The assumption of equal variances among the groups in analysis of variance is an expression of the assumption of homoscedasticity for linear models more generally. For ANOVA, this assumption can be tested via Levene’s test. The test is a function of the residuals and means within each group, though various modifications are used, including the Brown-Forsythe test. This uses the medians within group, rather than the mean, and is recommended when normality may be suspect.
We illustrate using the HELP data set available here, modeling depressive symptoms (assessed via CESD) as a function of abused substance.
In SAS, the tests are available as an option to the means statement in proc glm
data help; set "C:\book\help.sas7bdat"; run; proc glm data = help; class substance; model cesd = substance; means substance / hovtest=levene(type=abs) hovtest=bf; run;
The two requested tests are a version of Levene’s test that is produced in R, below, and the aforementioned Brown-Forsythe test. The relevant results are shown below.
Levene's Test for Homogeneity of CESD Variance ANOVA of Absolute Deviations from Group Means Sum of Mean Source DF Squares Square F Value Pr > F SUBSTANCE 2 272.4 136.2 2.61 0.0747 Error 450 23480.7 52.1793 Brown and Forsythe's Test for Homogeneity of CESD Variance ANOVA of Absolute Deviations from Group Medians Sum of Mean Source DF Squares Square F Value Pr > F SUBSTANCE 2 266.0 133.0 2.46 0.0864 Error 450 24310.9 54.0243
There’s some suggestion of a lack of homoscedasticity; it might be wise to consider methods robust to violations of this assumption.
In R, the test can be found in the levene.test() function in the lawstat package.
help = read.csv("http://www.math.smith.edu/r/data/help.csv") library(lawstat) with(help, levene.test(cesd, as.factor(substance), location="mean")) classical Levene's test based on the absolute deviations from the mean ( none not applied because the location is not set to median ) data: cesd Test Statistic = 2.6099, p-value = 0.07465 with(help, levene.test(cesd, as.factor(substance),location="median")) modified robust Brown-Forsythe Levene-type test based on the absolute deviations from the median data: cesd Test Statistic = 2.462, p-value = 0.08641
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