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some pointers here :

http://stats.stackexchange.com/questions/5680/can-i-trust-anova-results-for-a-non-normally-distributed-dv

http://stats.stackexchange.com/questions/56971/alternative-to-one-way-anova-unequal-variance

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Do you mean the normality and homoscedasticity of data or of errors?

If the former then it is not a big deal unless the errors are OK.

If the latter then you have a problem.

But I do not think that people care much about these issues. And generally I replicate the analysis using other methods (like nonparametric ones) when I suspect the eligibility of my findings.

Dear Cyril,

My understanding is that ANOVA is robust to non-normal data provided that there are equal group sizes, at least 20 degrees of freedom and the kurtosis is not significantly different from zero.

Dear Cyril, If you are talking about residuals( and you should be) the answer is never. see the books at the following links:

http://www.amazon.com/gp/product/0123869838/ref=s9_simh_gw_g14_i5_r?ie=UTF8&fpl=fresh&pf_rd_m=ATVPDKIKX0DER&pf_rd_s=desktop-1&pf_rd_r=0MKJNVRMSR6EFPYC8ZDE&pf_rd_t=36701&pf_rd_p=2437869742&pf_rd_i=desktop

http://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=sr_1_1?s=books&ie=UTF8&qid=1458705125&sr=1-1&keywords=Kutner+Applied+linear+statistical

Thank you all of us for your interesting answers and literature on the subject !

Be sure, I'll have a look at all of them.

@David : Indeed Pr., you just reinforce the inconsistency I found on that subject, which is at the origin of this question...

Regards

@Cyril what is the inconsistency?

@David what is wrong in my post?

If I caricature what I read is :

1) "As long as you get p-value below 0.05, normality and equal variance doesn't matter"

2) "If you compute ANOVA without verifying equal variance and normality, you forget all about Beta (and test power) and so you don't know what you are doing"

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@Cyril

i'd appreciate pointers towards (1) ... just for my own little shop of (statistical) horrors !

otherwise, David's point above is indeed crucial : what violation of normality are we talking about ? Variables or residuals ?

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Dear Cyril, There is no inconsistency in my response. If the residuals show those errors then you HAVE to fix them or you can't believe any inferences you might make. Pardon me but I don't see any inconsistency. Best wishes.

@Mehmet  Just because some foolish people ignore basic facts doesn't make that acceptable behavior. U seem to suggest that it is. Best wishes.

@Cyril If you mean the inconsistency about assumptions and validity of ANOVA found in many resources on internet, yes I would agree that on internet one can find many different and inconsistent propositions about ANOVA. I had experienced the same confusion before. Everybody understands and argues differently. But the truth : what is necessary for ANOVA and most linear models to be valid is the normality and homoscedasticity of residuals. That is it.

Coming back to why I said "People do not much care about the assumptions" has some reasons: i) pointing out the inconsistency observed, ii) pointing out that ANOVA is robust to non-conformity of assumptions at some degree, iii) one does not have many alternatives when the assumptions are not satisfied. I know there are some methods like transformations making variance stabilized etc, which seem odd to me. I see them as inefficiencies of the statistical method under consideration.

For one-way ANOVA actually one has Welch test and similars which enable to analyse data with heteroscedastic variances. But I tried to make the discussion general so far.

I had a previous entry on my blog about this subject which I recommend you to have a look, because it talks about other odd(!) issues.

http://statbits.blogspot.com/2015/12/of-anova-and-residuals.html

@Mehmet, One has many possibilities if the residuals show bad, e.g. transformations, robust methods, weighted regression, shrinkage estimators, etc depending on the individual problem encountered. It may take effort but usually one can fix the situation so that a reliable inference can be obtained. See the books I cited above and many others. Best wishes.

@David : Sorry for the misunderstanding. The "inconsistency" wasn't in your own answer. But was more about the word "never" you wrote, which is in contradiction with what we can found in some book and in many place in the internet (unfortunately...).

@Fabrice : Concerning book : the first one I remember is a french one (sorry...) : http://www.amazon.fr/Statistiques-avec-R-Pierre-Andr%C3%A9-Cornillon/dp/2753519927 in its first edition (I don't have the others). In the chapter concerning ANOVA, authors don't talk about any assumption before performing the test. I sent an e-mail to one of them and get an answer that I can resume in : "that doesn't really matter" (I don't have it anymore and I may resume it too much...).

Concerning website you can easily find this kind of assumption anywhere : "With violations of normality, continuing with the ANOVA should be ok if you have a large sample size and equal sized groups." http://www.statisticssolutions.com/manova-analysis-anova/

But as I say, I wrote caricature... but for me "forgetting all about beta" is equal to "only wanting alpha below 5% or 1% no matter what". Isn't it ?

@Mehmet : Thank you for your valuable insight in that subject.

@All : Concerning assumptions, I was initially talking about both of them : variables and residuals. However, (and I thank you for that) I didn't note previously (shame on me...) that the residuals was really the subject here...

Finally thank you all of you for your help on that question, I have now good literature and good thoughts on that subject. It is what I was seeking for !

@David yes there are many other methods, but they are not presented as alternatives by many lecturers,statisticians etc, that makes the beginner confine to those older standard methods. Those methods might not be known to many statisticians as well. 

I think it is worth and valid trying to perform a weighted regression in case the residuals of ANOVA are not homoscedastic and normal. Although this might come weird to many people.

I wish to learn your opinions about my previous comment.

What do you say on performing weighted regression when residuals are not homoscedastic?

Dear researchers,

Please, I would like to get the detailed algorithms for Robust ANOVA (RANOMA) instead of using softwares.

Your sincerely,

Elcio Oliveira

Dear E. C. de Oliveira,

I look at the "RANOVA", but still by doing this sort of things we don't know what we are doing exactly (as we never report the test power...).

If the data doesn't meet the assumptions needed, the issues are probably not coming from the test but more from the data themself.

So, in my experience (I ask this question 3 years ago now...), there are three solutions :

1) Look at the data and do a "pre-treatment" to make them meet the assumptions (log, sqrt etc.).

2) Get more and more data until they meet assumptions.

3) Look at your hypothesis or test: Is there really something to prove ? Do I really need a statistical test to prove my point ? Sometimes we don't ask the good question so don't get the "right" hypothesis.

Best regards.