2-way ANOVA Normality Requirements

My point of view has already been summarized by this post:

Normality isn't a huge issue. Your graphics look close enough to normality that we aren't worried too much (they really aren't perfect and the qqplot could be nicer but they're nothing to write home about). The sample size is large enough that the central limit theorem will fix any problems we had with non-normality.

Don't worry - you're fine.
So I think (that is: guess) that most statisticians would have a quick glance at your histogram and quantile-quantile-plot, and say: “okey it is almost normally distributed. Let’s run anova (analysis of variance).

But you asked a question. And I try to answer questions. In a situation where there are heavy tails gamlss might be a solution.

Also, I agree with what hjsmith said above. [(I hope I spelled that name correctly. Otherwise I will be cursed.)] In many journals you will not have the space for your graphs. But don’t get me wrong. I want to applaud you for bringing up the question.

[ohh, off topic, I appreciated the “thanks” above so I am glad that it seems that you find it as interesting with gamlss as I do. ]

I did spell the name hlsmith incorrectly. I do apologize! Maybe I will be both cursed and banned. :)
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