This brunnermunzel package is to perform (permuted)
Brunner-Munzel test for stochastic equality of two samples, which is
also known as the Generalized Wilcoxon test.
For Brunner-Munzel test (Brunner and Munzel 2000),
brunner.munzel.test function in lawstat
package is very famous. This function is extended to enable to use
formula, matrix, and
table as an argument.
Also, the function brunnermunzel.permutation.test for
permuted Brunner-Munzel test (Neubert and Brunner 2007) was provided.
brunnermunzel packageIn this section, we will use sample data from Hollander & Wolfe (1973), 29f. – Hamilton depression scale factor measurements in 9 patients with mixed anxiety and depression, taken at the first (x) and second (y) visit after initiation of a therapy (administration of a tranquilizer)“.
x <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)For formula interface, data was converted to
data.frame.
dat <- data.frame(
value = c(x, y),
group = factor(rep(c("x", "y"), c(length(x), length(y))),
levels = c("x", "y")))library(dplyr)
dat %>%
group_by(group) %>%
summarize_all(list(mean = mean, median = median))
#> # A tibble: 2 x 3
#> group mean median
#> <fct> <dbl> <dbl>
#> 1 x 1.77 1.68
#> 2 y 1.33 1.06library(brunnermunzel)
brunnermunzel.test(x, y)
#>
#> Brunner-Munzel Test
#>
#> data: x and y
#> Brunner-Munzel Test Statistic = -1.4673, df = 15.147, p-value = 0.1628
#> 95 percent confidence interval:
#> -0.02962941 0.59753064
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.2839506
brunnermunzel.test(value ~ group, data = dat)
#>
#> Brunner-Munzel Test
#>
#> data: value by group
#> Brunner-Munzel Test Statistic = -1.4673, df = 15.147, p-value = 0.1628
#> 95 percent confidence interval:
#> -0.02962941 0.59753064
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.2839506To perform permuted Brunner-Munzel test, use
brunnermunzel.test with “perm = TRUE” option,
or brunnermunzel.permutation.test function. This
“perm” option is used in also formula interface, matrix,
and table.
When perm is TRUE,
brunnermunzel.test calls
brunnermunzel.permutation.test in internal.
brunnermunzel.test(x, y, perm = TRUE)
#>
#> permuted Brunner-Munzel Test
#>
#> data: x and y
#> p-value = 0.1581
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.2839506
brunnermunzel.permutation.test(x, y)
#>
#> permuted Brunner-Munzel Test
#>
#> data: x and y
#> p-value = 0.1581
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.2839506Because statistics in all combinations are calculated in permuted Brunner-Munzel test (\({}_{n_{x}+n_{y}}C_{n_{x}}\) where \(n_{x}\) and \(n_{y}\) are sample size of \(x\) and \(y\), respectively), it takes a long time to obtain results.
Therefore, when sample size is too large [the number of combination
is more than 40116600 (\(=\)
choose(28, 14))], it switches to Brunner-Munzel test
automatically.
# sample size is 30
brunnermunzel.permutation.test(1:15, 3:17)
#> Warning in brunnermunzel.permutation.test.default(1:15, 3:17): Sample number is too large. Using 'brunnermunzel.test'
#>
#> Brunner-Munzel Test
#>
#> data: x and y
#> Brunner-Munzel Test Statistic = 1.1973, df = 28, p-value = 0.2412
#> 95 percent confidence interval:
#> 0.4115330 0.8373559
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.6244444force optionWhen you want to perform permuted Brunner-Munzel test regardless
sample size, you add “force = TRUE” option to
brunnermunzel.permutation test.
brunnermunzel.permutation.test(1:15, 3:17, force = TRUE)
#>
#> permuted Brunner-Munzel Test
#>
#> data: 1:15 and 3:17
#> p-value = 0.2341alternative optionbrunnermunzel.test also can use
“alternative” option as well as t.test and
wilcox.test functions.
To test whether the average rank of group \(x\) is greater than that of group \(y\), alternative = "greater"
option is added. In contrast, to test whether the average rank of group
\(x\) is lesser than that of group
\(y\),
alternative = "less" option is added.
The results of Brunner-Munzel test and Wilcoxon sum-rank test
(Mann-Whitney test) with alternative = "greater" option are
shown. In this case, median of \(x\) is
1.68, and median of \(y\) is 1.06.
brunnermunzel.test(x, y, alternative = "greater")
#>
#> Brunner-Munzel Test
#>
#> data: x and y
#> Brunner-Munzel Test Statistic = -1.4673, df = 15.147, p-value = 0.08138
#> 95 percent confidence interval:
#> -0.02962941 0.59753064
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.2839506
wilcox.test(x, y, alternative = "greater")
#> Warning in wilcox.test.default(x, y, alternative = "greater"): cannot compute
#> exact p-value with ties
#>
#> Wilcoxon rank sum test with continuity correction
#>
#> data: x and y
#> W = 58, p-value = 0.06646
#> alternative hypothesis: true location shift is greater than 0When using formula, brunnermunzel.test with
alternative = "greater" option tests an alternative
hypothesis “1st level is greater than 2nd level”.
