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The Brown-Forsythe test and the Levene test

Both tests: the Levene test (Levene, 1960 ¹⁾) and the Brown-Forsythe test (Brown and Forsythe, 1974 ²⁾) are used to verify the hypothesis determining the equality of variance of an analysed variable in several ( $k>=2$ ) populations.

Basic assumptions:

measurement on an interval scale,
normality of distribution of an analysed feature in each population,
an independent model.

Hypotheses:

$\begin{array}{cl} \mathcal{H}_0: & \sigma_1^2=\sigma_2^2=...=\sigma_k^2,\\ \mathcal{H}_1: & $not all $\sigma_j^2$ are equal $(j=1,2,...,k)$,$ \end{array}$

where:

$\sigma_1^2$ , $\sigma_2^2$ ,…, $\sigma_k^2$ $-$ variances of an analysed variable of each population.

The analysis is based on calculating the absolute deviation of measurement results from the mean (in the Levene test) or from the median (in the Brown-Forsythe test), in each of the analysed groups. This absolute deviation is the set of data which are under the same procedure performed to the analysis of variance for independent groups. Hence, the test statistic is defined by:

$\begin{displaymath} F=\frac{MS_{BG}}{MS_{WG}}, \end{displaymath}$

The test statistic has the F Snedecor distribution with $df_{BG}$ and $df_{WG}$ degrees of freedom. The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

Note

The Brown-Forsythe test is less sensitive than the Levene test, in terms of an unfulfilled assumption relating to distribution normality.

The settings window with the Levene, Brown-Forsythe tests' can be opened in Statistics menu→Parametric tests→Levene, Brown-Forsythe.

¹⁾

Levene H. (1960), Robust tests for the equality of variance. In I. Olkin (Ed.) Contributions to probability and statistics (278-292). Palo Alto, CA: Stanford University Press

²⁾

Brown M.B., Forsythe A. B. (1974a), Robust tests for equality of variances. Journal of the American Statistical Association, 69,364-367

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The Brown-Forsythe test and the Levene test

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