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Brunner–Munzel Test
The Brunner–Munzel test is used to test the stochastic equality of two independent samples, similar to the Wilcoxon–Mann–Whitney test. The Brunner–Munzel test is sometimes called the generalized Wilcoxon test.
The brunnermunzel.test function in the brunnermunzel package can be used for this test. There is also an option to conduct the test by permutation.
van Elteren test
The van Elteren test is a stratified version of the Wilcoxon–Mann–Whitney test. That is, it compares two groups across another variable representing strata.
The sanon package conducts a stratified Wilcoxon–Mann–Whitney test. It can also handle models with multiple stratifying variables.
A stratified test can also be conducted with the coin package.
Another approach is to use aligned ranks transformation anova.
Note that all these approaches are slightly different and return different results.
Packages used in this chapter
The packages used in this chapter include:
• brunnermunzel
• rcompanion
• sanon
• coin
• ARTool
The following commands will install these packages if they are not already installed:
if(!require(brunnermunzel)){install.packages("brunnermunzel")}
if(!require(rcompanion)){install.packages("rcompanion")}
if(!require(sanon)){install.packages("sanon")}
if(!require(coin)){install.packages("coin")}
if(!require(ARTool)){install.packages("ARTool")}
Brunner–Munzel test example
Data = read.table(header=TRUE, stringsAsFactors=TRUE, text="
Speaker Likert
Pooh 3
Pooh 5
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 4
Pooh 5
Pooh 5
Piglet 2
Piglet 4
Piglet 2
Piglet 2
Piglet 1
Piglet 2
Piglet 3
Piglet 2
Piglet 2
Piglet 3
")
Brunner Munzel test
Note that the results also give the Vargha and Delaney A statistic and confidence interval for this statistic.
library(brunnermunzel)
brunnermunzel.test(Likert ~ Speaker, data=Data)
Brunner-Munzel Test
Brunner-Munzel Test Statistic =
10.354, df = 12.688, p-value = 1.495e-07
95 percent confidence interval:
0.8558721 1.0441279
sample estimates:
P(X<Y)+.5*P(X=Y)
0.95
Brunner Munzel test by permutation
library(brunnermunzel)
brunnermunzel.test(Likert ~ Speaker, data=Data, perm=TRUE)
permuted Brunner-Munzel Test
p-value = 0.0002382
sample estimates:
P(X<Y)+.5*P(X=Y)
0.95
Comparison to the Wilcoxon–Mann–Whitney test
wilcox.test(Likert ~ Speaker, data=Data, correct=FALSE)
Wilcoxon rank sum test
W = 5, p-value = 0.0004065
Vargha and Delaney’s A
library(rcompanion)
1 - vda(Likert ~ Speaker, data=Data, ci=TRUE)
VDA lower.ci upper.ci
1 0.95 1 0.843
### Note these values essentially match the values
given by
### the brunnermunzel output, with some variability in the
### confidence interval for bootstrapping, and the fact that
### the brunnermunzel output allows for
values to exceed 1.
van Elteren test example
Data1 = read.table(header=TRUE, stringsAsFactors=TRUE, text="
Instructor Supplement Sodium
'Brendon Small' A 1200
'Brendon Small' A 1400
'Brendon Small' A 1350
'Brendon Small' A 950
'Brendon Small' A 1400
'Brendon Small' B 1150
'Brendon Small' B 1300
'Brendon Small' B 1325
'Brendon Small' B 1425
'Brendon Small' B 1500
'Brendon Small' C 1250
'Brendon Small' C 1150
'Brendon Small' C 950
'Brendon Small' C 1150
'Brendon Small' C 1600
'Brendon Small' D 1300
'Brendon Small' D 1050
'Brendon Small' D 1300
'Brendon Small' D 1700
'Brendon Small' D 1300
'Melissa Robins' A 900
'Melissa Robins' A 1100
'Melissa Robins' A 1150
'Melissa Robins' A 950
'Melissa Robins' A 1100
'Melissa Robins' B 1150
'Melissa Robins' B 1250
'Melissa Robins' B 1250
'Melissa Robins' B 1225
'Melissa Robins' B 1325
'Melissa Robins' C 1125
'Melissa Robins' C 1025
'Melissa Robins' C 950
'Melissa Robins' C 925
'Melissa Robins' C 1200
'Melissa Robins' D 1100
'Melissa Robins' D 950
'Melissa Robins' D 1300
'Melissa Robins' D 1400
'Melissa Robins' D 1100
")
xtabs(~ Instructor + Supplement, data=Data1)
Supplement
Instructor A B C D
Brendon Small 5 5 5 5
Melissa Robins 5 5 5 5
### Counts of observations per Instructor and Supplement
sanon package
library(sanon)
Out = sanon(Sodium ~ grp(Instructor) + strt(Supplement), data=Data1)
summary(Out)
Estimate Std.Err Chisq Pr(>Chisq)
Sodium -0.2650 0.0809 10.7 0.0011 **
Out
Sample size: 40
Strata ( Supplement ): A, B, C, D
Response levels:
[Sodium; 19 levels] (lower) 900, 925, 950, ..., 1500, 1600, 1700 (higher)
Design Matrix:
[,1]
Sodium 1
coin package
library(coin)
wilcox_test(Sodium ~ Instructor | Supplement, data=Data1)
Asymptotic Wilcoxon-Mann-Whitney Test
Instructor (Brendon Small, Melissa Robins)
stratified by Supplement
Z = 2.9858, p-value = 0.002829
Comparison to unstratified Wilcoxon-Mann-Whitney
wilcox_test(Sodium ~ Instructor, data=Data1)
Asymptotic Wilcoxon-Mann-Whitney Test
Z = 2.851, p-value = 0.004359
Aligned ranks transformation anova
library(ARTool)
model = art(Sodium ~ Instructor + Supplement + Instructor:Supplement,
data=Data1)
anova(model)
Analysis of Variance of Aligned Rank Transformed Data
Df Df.res F value Pr(>F)
1 Instructor 1 32 9.74449 0.0037985 **
2 Supplement 3 32 2.55669 0.0725482 .
3 Instructor:Supplement 3 32 0.44091 0.7253272
References
Kawaguchi, A, and Koch, G.G. 2015. sanon: An R Package for Stratified Analysis with Nonparametric Covariable Adjustment. JSS Journal of Statistical Software: 67(9). doi: 10.18637/jss.v067.