What's the feature?
It's known that the U statistic (Mann-Whitney and the Wilcoxon version) can "run into trouble" due to unequal variances and fail to maintain the nominal significance level and power, just the same way the classic t-test. To address it, Brunner and Munzel proposed their studentised version of the MWW test the same way Welch and Satterthwaite (and Janssen - for the permutation version) proposed theirs for the Gosset's t statistic.
Both Welch-t (Lakens, 2017) and Brunner-Munzel (Karch, 2021) have been recommended for replacing their counterparts classic t and MWW in daily use.
This matters also in randomized studies. While randomization enables statistical testing under true (virtual) H0, but it does not guarantee that -in a concrete realization- the arms will be balanced in any kind of meaning, including dispersions.
Brunner-Munzel is about the same quantity as MWW - stochastic superiority (aka probabilistic index), i.e. P(B>A)+0.5*P(B=A), only accounting for unequal dispersions (probably I should rather say "scales", but I hope we can roughly stay with dispersions expressed by variances) for inference.
In R the most common implementation can be found in the brunnermunzel package.
> (tmp <- brunnermunzel::brunnermunzel.test(runif(10), rnorm(10)))
Brunner-Munzel Test
data: runif(10) and rnorm(10)
Brunner-Munzel Test Statistic = -0.83653, df = 10.642, p-value = 0.4212
95 percent confidence interval:
0.06296806 0.69703194
sample estimates:
P(X<Y)+.5*P(X=Y)
0.38
> str(tmp)
List of 7
$ method : chr "Brunner-Munzel Test"
$ statistic: Named num -0.837
..- attr(*, "names")= chr "Brunner-Munzel Test Statistic"
$ data.name: chr "runif(10) and rnorm(10)"
$ parameter: Named num 10.6
..- attr(*, "names")= chr "df"
$ estimate : Named num 0.38
..- attr(*, "names")= chr "P(X<Y)+.5*P(X=Y)"
$ p.value : num 0.421
$ conf.int : Named num [1:2] 0.063 0.697
..- attr(*, "names")= chr [1:2] "lower" "upper"
..- attr(*, "conf.level")= num 0.95
- attr(*, "class")= chr "htest"Luckily, both permuted and asymptotic versions return a htest-class result, so broom::tidy() can handle it:
> broom::tidy(tmp <- brunnermunzel::brunnermunzel.test(runif(10), rnorm(10)))
# A tibble: 1 × 7
estimate statistic p.value parameter conf.low.lower conf.high.upper method
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <chr>
1 0.57 0.482 0.640 10.9 0.250 0.890 Brunner-Munzel Test
> broom::tidy(tmp <- brunnermunzel::brunnermunzel.permutation.test(runif(10), rnorm(10)))
# A tibble: 1 × 3
estimate p.value method
<dbl> <dbl> <chr>
1 0.32 0.237 permuted Brunner-Munzel Test
What's the feature?
It's known that the U statistic (Mann-Whitney and the Wilcoxon version) can "run into trouble" due to unequal variances and fail to maintain the nominal significance level and power, just the same way the classic t-test. To address it, Brunner and Munzel proposed their studentised version of the MWW test the same way Welch and Satterthwaite (and Janssen - for the permutation version) proposed theirs for the Gosset's t statistic.
Both Welch-t (Lakens, 2017) and Brunner-Munzel (Karch, 2021) have been recommended for replacing their counterparts classic t and MWW in daily use.
This matters also in randomized studies. While randomization enables statistical testing under true (virtual) H0, but it does not guarantee that -in a concrete realization- the arms will be balanced in any kind of meaning, including dispersions.
Brunner-Munzel is about the same quantity as MWW - stochastic superiority (aka probabilistic index), i.e. P(B>A)+0.5*P(B=A), only accounting for unequal dispersions (probably I should rather say "scales", but I hope we can roughly stay with dispersions expressed by variances) for inference.
In R the most common implementation can be found in the brunnermunzel package.
Luckily, both permuted and asymptotic versions return a
htest-class result, so broom::tidy() can handle it: