I have used the non-parametric Kruskal Wallis test as an alternative to the ANOVA on not-normally distributed data and I was wondering if it would be possible/useful to carry out also the LSD test.
The LSD test ist based on using the pooled variance estimate (usually using the MSE from the ANOVA). That makes no sense if you don't consider the variance as a useful model parameter.
People usually use Mann-Whitney/Wilxocon U-tests to get the significance of ranking differences between pairs of samples.
The family-wise error rate is controlled if you use Bonferroni/Holm correction of the p-values (from the U-tests). The KW-test is not required if you are interested in pairwise comparisons.
Make sure that you test a meaningful hypothesis and that you interpret the test results correctly (it's about stochastic equivalence and not about means or medians, as many people wrongly write).
You have several options for handling your non normal data. Several tests, including the one sample Z test, T test and ANOVA assume normality. You may still be able to run these tests if your sample size is large enough (usually over 20 items).
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