Levene's Test for Homogeneity of Variance

Levene's test is a statistical method for testing the null hypothesis that two or more groups have the same variance. It is commonly used in analysis of variance (ANOVA) to check if the assumption of equal variances is valid. The null hypothesis is that the variances in each group are equal, while the alternative hypothesis is that at least one group has a different variance.

Algorithm

The Levene's test is a hypothesis test that compares the mean of absolute deviations from the median of each group. The test statistic is calculated as follows:

$$ W = \frac{(N-k)}{(k-1)}\frac{\sum_{i=1}^k n_i (\bar{z_{i\cdot}} - \bar{z_{\cdot\cdot}})^2}{\sum_{i=1}^k \sum_{j=1}^{n_i} (z_{ij} - \bar{z_{i\cdot}})^2}, $$

where

• $N$ is the total sample size
• $k$ is the number of groups
• $n_i$ is the sample size of the ith group
• $\bar{z_{i\cdot}}$ is the mean of the group
• $\bar{z_{\cdot\cdot}}$ is the overall mean
• $z_{ij}$ is the jth observation in the ith group
• $z_{i\cdot}$ is the mean of the ith group

The test statistic follows an F-distribution with degrees of freedom $(k-1, N-k)$ under the null hypothesis.

Python Implementation

Here is an implementation of Levene's test in Python:

from scipy.stats import levene

def levene_test(groups, alpha=0.05):
    """
    Perform Levene's test for homogeneity of variance.
    
    Parameters:
    groups (list): List of arrays, each corresponding to a group.
    alpha (float): Significance level.
    
    Returns:
    p-value (float): Test result.
    reject (bool): Test decision.
    """
    stat, pval = levene(*groups)
    reject = pval < alpha
    return pval, reject

The function takes an array of arrays, where each array is a group of observations. It returns the p-value and the decision whether to reject the null hypothesis or not.

Example Usage

>>> import numpy as np
>>> group1 = np.array([1, 2, 3, 4, 5])
>>> group2 = np.array([1, 2, 3, 4, 6])
>>> pval, reject = levene_test([group1, group2])
>>> pval
0.3863574709646789
>>> reject
False

In this example, we have two groups with slightly different variances. The test result shows that there is no statistically significant difference in variances between the two groups, so we can assume that the assumption of equal variances is valid for this data.