Dear Gabriel,

I think the determination of whether to use a non-parametric test will depend on the error distribution (i.e., the assumption of normality) and not the experimental design. In R at least the Wilcoxon test is common to compare to means with non-normal errors. 

Would an alternative be to look at generalized linear models?

Steve

 Dear Dr. Stephen Crittenden,

I'm sorry, I didn't expressed myself properly. I already analyzed the dritibution of my data. They do have a central tendency but are skewed, and I thought that it would be  more appropriate to use nonparametric methods. And I would like to know if there is a nonparametric test adequate to my experimental design.

Thanks,

Cheers

Did you check normality for the data or for errors? Errors should be normal not the data.

There is nothing like a "non-parametric alternative" to parametric analyses. Even in the simplest case for two samples, the often heard statement that Wilcoxon test should be an alternative to a t-test is nonsense. These tests are about entirely different models and hypotheses. The confusion may come from the fact that under very particular circumstances, the tested hypotheses are essentially the same - but just then the t-test would be tha tailor-made solution to the problem.

It is usually better to understand your data and chose an appropriate model with interpretable coefficients rather than turning to some mode of analysis that spits out a p-value without making clear what the tested hypothesis actually is.

Dear Dr. Wilhelm and Iyisoy,

Thank you for your reply. I only checked my data graphically, and not the errors. I checked if it looks like a normal distribution and the skewness value. I did not used a normality test (like Shapiro–Wilk) because I heard that there is some debate about its validity.

Should I check normality for errors then? I'll search on the internet about it. If you have a link to some text I would appreciate it.

About the "non-parametric alternative", I'm sorry for my ignorance. I learned in Statistics that you should check if your data can be analyzed by paremetric test. If not, you should look to an alternative. But I understand that it probably is an oversimplification. I do understand my data, and chose the apropriate analysis (repeated measures ANOVA). But since the skewness values are above 1 and the data doesn't look like a normal distribution (graphically), I was wondering if there is an alternative method for me to analyse my data. And this method would better not assume any distribution, because my data doesn't seems to fit an especific model of distribution.

Cheers

Check normality for errors. First save errors after your model. Then check normality by using any method you prefer, inspecting histogram, skewness & curtosis, Q-Q plot, a statistical test or all.

You can use a mixed effects model which is more flexible than repeated measures ANOVA. There are also some nonparametric methods but I do not think you need to use them.

If you do understand your data and if you have chosen ANOVA to analyze your (possibly transformed) data, then there is actually no need to be concerned with the assumed error distribution (if you understood your data, the assumption is obvious, and even if the current observed data seems to adhere only badly to this assumption, it is only attributable to "unlucky sampling"). That you want to check the distribution of your data indicates that you are not sure about its distibution and hence that you don't understand your data (or the "data-generating process"). Thats not a bad thing. It is an important and difficult part of the data analysis to learn and understand one's data (and to develop a resonable and sensible idea about the data-generating process). This process usually leads to some meaningful model (in which the restriction of the coefficients may then be tested by ANOVA, is desired). 

Having a look at the Q-Q plot of the residuals is usually absolutely enough to check if there is something unexpected but possibly relevant. A clear non-linear structure/pattern in this plot will indicate the possibility that the model and/or the error structure might not be well chosen.

Checking histograms has the disadvantage that you need quite a lot of data to get some reasonably good impression and that the shape is easily changing with the choice of the bin widths. Statistical tests on assumptions are logically flawed ("significant" results don't neccesarly mean that the prospected analysis is unsuited, and "non-significant" results don't neccesarily mean that the analysis woud be ok - so in no way you get any useful answer to your problem!).

Dear Dr. Wilhelm and Iyisoy,

I just did the Q-Q plot and all of my data seems to fit nicely. So I guess problem solved. Thank you all for sharing your expertise with me.

All the best.