Considering pre-tests in family-wise error correction doing an ANOVA When I do multiple tests on a dataset, I need to correct for the family-wise error (e.g. using Bonferroni method when performing multiple pairwise t-tests). If I do an ANOVA, I need to make sure that several assumptions are met (independence of observations, normal distributions of the residuals, homoscedasticity), which I can do using several statistical tests (e.g., Shapiro–Wilk test, Levene's test). Do I need an significance-level correction for this as well?
 A: It's good to be concerned about multiple testing, but it's possible to go overboard with it. I'd say this is a case of that. 
I'd also not recommend doing a test for every single model assumption. Given sufficient data, you will find that many assumptions do not hold according to standard tests. This does not necessarily mean that the statistical procedure is invalid, because many basic tests (like ANOVAs) are robust to small deviations from those assumptions. There's a reason why the most common advice is to plot your model output and diagnostics (e.g. QQ plots), and not just look at tests of model assumptions. The information conveyed in such plots is much richer and with practice, you will be able to discern whether such deviations are likely to be a problem when interpreting your model output.
Fundamentally, I'd advise you to not put too much stock into p-values. I won't go so far as to say they are entirely useless, though you will find good arguments for why we would be better off without them. But if your analysis or decision relies strongly on a specific p-value, you are likely to be misled or to take poor decisions. 
