I have two vastly unequal groups (n1=18 and n2=400), and want to test mean differences between the groups for some 700 parameters (more specifically, RNA expression levels for 700 genes).
While I understand that Welch's t works well for non-normally distributed samples if both groups are large enough, this is not the case here. Testing for normality 700*2 times (or examining QQ plots) would take a lot and would almost certainly show that some variables are normally distributed and some are not.
In this case, would you go for
- Welch t-tests for everything (which would probably be breaking assumptions),
- Mann-Whitney tests for everything (which have a lower power and might lead to not getting too many significant results), or
- choosing the appropriate test for each variable, according to normality testing? This would probably not be a good idea, since I have to adjust the p-values using the FDR method, and I don't think that is correct if the p-values are generated by completely different tests