The Tukey's HSD makes the assumption that your dependent variable is normally distributed and so is not appropriate as a post-hoc test following a non-parametric omnibus test like KW.  The only real non-parametric post-hoc test for unpaired data is the Dunn'a test which is basically equivalent to the Bonferroni correction but applied to a series Mann-Whitney U tests rather than t tests.  I hope that helps.

The Tukey's HSD makes the assumption that your dependent variable is normally distributed and so is not appropriate as a post-hoc test following a non-parametric omnibus test like KW.  The only real non-parametric post-hoc test for unpaired data is the Dunn'a test which is basically equivalent to the Bonferroni correction but applied to a series Mann-Whitney U tests rather than t tests.  I hope that helps.

Rowan, a significant omnibus test is not required to perform post-hoc tests while controlling the family-wise error-rate (FWER) (only Fisher's LSD for k=3 groups requires a significant ANOVA-p to control the FWER, and maybe som anyway not-so-recommended post-hoc procedures). Tukey's HDS controls the FWER, whether or not an omnibus-test is significant. But this is only a gloss to your question.

More important is the conceptional problem that Kruskal-Wallis is about distributions and Tukey's HSD is about means. So you mix different models/hypotheses, what is not a good idea.

The special case for pair-wise comparisons of distributions is the (Mann-Withney-)Wilcoxon test. The FWER can be controlled by adjusting the p-values by the method of Bonferrony, Holm, Sidak and others. The adjustment by the method of Benjamini/Hochberg will control the false-discovery rate (FDR).

The calculation of Tukey's HSD is shown in many sites in teh internet. Have you ever searched for it? You will also fiond tables with critical values for q.

http://web.mst.edu/~psyworld/tukeyssteps.htm

http://academic.udayton.edu/gregelvers/psy217/labs/lab10/TukeyMC.pdf

https://stat.duke.edu/courses/Spring98/sta110c/qtable.html

http://www.sagepub.com/fitzgerald/study/materials/appendices/app_h.pdf

A possible middle ground approach would be to convert the raw data to normal scores (or z values), and then apply the parametric F test to it. It is then a robust F test that compares the means (of the normal scores). The results will be similar to the results of the KW test. One advantage of the robust F test is the possibility to use post-hoc tests very easily.

The KW Test is similar to having a test statistic SS Treatment/MS Error, where the ranks are used in place of the raw data. It is not surprising that results for the KW Test are similar to the results o an F test based on ranks.

Note that the KW Test is very similar to an F Test that is based on ranks. The KW Test is simply SS Treatments/ MSE (all based on ranks).

The use of normal scores (ranks) allows me to extend such tests to 3-way or 4-way ANOVA's, say.

Hi Rowan, A nice alternative for a posthoc test after  the KW is the Conover test, I use the free software package : brightstat for KW+Conover

http://brightstat.com/

Thank you for your comments they are all quite helpful to me not just for this study but my overall understanding of non-parametric post-hoc statistics. It appears this is an area with little consensus.

Statisticians often offer different solutions from each other. I teach this fact in my graduate courses so students better understand why they see different approaches being suggested in the literature (for similar situations). Mathematics students can find this confusing when they are used to "solid facts" in Mathematics. The use of statistics is also an art.