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  • The type of non-normality and the type of test is important here. Also, if you are using an ANOVA or regression model, the important thing is the normality of the residuals not the normality of the raw data.
  • It also depends how risk averse you are. Some people immediately flee for the security of non-parametric tests when the slightest thing looks bad. Others stay with parametric tests until things get really extreme. Some people will try transformations are every opportunity and others shun any transformations at all.
  • Look at the standard used in journals in your area to get a feel for whether you should be extra cautious or not.
  • The simplicity of using the same test for all the variables may be the most important consideration here. But whatever choice you make, resign yourself to the fact that the referees of the journal you submit your results to will ask you to change to the other approach.
  • Some people advocate running two or more tests simultaneously as a sort of sensitivity analysis. If you get pretty much the same results using a t-test and the Mann-Whitney-Wilcoxon test, then you can sleep well at night. If the two tests differ then you can investigate why they differ.

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