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This post is the result of work performed in collaboration with my colleague Eleonora Paparelli (who actually did most of the work!). We wanted to compare several distributions using Wilcoxon test and summarize results (i.e. indicate the comparisons showing significant differences).
R base includes pairwise.wilcox.test to perform Wilcoxon rank sum test between all pairs of samples in a study.
A common way to represent significance in pairwise comparisons is the use of letters. Samples sharing a letter are not different from each other. Samples not sharing letters are different.
Library multcompView in R can take a square matrix of p-values and return letters for all samples.
Sadly enough, pairwise.wilcox.test returns a triangular matrix, so I had to write a small function – named tri.to.squ – to take the output of pairwise.wilcox.test and convert it in a suitable input for the multcompLetters function of multcompView.
Now we can easily plot the distributions as box plot and add the letters as text.

```library(multcompView)

tri.to.squ<-function(x)
{
rn<-row.names(x)
cn<-colnames(x)
an<-unique(c(cn,rn))
myval<-x[!is.na(x)]
mymat<-matrix(1,nrow=length(an),ncol=length(an),dimnames=list(an,an))
for(ext in 1:length(cn))
{
for(int in 1:length(rn))
{
if(is.na(x[row.names(x)==rn[int],colnames(x)==cn[ext]])) next
mymat[row.names(mymat)==rn[int],colnames(mymat)==cn[ext]]<-x[row.names(x)==rn[int],colnames(x)==cn[ext]]
mymat[row.names(mymat)==cn[ext],colnames(mymat)==rn[int]]<-x[row.names(x)==rn[int],colnames(x)==cn[ext]]
}

}
return(mymat)
}
first.set<-cbind(rnorm(100,mean=1.8),1)
second.set<-cbind(rnorm(100,mean=0.9),2)
third.set<-cbind(rnorm(100,mean=1),3)

full<-rbind(first.set,second.set,third.set)

pp<-pairwise.wilcox.test(full[,1], full[,2], p.adjust.method = "none", paired = FALSE)
mymat<-tri.to.squ(pp\$p.value)
myletters<-multcompLetters(mymat,compare="<=",threshold=0.05,Letters=letters)

boxplot(full[,1] ~ full[,2],ylab="Something nice",names=c("Set 1","Set 2","Set 3"),ylim=c(min(full[,1]),0.3+max(full[,1])))
text(c(1,2,3),0.2+max(full[,1]),c(myletters\$Letters,myletters\$Letters,myletters\$Letters))

```  