Brown–Forsythe test

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. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting F-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians. 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. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting F-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians. The transformed response variable is constructed to measure the spread in each group. Let where y ~ j {displaystyle { ilde {y}}_{j}} is the median of group j. The Brown–Forsythe test statistic is the model F statistic from a one way ANOVA on zij: 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 z ~ ⋅ j {displaystyle { ilde {z}}_{cdot j}} are the group means of the z i j {displaystyle z_{ij}} and z ~ ⋅ ⋅ {displaystyle { ilde {z}}_{cdot cdot }} is the overall mean of the z i j {displaystyle z_{ij}} . This F-statistic follows the F-distribution with degrees of freedom d 1 = p − 1 {displaystyle d_{1}=p-1} and d 2 = N − p {displaystyle d_{2}=N-p} under the null hypothesis. 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.