# When to use Brown-Forsythe Test?

I have been researching the differences between Welch ANOVA and Brown-Forsythe Test. I know that Welch ANOVA is used for more than two groups comparing whether there is statistically meaningful difference in their means when variance of homogenity assumption is not held.

My question is, is Brown-Forsythe Test an alternative to Welch ANOVA when homogenity of variances assumption is not met, or is it an alternative test for Levene Test for comparing homogenity of variances when normality assumption is not met?

Many sources say different things about this test. Nearly half of the sources approve that Brown-Forsythe Test is used for testing equality of means, and the other part approves that it is used for testing equality of variances.

Any help would be greatly appreciated.

• Which sources and what do they actually say? Nov 20, 2015 at 7:29
• As in answer by user95572 . Brown-Forsythe wrote about anova for both equality of means and for equality of variances in 1974, so BF 1974 can refer to either test or either article. May 21, 2020 at 21:11

Levene's test for homogeneity of variance involves determining an absolute deviation score from group means for each of the n, while Brown-Forsythe's test for homogeneity of variance involves determining an absolute deviation score from group medians for each of the n.

From there, both use a single-factor between-subjects analysis of variance to contrast the means of the deviation scores.

Both are robust against normality assumption violations. However Hartley's F max Test, another popular homogeneity of variance test, is not.

From: Sheskin, D. (2011). Handbook of parametric and nonparametric statistical procedures: 11th edition. Boca Raton: Chapman & Hall/CRC.

Struggling with this myself. It's a very late answer, but in case anyone comes across this thread maybe it'll help.

Levene's test always comes up as a variance comparison test.

A modification to Leven's test is Brown-Forsythe (the whole mean vs median thing) so it also appears under the name Levene's modified test = Brown-Forsthe variance comparison test.

There is also a a Brown-Forsthe test for equality of means that is the next step when variance assumptions are violated. The result here is not the same as the results of a Brown-Forsythe test of variance.

I'm struggling to make these comparisons myself since I don't have all the related software, but so far it looks like:

In SPSS, ANOVA with the Brown-Forsythe option selected gives you the equality of means test

For Brown-Forsythe variance test the following programs do this:

In SAS; hovtest The HH library in R; hovBF, The lawstat library you can specify using median or mean; levene.test(measurements,category,location="median") = Brown-Forsythe variance levene.test(measurements,category,location="mean") for straight up Levene's variance.

UPDATE Just found an SPSS script that provides both Levene's as well as Brown-Forsythe tests of variance

EXAMINE VARIABLES=ZCu BY Location
/COMPARE GROUPS
/STATISTICS DESCRIPTIVES
/CINTERVAL 95
/MISSING LISTWISE
/NOTOTAL


-Output table for Test of Homogeneity of Variance - means is Levene and median is Brown-Forsythe test

Confirmed using GraphPad Prism Brown-Forsythe test (which is also for variance)

• After you merge, it would be useful if you could locate a reference for the mean comparison you mention. Nov 20, 2015 at 7:31
• BF for equality of means is Brown, Morton B., and Alan B. Forsythe. "The small sample behavior of some statistics which test the equality of several means." Technometrics 16, no. 1 (1974): 129-132. They wrote also the variance anova in 1974 May 21, 2020 at 21:07

Just found an SPSS script that provides Levene as wells as Brown-Forsythe tests of variance

EXAMINE VARIABLES=ZCu BY Location