To test or not to test: Preliminary assessment of normality when comparing two independent samples

Equally sized samples were drawn from exponential, uniform, and normal distributions. The two-sample t test was conducted if either both samples (Strategy I) or the collapsed set of residuals from both samples (Strategy II) had passed the preliminary Shapiro-Wilk test for normality; otherwise, Mann-Whitney’s U test was conducted. By simulation, we separately estimated the conditional Type I error probabilities for the parametric and nonparametric part of the two-stage procedure. Finally, we assessed the overall Type I error rate and the power of the two-stage procedure as a whole.Preliminary testing for normality seriously altered the conditional Type I error rates of the subsequent main analysis for both parametric and nonparametric tests. We discuss possible explanations for the observed results, the most important one being the selection mechanism due to the preliminary test. Interestingly, the overall Type I error rate and power of the entire two-stage procedure remained within acceptable limits.The two-stage procedure might be considered incorrect from a formal perspective; nevertheless, in the investigated examples, this procedure seemed to satisfactorily maintain the nominal significance level and had acceptable power properties.