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?
Solved – Is normality important for ANOVA when N>30
anovanormality-assumption
Related Solutions
The difficulty is that skewness and kurtosis are dependent; their effects can't be completely separated.
The problem is that if you want to examine the effect of a highly skew distribution, you must also have a distribution with high kurtosis.
In particular, kurtosis* $\geq$ skewness$^2+1$.
* (ordinary scaled fourth moment kurtosis, not excess kurtosis)
Khan and Rayner (which is mentioned in the earlier answer) work with a family that allows some exploration of the impact of skewness and kurtosis, but they cannot avoid this issue, so their attempt to separate them severely limits the extent to which the effect of skewness can be explored.
If one holds the kurtosis ($\beta_2$) constant, one cannot make the skewness more than $\sqrt{\beta_2-1}$. If one wishes to consider unimodal distributions, the skewness is even more restricted.
For example, if you want to see the effect of high skewness - say skewness > 5, you cannot get a distribution with kurtosis less than 26!
So if you want to investigate the impact of high skewness, you are unable to avoid investigating the impact of high kurtosis. Consequently if you do try to separate them, you in effect hold yourself unable to assess the effect of increasing skewness to high levels.
That said, at least for the distribution family they considered, and within the limits that the relationship between them poses, the investigation by Khan and Rayner does seem to suggest that kurtosis is the main problem.
However, even if the conclusion is completely general, if you happen to have a distribution with (say) skewness 5, it's likely to be little comfort to say "but it's not the skewness that's the problem!" -- once your skewness is $>\sqrt{2}$, you can't get a kurtosis to be that of the normal, and beyond that, minimum possible kurtosis grows rapidly with increasing skewness.
Thanks for posting the data. Posting shows that the box plots concealed, although not intentionally, the sample sizes and important detail too. Whenever I see skewness on a positive response, my first instinct is to reach for logarithms, as they so often work well. Here, however, logarithms drastically over-transform, and plotting everything shows up a small surprise, namely that the two lowest values need care and attention.
The graph here is a quantile-box plot in which the original data points are plotted in order on scales consistent with the box idea (i.e. about half the points are inside the box and about half outside, the "about" being a side-effect of sample sizes like 11).
A more cautious square root transformation seems about right.
Personally I regard preliminary tests for normality and so forth as over-rated stuff left over from the 1960s. I feel far too queasy about forking paths of the form: pass the test OK, fail the test do something quite different, particularly with small sample sizes. Once you have a scale on which you have approximate symmetry and approximate equality of variances, linear models will work well.
Similarly, skewness and kurtosis from small samples can hardly be trusted. (Actually, skewness and kurtosis from large samples can hardly be trusted.) For some of the reasons see e.g. this paper
Indeed, some fits with generalised linear models with cohort and gender as indicator predictor variables show that results seem consistent over identity, root and log links, even despite the evidence of the first graph. If this were my problem I would push forward with a square root link function. In other words, although transformations are informative about the best scale to work on, you let the link function of a generalised linear model do the work.
Campaign slogan: Conventional box plots with a few groups leave out detail that could easily be interesting or useful and don't make full use of the space available. Use graphs that show more!
EDIT:
Here is token output: predicted values using generalised linear model, root link, normal family, interaction between cohort and females:
+--------------------------------------+
| cohort females predicted Freq. |
|--------------------------------------|
| 1 males 2.056 12 |
| 1 females 5.024 12 |
| 2 males 12.712 11 |
| 2 females 15.348 11 |
+--------------------------------------+
Best Answer
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.