Is normality important for ANOVA when N>30? I want to use repeatead measures ANOVA with sample size approximately n=40. Normality is violated according to S-W test. After log transformation although Q-Q plots seem better, the distribution is still not normal (there are some outliers) and for S-W test of normality, again p<0.05. What should I do? Can I just ignore it?
 A: By and large, repeated measures ANOVAs don't so much require normality as approximate normality because they are pretty robust to normality violations. Violations of sphericity are generally a larger problem than violations of normality. 
I'd say even with n=40, if your plots look like they don't deviate too much from normal then you should be fine. If they look like they are slightly skewed versions of normal distributions then testing out some transformations (as you have done) to get them more normal should be sufficient. 
Of course, I have no idea what your data looks like so if it is a huge deviation from normal then perhaps an RM ANOVA isn't the way to go. In this case, you could use a non-parametric alternative to the RM ANOVA. Non-parametric tests have the advantage of not assuming normality. They only assume that your groups have the same distribution (whatever that distribution happens to be). I think Friedman's ANOVA is a non-parametric alternative to the RM ANOVA but I have no experience with it.
