If I have three sets of continuous data which are not normally distributed, can I still compare them with ANOVA? I have three sets of data from some experiments. 
I fitted each set to different distributions, and each one fits a different distribution. For example, Gamma, Weibull, and Lognormal. If I want to compare these sets what methods would you suggest? 
I am not sure, but I think I read somewhere that I can use Student t test and ANOVA only if I have the normal distribution. Therefore in my case what would be the approach that I should take? 
Similar questions were asked in this link and this link but they were not clear for me. 
Thank you
 A: Forget about any comparisons where you transform variables differently or use different distributions.  Go back to fundamentals.  If you want to test a general hypothesis such as stochastic ordering, use a semiparametric model or corresponding rank test such as the Kruskal-Wallis test.  These are distribution-free and transformation-invariant.
A: I think you are answering your own question.  
As you say, your three sets are different.  They may have similar metrics, like mean and variance, but they have different distributions.
I would use T-test and ANOVA if I want to assert that two of the sets are basically batches of the same thing.
If two sets have different distributions, then I know they are batches of a different type.
But give a thought as to what you want to assert about the batches.  How much they vary with eachother?  Then perhaps what you want is ANCOVA.
A: ANOVA is quite robust to violations of normality if the variances are roughly equal and the sample sizes are similar as well. Both are not given here which is a problem.
You could use Welch t-tests instead of ANOVA and all those problems are solved. A Welch t-test does not need equal variances or sample size. You would do a t-test for group 1 against 2, 1 against 3 and 2 against 3. For each test, you allocate 1/3 of your alpha level (Bonferroni correction).
However, per central limit theorem, you need 30 samples per group to do t-tests on non-normal data. This remains a problem unless you can have this amount of data.
