Levene's test

In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). If the resulting p-value of Levene's test is less than some significance level (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.

Some of the procedures typically assuming homoscedasticity, for which one can use Levene's tests, include analysis of variance and t-tests.

Levene's test is often used before a comparison of means. When Levene's test shows significance, one should switch to more generalized tests that is free from homoscedasticity assumptions (sometimes even non-parametric tests).

Levene's test may also be used as a main test for answering a stand-alone question of whether two sub-samples in a given population have equal or different variances.

The test statistic, W, is defined as follows:

where

(Both definitions are in use though the second one is, strictly speaking, the Brown–Forsythe test – see below for comparison.)

The significance of ${\displaystyle W}$ is tested against ${\displaystyle F(\alpha ,k-1,N-k)}$ where ${\displaystyle F}$ is a quantile of the F-test distribution, with ${\displaystyle k-1}$ and ${\displaystyle N-k}$ its degrees of freedom, and ${\displaystyle \alpha }$ is the chosen level of significance (usually 0.05 or 0.01).

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