In contrast, brunnermunzel.test with
alternative = "less" option tests an alternative hypothesis
“1st level is lesser than 2nd level”.
dat$group
#> [1] x x x x x x x x x y y y y y y y y y
#> Levels: x ybrunnermunzel.test(value ~ group, data = dat, alternative = "greater")$p.value
#> [1] 0.08137809
wilcox.test(value ~ group, data = dat, alternative = "greater")$p.value
#> Warning in wilcox.test.default(x = DATA[[1L]], y = DATA[[2L]], ...): cannot
#> compute exact p-value with ties
#> [1] 0.06645973brunnermunzel.test(x, y, alternative = "less")$p.value
#> [1] 0.9186219
wilcox.test(x, y, alternative = "less")$p.value
#> Warning in wilcox.test.default(x, y, alternative = "less"): cannot compute exact
#> p-value with ties
#> [1] 0.9442044est optionNormally, brunnermunzel.test and
brunnermunzel.permutation test return the estimate \(P(X<Y) + 0.5 \times P(X=Y)\). When
‘est = "difference"’ option is used, these functions return
mean difference [\(P(X<Y) -
P(X>Y)\)] in estimate and confidence interval.
Note that \(P(X<Y) - P(X>Y) = 2p - 1\) when \(p = P(X<Y) + 0.5 \times P(X=Y)\).
This change is proposed by Dr. Julian D. Karch.
brunnermunzel.test(x, y, est = "difference")
#>
#> Brunner-Munzel Test
#>
#> data: x and y
#> Brunner-Munzel Test Statistic = -1.4673, df = 15.147, p-value = 0.1628
#> 95 percent confidence interval:
#> -1.0592588 0.1950613
#> sample estimates:
#> P(X<Y)-P(X>Y)
#> -0.4320988
brunnermunzel.permutation.test(x, y, est = "difference")
#>
#> permuted Brunner-Munzel Test
#>
#> data: x and y
#> p-value = 0.1581
#> sample estimates:
#> P(X<Y)-P(X>Y)
#> -0.4320988In some case, data is provided as aggregated table. Both
brunnermunzel.test and
brunnermunzel.permutation.test accept data of
matirix and table class.
| Normal | Moderate | Severe | |
|---|---|---|---|
| A | 5 | 3 | 2 |
| B | 1 | 3 | 6 |
dat1 <- matrix(c(5, 3, 2, 1, 3, 6), nr = 2, byrow = TRUE)
dat2 <- as.table(dat1)
colnames(dat2) <- c("Normal", "Moderate", "Severe")dat1 # matrix class
#> [,1] [,2] [,3]
#> [1,] 5 3 2
#> [2,] 1 3 6
dat2 # table class
#> Normal Moderate Severe
#> A 5 3 2
#> B 1 3 6brunnermunzel.test(dat1)
#>
#> Brunner-Munzel Test
#>
#> data: Group1 and Group2
#> Brunner-Munzel Test Statistic = 2.4447, df = 17.394, p-value = 0.02542
#> 95 percent confidence interval:
#> 0.5359999 0.9840001
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.76
brunnermunzel.test(dat2)
#>
#> Brunner-Munzel Test
#>
#> data: A and B
#> Brunner-Munzel Test Statistic = 2.4447, df = 17.394, p-value = 0.02542
#> 95 percent confidence interval:
#> 0.5359999 0.9840001
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.76brunnermunzel.permutation.test(dat1)
#>
#> permuted Brunner-Munzel Test
#>
#> data: Group1 and Group2
#> p-value = 0.05116
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.76
brunnermunzel.permutation.test(dat2)
#>
#> permuted Brunner-Munzel Test
#>
#> data: A and B
#> p-value = 0.05116
#> sample estimates:
#> P(X<Y)+.5*P(X=Y)
#> 0.76brunnermunzel.test functionbrunnermunzel.test function is derived from
brunner.munzel.test function in lawstat
package (Maintainer of this package is Vyacheslav Lyubchich; License is
GPL-2 or GPL-3) with modification. The authors of this function are
Wallace Hui, Yulia R. Gel, Joseph L. Gastwirth and Weiwen Miao.
combination subroutine by FORTRAN77FORTRAN subroutine combination in combination.f is
derived from the program by shikino (http://slpr.sakura.ne.jp/qp/combination)(CC-BY-4.0) with
slight modification.
Without this subroutine, I could not make
brunnermunzel.permutation.test. Thanks to shikono for your
useful subroutine.