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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.
Stratified Kruskal–Wallis test
A stratified Kruskal–Wallis test can be used to compare multiple groups across another stratifying variable. This test can be conducted with the free1way function or with the kruskal_test function in the coin package.
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.
Stratified Wilcoxon–Mann–Whitney 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
free1way function
free1way(Sodium ~ Instructor | Supplement, data=Data1)
Stratified 2-sample Wilcoxon test against proportional
odds alternatives
Perm Z = -2.8086, p-value = 0.004976
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
Stratified Kruskal–Wallis example
Data2 = read.table(header=TRUE, stringsAsFactors=TRUE, text="
Instructor Town Sodium
'Brendon Small' Squiggleville 1200
'Brendon Small' Squiggleville 1400
'Brendon Small' Squiggleville 1350
'Brendon Small' Metalocalypse 950
'Brendon Small' Squiggleville 1400
'Brendon Small' Squiggleville 1150
'Brendon Small' Squiggleville 1300
'Brendon Small' Metalocalypse 1325
'Brendon Small' Metalocalypse 1425
'Brendon Small' Squiggleville 1500
'Brendon Small' Squiggleville 1250
'Brendon Small' Metalocalypse 1150
'Brendon Small' Metalocalypse 950
'Brendon Small' Squiggleville 1150
'Brendon Small' Metalocalypse 1600
'Brendon Small' Metalocalypse 1300
'Brendon Small' Metalocalypse 1050
'Brendon Small' Metalocalypse 1300
'Brendon Small' Squiggleville 1700
'Brendon Small' Squiggleville 1300
'Coach McGuirk' Squiggleville 1100
'Coach McGuirk' Squiggleville 1200
'Coach McGuirk' Squiggleville 1250
'Coach McGuirk' Metalocalypse 1050
'Coach McGuirk' Metalocalypse 1200
'Coach McGuirk' Metalocalypse 1250
'Coach McGuirk' Squiggleville 1350
'Coach McGuirk' Squiggleville 1350
'Coach McGuirk' Squiggleville 1325
'Coach McGuirk' Squiggleville 1525
'Coach McGuirk' Squiggleville 1225
'Coach McGuirk' Squiggleville 1125
'Coach McGuirk' Metalocalypse 1000
'Coach McGuirk' Metalocalypse 1125
'Coach McGuirk' Squiggleville 1400
'Coach McGuirk' Metalocalypse 1200
'Coach McGuirk' Squiggleville 1150
'Coach McGuirk' Squiggleville 1400
'Coach McGuirk' Squiggleville 1500
'Coach McGuirk' Squiggleville 1200
'Melissa Robins' Metalocalypse 900
'Melissa Robins' Metalocalypse 1100
'Melissa Robins' Metalocalypse 1150
'Melissa Robins' Metalocalypse 950
'Melissa Robins' Metalocalypse 1100
'Melissa Robins' Metalocalypse 1150
'Melissa Robins' Squiggleville 1250
'Melissa Robins' Squiggleville 1250
'Melissa Robins' Squiggleville 1225
'Melissa Robins' Squiggleville 1325
'Melissa Robins' Metalocalypse 1125
'Melissa Robins' Metalocalypse 1025
'Melissa Robins' Metalocalypse 950
'Melissa Robins' Metalocalypse 925
'Melissa Robins' Squiggleville 1200
'Melissa Robins' Metalocalypse 1100
'Melissa Robins' Metalocalypse 950
'Melissa Robins' Metalocalypse 1300
'Melissa Robins' Squiggleville 1400
'Melissa Robins' Metalocalypse 1100
")
### Order factors by the order in data frame
### Otherwise, R will alphabetize them
Data2$Instructor = factor(Data2$Instructor,
levels=unique(Data2$Instructor))
Data2$Town = factor(Data2$Town,
levels=unique(Data2$Town))
coin package
library(coin)
kruskal_test(Sodium ~ Instructor | Town, data=Data2)
Asymptotic Kruskal-Wallis Test
chi-squared = 6.2652, df = 2, p-value = 0.04361
free1way function
free1way(Sodium ~ Instructor | Town, data=Data2)
Stratified 3-sample Kruskal-Wallis test against
proportional odds alternatives
Perm chi-squared = 4.4812, df = 2, p-value = 0.1064
Aligned ranks transformation anova
library(ARTool)
model = art(Sodium ~ Instructor + Town + Instructor:Town, data=Data2)
anova(model)
Analysis of Variance of Aligned Rank Transformed Data
Df Df.res F value Pr(>F)
1 Instructor 2 54 6.05487 0.004241 **
2 Town 1 54 22.55388 1.5476e-05 ***
3 Instructor:Town 2 54 0.59009 0.557809
Compare to unstratified Kruskal–Wallis
kruskal_test(Sodium ~ Instructor, data=Data2)
Asymptotic Kruskal-Wallis Test
chi-squared = 9.6704, df = 2, p-value = 0.007945
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.