Brown-Forsythe Test
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A Brown-Forsythe Test is a statistical test for the equality of group variances.
- Context:
- It is similar to the Levene's test except, this uses the median instead of the mean.
- …
- Counter-Example(s):
- See: Analysis of variance, Statistical Test, Bartlett's Test, F-test of Equality of Variances, Levene’s Test.
References
2016
- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Brown%E2%80%93Forsythe_test Retrieved 2016-07-30
- 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.
- Transformation
- The transformed response variable is constructed to measure the spread in each group. Let
- [math]\displaystyle{
z_{ij}=\left\vert y_{ij} - \tilde{y}_j \right\vert
}[/math]
- where [math]\displaystyle{ \tilde{y}_j }[/math] is the median of group j. The Brown–Forsythe test statistic is the model F statistic from a one way ANOVA on zij:
- [math]\displaystyle{ F = \frac{(N-p)}{(p-1)} \frac{\sum_{j=1}^{p} n_j (\tilde{z}_{\cdot j}-\tilde{z}_{\cdot\cdot})^2} {\sum_{j=1}^{p}\sum_{i=1}^{n_j} (z_{ij}-\tilde{z}_{\cdot j})^2} }[/math]
- where p is the number of groups, nj is the number of observations in group j, and N is the total number of observations. Also [math]\displaystyle{ \tilde{z}_{\cdot j} }[/math] are the group means of the [math]\displaystyle{ z_{ij} }[/math] and [math]\displaystyle{ \tilde{z}_{\cdot\cdot} }[/math] is the overall mean of the [math]\displaystyle{ z_{ij} }[/math].
- If the variances are indeed heterogeneous, techniques that allow for this (such as the Welch one-way ANOVA) may be used instead of the usual ANOVA.
- Good [1994,2005], noting that the deviations are linearly dependent, has modified the test so as to drop the redundant deviations.