Monday, June 25, 2012

Example 9.36: Levene's test for equal variances

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";

proc glm data = help;
class substance;
model cesd = substance;
means substance / hovtest=levene(type=abs) hovtest=bf;

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("")
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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nacnudus said...

I note you force SAS to use type=abs, rather than the default type=square (of the residuals). Is there any way to use the squared residuals in the R Levene test?

Ken Kleinman said...

Not in the function that's provided with the lawstat package, as far as I know. There may well be implementations in other packages, and it's likely that generating the test from scratch would not be too difficult, especially if you begin with the code included in the lawstat package.