Reading this question and this question and the answers and comments, I have another question. Does the effect size of a normality test determine whether the normality assumption can be ignored for parametric tests with large samples?
The two examples I am particularly interested in right now are: normalizing data for an array of machine learning algorithms, some of which may employ parametric methods; comparing data using parametric tests like Student's t-test or MANOVA. I'm specifically using the Shapiro-Wilk test of normality, but I am also interested in other tests of normality.
I'm understanding that testing the normality of large samples with the Shapiro-Wilk test may produce statistically significant rejections of the null that are actually of no practical significance. The test may determine that the distribution deviates from normality, while in reality it is "normal enough" for all intents and purposes.
What do I mean by "normal enough" you ask? Thanks for asking. That's kind of the crux of the question, isn't it? How do we determine what's "normal enough" for a parametric test? Student's t-test is a good example, but I am interested to hear about how the answer may depend on the parametric test in question.
My hunch is that figuring out the effect size might somehow lend a practical interpretation to the statistical significance of the Shapiro-Wilk test and/or other tests of normality. But, I'm having trouble fleshing the idea out, and I wonder if anyone can help me with that, or perhaps set me straight if I'm barking up the wrong tree.