Hi Opnightfall1771,
The Tukey HSD test returns p-values that have been corrected for the number of independent pair-wise comparisons that are possible given the number of factor levels -- so, those p-values may be interpreted directly and require no further corrections.
The Kruskal-Wallis test you mentioned is the generalization of the Wilcoxon (or Mann-Whitney test) to factors with more than 2 levels, however if you are performing pair-wise tests the analyses are just Wilcoxon tests. The results from selecting (in Fit Y by X) Nonparametric > Nonparametric Multiple Comparisons > Wilcoxon Each Pair are not corrected for multiple comparisons. This is simply the nonparametric version of the "Each Pair Student’s t" option. You could use Bonferroni corrections with those p-values, however this will quickly become overly conservative with many factor levels and you would be better off using a more efficient analysis, such as the Steel-Dwass All Pairs (available under Nonparametric Multiple Comparisons), which is the nonparametric equivalent of the Tukey HSD. The p-values returned in that analysis are corrected p-values, just like the p-values generated from running a Tukey HSD.
I hope this helps!
Julian
