I'm running statistics on some assays where I have 4 groups for comparison. I'm testing for normality using the Shapiro-Wilk test. Often, some of the 4 groups pass the test, while the other groups don't. Can I still use ANOVA if only 2-3 of the 4 groups are normally distributed? Sample size is >40 per group.
ANOVA does not assume normality of the data, only of the errors, which we can approximate with the residuals. If the only independent variable in your study is the group, then this becomes equivalent.
In any case, since there are now easily implementable methods that do not rely on this assumption, I suggest using them. They include quantile regression, robust regression, regression trees (and their offshoots) and so on.