Brown-Forsythe Test for Equality of Means

The Brown-Forsythe test for equality of means is a one-way ANOVA alternative that divides the between-group sum of squares by a variance-weighted error term, allowing unequal group variances.

평균 동질성에 대한 Brown-Forsythe 검정은 집단 간 제곱합을 분산 가중 오차항으로 나누어, 분산이 달라도 셋 이상 집단의 평균을 비교하는 분산분석 대안이다.

Is there an alternative to Welch's ANOVA for comparing means when the group variances differ?

분산이 다른 집단의 평균 비교에서 Welch 말고 다른 방법도 있나요?

The Brown-Forsythe F* statistic uses the same numerator as the classic ANOVA but a denominator built from each group's variance weighted by one minus its share of the sample. Its denominator degrees of freedom come from a Satterthwaite approximation.

When to use Brown-Forsythe Test for Equality of Means

• Several independent group means are compared under unequal variances.
• A heteroscedasticity-robust check of a one-way ANOVA result is needed.
• Reviewers ask for both Welch and Brown-Forsythe results.

When to use something else

• Only the equality of variances is of interest; use Levene's test with the median (the Brown-Forsythe variance test).
• Data are heavily skewed with outliers; use a robust ANOVA on trimmed means.

Assumptions and requirements

• Observations are independent.
• The outcome is approximately normal within each group.
• Each group has at least two observations.

What the analysis reports

• F* statistic
• Numerator and Satterthwaite denominator degrees of freedom
• p value
• Group means and variances

How to report it

The Brown-Forsythe test for equality of means was significant, F*([df1], [df2]) = [F], p = [p].

Related methods

• Welch's ANOVA
• One way ANOVA
• Levene's Test

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