The permutation F test; alternative forms of the permutation F statistic
The Kruskal-Wallis test
The KW statistic has the form of SST based on ranks, so a permutation KW test can be based on SST(ranks). Alternatively, we can obtain a large sample approximation to the permutation distribution by using a scaling factor of 1/SR2 , where SR2 = N(N+1)/12 for ranks without ties.
For data sets with ties, SR2 is calculated as the sample variance of the adjusted ranks.
Conditions favoring the K-W test; use of general scores.
Multiple comparisons: there are two general approaches presented
i) Modify the traditional Bonferroni, Fisher's LSD, or Tukey procedures for ranks (using the appropriate form of SR2 )
ii) Use a permutation version of Bonferroni, Fisher's LSD, or Tukey.
A convenient way to perform permutation multiple comparisons while adjusting for multiple tests is available in SAS in Proc Multtest.