One of the assumptions of ANOVA is normally distributed residuals. However, as noted here some authors have stated that this assumption is barely important at all. Simulation results have shown that, provided sufficient sample size, deviations from normality can be tolerated without a large adverse effect. In practice I often hear people recite that 'the test is robust to violations of the assumption of normality'.
My question is this: When making inferences at the group level (not point estimates for individual observations), why is it not the case that the assumption is about the normality of the error distribution (i.e. the hypothetical distribution that we quantify with terms like the Standard Error of the Mean) as opposed the specifically observed residuals?
Isn't the most proximal cause of Type I inflation or deflation, non-symmetrical errors in estimating the mean, etc, really a reflection of the non-normality of the error distribution?
I grant that, clearly, there is an equivalence here that may lead to this being a bit of semantics. Non-normally distributed residuals almost inevitability are going to lead to a somewhat non-normal error distribution of the mean estimate. However, I think the simulation results I referred to above (and our general knowledge of the Central Limit Theorem) hold out the idea that if the means being described have a sufficient number of samples, then the error distribution can become reasonably normal and thus nothing bad happens.
Moreover, if the assumption was about the distribution of errors it would neatly describe why it is that the violation of this assumption can frequently be papered over with larger sample sizes. It would also highlight characteristics of datasets where 'larger' is so huge as to make it impractical (e.g. zero inflated datasets).
Is there something I am missing that makes it critical that we describe the assumption as 'normality of residuals'?