Center − Which data function to utilize for the test. "Median" is the default.

• Proportiontocut − When the center is "trimmed," the number of data points to remove from each end is indicated.

Explanation

In the levene() function, users must pass the one-dimensional sample data of various lengths along with the parameter center as "Median." This function then returns the statistic and p_value of the provided samples.

Algorithm

• Import the levene function from scipy.

• Create the data samples for which you want to perform the Brown-Forsythe test.

• Pass the sample data to the levene function to perform the test.

• Get statistic and p_value in return from the function.

You can use the stats. Levene method from the scipy library to perform a Brown-Forsythe test.

from scipy.stats import levene

group1 = [1, 2, 3, 4, 5]
group2 = [2, 3, 4, 5, 6]
group3 = [3, 4, 5, 6, 7]

statistic, pvalue = levene(group1, group2, group3)
print("statistic: ", statistic)
print("p-value: ", pvalue)

Output

statistic:  0.0
p-value:  1.0

Here, as you can see the p-value is 1 which is greater than 0.05. This means that we can accept the null hypothesis. Therefore, the variances of both the groups are the same. Thus, the alternate hypothesis is rejected.

Along with implementing the Brown-Forsythe question, we also one to clear one confusion that machine learning engineers usually have. It is how Brown-Forsythe and ANOVA tests are related to each other.

How is the Brown ñ Forsythe Test and the ANOVA Test Related?

The Brown-Forsythe and ANOVA (Analysis of Variance) tests are related because they test for differences in group means. However, they test for different assumptions and have different applications.

ANOVA is a statistical method used to test whether there are significant differences between the means of two or more groups. It presumes that the groups' variances are equal and that the data is normally distributed. ANOVA is used to determine whether the means of two or more groups are equal, and it compares the variances of the groups.

The Brown-Forsythe test is a variation of Levene's test, which uses the absolute deviations from the mean, while the Brown-Forsythe test uses the deviations from the median. On the other hand, the Brown-Forsythe test is a test for homogeneity of variances, which is a necessary assumption for ANOVA. It is used to determine whether the variances of two or more groups are equal.

In practice, it is common to perform the Brown-Forsythe test before ANOVA to check if the assumption of equal variances is met. If the variances are not equal, it may be appropriate to use a non-parametric test like the Kruskal-Wallis test or Welch's ANOVA test instead of the regular one.

Use cases of Brown ñ Forsythe Test

The Brown-Forsythe test is used in various fields such as biology, medicine, psychology, social sciences, and engineering to test for equal variances in different groups. Some common use cases include −

• Comparing the variances of two or more samples − The Brown-Forsythe test can determine if the variances of two or more samples are equal. For example, in a medical study, the test can be used to compare the variances of blood pressure measurements in different groups of patients.

• Testing for homogeneity of variances before performing ANOVA − As the Brown-Forsythe test is a test for homogeneity of variances, it can be used to check if the assumption of equal variances is met before performing an ANOVA. This ensures that the results of the ANOVA are valid.

• Testing for equal variances in non-normally distributed data − The Brown-Forsythe test is more robust to non-normality than Levene's test. It can be used to test for equal variances in non-normally distributed data.

• Comparing variances in repeated-measures designs − When conducting experiments with repeated-measures designs, it can be useful to use the Brown-Forsythe test to check for homogeneity of variances across groups.

• Quality control in manufacturing − The Brown-Forsythe test can be used to check for equal variances in different production batches to ensure the product's quality is consistent.

Conclusion

In conclusion, the Brown-Forsythe Test is a useful statistical method for detecting the presence of heteroscedasticity in a dataset. It can be easily implemented in Python using the scipy library. The test results can inform decisions about the appropriate statistical analysis to perform on the data. By understanding the test's assumptions and interpreting the results, researchers can better understand the distribution of their data and make informed decisions about their analysis.