# Levene and Brown-Forsythe Tests for Homogeneity of Variances (HOV)

An important assumption in analysis of variance (ANOVA and the t-test for mean differences) is that the variances in the different groups are equal (homogeneous). Two powerful and commonly used tests of this assumption are the Levene test and the Brown-Forsythe modification of this test. However, it is important to realize that 1) the homogeneity of variances assumption is usually not as crucial as other assumptions for ANOVA, in particular in the case of balanced (equal n) designs (see also ANOVA: Homogeneity of Variances and Covariances), and 2) that the tests described below are not necessarily very robust themselves (e.g., Glass and Hopkins, 1996, p. 436, call these tests "fatally flawed;" see also the description of these tests below). If you are concerned about a violation of the HOV assumption, it is always advisable to repeat the key analyses using nonparametric methods.

Levene's test (homogeneity of variances). For each dependent variable, an analysis of variance is performed on the absolute deviations of values from the respective group means. If the Levene test is statistically significant, the hypothesis of homogeneous variances should be rejected.

Brown & Forsythe's test (homogeneity of variances). Recently, some authors (e.g., Glass and Hopkins, 1996) have called into question the power of the Levene test for unequal variances. Specifically, the absolute deviation (from the group means) scores can be expected to be highly skewed; thus, the normality assumption for the ANOVA of those absolute deviation scores is usually violated. This poses a particular problem when there is unequal n in the two (or more) groups that are to be compared. A more robust test that is very similar to the Levene test has been proposed by Brown and Forsythe (1974). Instead of performing the ANOVA on the deviations from the mean, one can perform the analysis on the deviations from the group medians. Olejnik and Algina (1987) have shown that this test will give quite accurate error rates even when the underlying distributions for the raw scores deviate significantly from the normal distribution. However, as Glass and Hopkins (1996, p. 436) have pointed out, both the Levene test and the Brown-Forsythe modification suffer from what those authors call a "fatal flaw," namely, that both tests rely on the homogeneity of variances assumption (of the absolute deviations from the means or medians); and hence, it is not clear how robust these tests are themselves in the presence of significant variance heterogeneity and unequal n.