See the attached for my recommendation. Bonferonni is probably easiest but all methods are available in R as it says. Best wishes David Booth

There are several listed here, with references, those functions starting with "frdAllPairs". Conover and Nemenyi are common. But if it's available, the exact test described in "frdAllPairsExactTest" might be advantageous. https://cran.r-project.org/web/packages/PMCMRplus/PMCMRplus.pdf

Bonferroni correction is the easiest method and very easy to calculate. Something to keep in mind is that the Bonferroni reduces Type I error (False positive) as the number of tests increases, this results to increase Type II error (increase the chances of false negative. i.e., decreased the test power). SO, there are other methods such as Tukey's which is less restrict compared with the Bonferroni

Wilcoxon signed-rank test for pair wise comparison

The Friedman test is an extension of the Wilcoxon signed-rank test and therefore it can be used for performing post-hoc test. The p value can be corrected using Bonferroni correction for multiple hypothesis testing at a 5% cut-off

Read more here https://www.reneshbedre.com/blog/friedman-test.html

Renesh Bedre and Mohammed Grawish ,

Friedman's test is not really an extension of the Wilcoxon signed rank test, and the latter may not be a great choice as a post-hoc test for Friedman, considering the differences in the way the two tests handle the data.

You are much better off using post-hoc tests designed for Friedman's test. As mentioned above, Conover and Nemenyi are common.

There are several available in Python:

https://stats.stackexchange.com/questions/560666/for-friedman-test-what-are-the-pros-and-cons-of-different-post-hoc-comparison-m

And in R:

https://cran.r-project.org/web/packages/PMCMRplus/index.html

In addition, using pairwise have the undesirable property of each test ignoring all the data except what's being considered in the pairwise test. This can lead to unsatisfactory results. For an extreme case of this, you might read the section on Schwenk dice here, with the caveat that I wrote it: https://rcompanion.org/handbook/F_08.html

Salvatore S. Mangiafico

Thank you for your insights and for sharing resources. I have read several papers and articles (some of them I have cited in my blog article) which mentions that Friedman's test is an extension of the Wilcoxon signed rank test. But upon further reading, there is no universal agreement on this.

Indeed Conover test is designed for unreplicated blocked data and good option as a post-hoc test for significant Friedman tests.

I have also updated the Conover test in my blog article here https://www.reneshbedre.com/blog/friedman-test.html

Hi, Renesh Bedre . Friedman's test is really not like the signed rank test. You can look at the math behind each of these tests to get a sense of that. If anything, I might say Quade Test is like Wilcoxon signed rank and Friedman is like the sign test. (The latter is suggested by this reference that you cite: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC468904/ . Getting reliable information on non-parametric tests can be devilish because they can be fairly presented as having different hypotheses depending on what assumptions are used, and authors often speak loosely for a lay audience. The following is an example that may help to elucidate the difference between the Friedman / sign approach and the Quade / signed-rank approach.

A = rep(c( 1, 3, 5, 7, 9,11,13,15,17,19),3)

B = rep(c(31,33,35,37,39, 2, 4, 6, 8,10),3)

Y = c(A, B)

G = factor(c(rep("A", length(A)), rep("B", length(B))))

Block = c(factor(1:length(A)), factor(1:length(A)))

friedman.test(Y ~ G | Block)

quade.test(Y ~ G | Block)

wilcox.test(A, B, paired=TRUE, correct=FALSE)

library(DescTools)

SignTest(x = A,

y = B)

Diff = A - B

hist(Diff)

hist(sign(Diff))