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Levene's and Brown-Forsythe's tests

for homogeneity of variance

Levene's test  Brown-Forsythe's tests 

Levene's test

Levene's test is one of the more widely used tests of homogeneity of variances carried out prior to performing an analysis of variance. It tests the null hypothesis that the population variances are equal by carrying out an analysis of variance on the absolute deviations of observations from the group mean.

The test statistic is an F-ratio calculated as below:

Algebraically speaking -

F   =   (N − k) Σ ni (Zi − Z..)2
(k − 1) Σ (Zij − Zi)2
where
  • ni is the number of observations in each group,
  • k is the number of groups,
  • N is the total number of observations,
  • Zij are the absolute deviations (|Yij -i|) where i is the mean of group i,
  • Z.. is the mean of all the absolute deviations (Zij),
  • Zi is the mean of the absolute deviations (Zij) for group i.

The F-ratio is tested against the upper critical value of the F distribution with k-1 and N-k degrees of freedom.

The (original) Levene's test (detailed above) gives the best power for symmetric, moderate tailed distributions. However, such distributions tend to be rather rare for biological data - so generally the Brown-Forsythe versions of the test are preferred.

 

 

Brown-Forsythe's tests

Levene's test was extended by Brown and Forsythe (1974). Instead of carrying out the ANOVA on absolute deviations from the mean of each group, it is done on the absolute deviations of observations from either the median or the 10% trimmed mean of each group.

Algebraically speaking -

F   =   (N − k) Σ ni (Zi − Z..)2
(k − 1) Σ (Zij − Zi)2
where
  • Zij are the absolute deviations either from the median of group i (|Yij -Y0.5i|) or from the 10% trimmed mean of group i (|Yij -Yi'|),
  • and all other symbols are as given for Levene's test.

These modifications make the test more robust to nonnormality. The use of deviations from the median is recommended for skewed distributions.