How robust is ANOVA to violations of normality? I have several dependant variables that are non normal in distribution: Kolmogorov-Smirnov test is significant, skewness ranges up to 8 for some variable, and kurtosis is generally about 2 but in a couple of cases raises to 12! It is a repeated-measures (RM) ANOVA model for the normally distributed variables.  
A couple of questions:


*

*How robust is RM ANOVA to violations of normality, when there is an equal number of observations per group?

*If I use non-parametric tests, do I need to correct for multiple testing that impacts type 1 error?

 A: Don't look at it as a binary thing:  "either I can trust the results or I can't."  Look at it as a spectrum.  With all assumptions perfectly satisfied (including the in most cases crucial one of random sampling), statistics such as F- and p-values will allow you to make accurate sample-to-population inferences.  The farther one gets from that situation, the more skeptical one should be about such results.  You've got a substantial degree of nonnormality; that's one strike against accuracy.   Now how about the other assumptions underlying the use of ANOVA?  Size it all up the best you can, and document in a footnote or a technical section what you find.  You also should look at this page, as @William pointed out.
As to your last question, I don't believe you need to change your strategy vis-a-vis multiple comparisons just because you move from a parametric to a nonparametric test.  If you want to describe the rationale for your current approach, I'm sure people will be glad to comment on it.
A: Let me state a couple of things.  First, I think it's best to understand repeated measures ANOVA as actually a multi-level model in disguise, and that may create additional complexities here.  I should let one of CV's contributors who are more expert on multi-level models address that issue.  
However, in general, it's worth noting that not all assumptions are created equal.  People tend to think that the normality assumption is vital, whereas I think of it as the least important.  Heterogeneity is a bigger deal.  Skew is potentially more damaging than kurtosis, but if it isn't too large, and all groups are skewed in the same direction, it may not be lethal.  Basically, whether or not the residuals are normal has to do with whether the p-values are accurate, but the parameter estimates should remain unbiased.  On the other hand, heterogeneity of variance has to do with the efficiency of the OLS estimator.  
A: My understanding is that ANOVA including repeated measures is robust to violations to normality of errors assumptions.  However there is indications that the errors should be equal in their variation across different factor levels.
Can I trust ANOVA results for a non-normally distributed DV?
