Brown–Forsythe test

In statistics, when a usual one-way ANOVA is performed, it is assumed that the group variances are statistically equal. If this assumption is not valid, then the resulting F-test is invalid. The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an ANOVA on a transformation of the response variable. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations from the median.

The transformed response variable is constructed to measure the spread in each group. Let

where ${\displaystyle {\tilde {y}}_{j}}$ is the median of group j. The Brown–Forsythe test statistic is the model F statistic from a one way ANOVA on z_ij:

where p is the number of groups, n_j is the number of observations in group j, and N is the total number of observations. Also ${\displaystyle {\tilde {z}}_{\cdot j}}$ are the group means of the ${\displaystyle z_{ij}}$ and ${\displaystyle {\tilde {z}}_{\cdot \cdot }}$ is the overall mean of the ${\displaystyle z_{ij}}$.

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