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.