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.

SAS
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.


R
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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