Hypothesis Tests

Levene Test (Brown-Forsythe)

Robust test for equality of variances of two samples, insensitive to non-normality.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 2 Variances (Levene's Test)

Description

The Levene test in the Brown-Forsythe variant tests equality of variances by comparing absolute deviations from the respective medians. It is much more robust than the F-test against departures from normality and is therefore the preferred choice when normality cannot be assumed.

Formulas

W = \frac{( N - k ) \sum _{i = 1}^{k} n _{i} ( z ˉ _{i \cdot} - z ˉ _{\cdot\cdot} ) ^{2}}{( k - 1 ) \sum _{i = 1}^{k} \sum _{j = 1}^{n_{i}} ( z _{ij} - z ˉ _{i \cdot} ) ^{2}}

Test statistic W with z_{ij} = |x_{ij} − Median_i| (Brown-Forsythe variant)

H₀

H_{0} : σ_{1}^{2} = σ_{2}^{2}

Null hypothesis: The population variances are equal

Assumptions

Samples are independent
Continuous data

Limitations

Less powerful than the F-test when data are truly normal

References

Brown, M. B. & Forsythe, A. B. (1974), Robust tests for the equality of variances, JASA, 69(346), 364–367
Levene, H. (1960), Robust tests for equality of variances, in Olkin et al. (Eds.), Contributions to Probability and Statistics