Brunner-Munzel Test

The Brunner-Munzel test compares two independent groups through the relative effect P(X < Y) + 0.5 P(X = Y) without assuming equal variances or equal distribution shapes.

Brunner-Munzel 검정은 등분산이나 같은 분포 모양을 가정하지 않고 상대효과 P(X<Y)+0.5P(X=Y)로 두 독립 집단을 비교하는 비모수 검정입니다.

My two groups have very different spreads; is one group stochastically larger than the other?

두 집단의 퍼짐 정도가 크게 다른데, 한 집단의 값이 다른 집단보다 확률적으로 더 큰지 알 수 있나요?

The Brunner-Munzel test estimates the probability that a value from the first group is smaller than one from the second, counting ties as one half, and tests whether it differs from 0.5. Its standard error comes from rank placements in each group, so unlike the Mann-Whitney test it remains valid when variances or shapes differ.

When to use Brunner-Munzel Test

• Two independent groups have unequal variances or different distribution shapes.
• The outcome is ordinal or skewed and you want a probability-scale effect size.
• You would otherwise use a Mann-Whitney test but its equal-shape assumption is doubtful.

When to use something else

• Each group has fewer than about 10 observations; use the permutation version of the test.
• Data are normal and you care about mean differences; use Welch's t test.

Assumptions and requirements

• Observations are independent within and between groups.
• The outcome is at least ordinal.
• Groups are large enough for the t approximation (about 10 or more each).

What the analysis reports

• Brunner-Munzel statistic with Satterthwaite-type degrees of freedom and p value
• Relative effect p̂ with its confidence interval
• Median and mean rank for each group

How to report it

A Brunner-Munzel test found a relative effect p̂ = [value], 95% CI [[value], [value]], W([df]) = [value], p = [value].

Related methods

• Mann-Whitney U Test
• Welch's t-test
• Permutation Test for Two Groups

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