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

Abstract

Background: Student's two-sample t test is generally used for comparing the means of two independent samples, for example, two treatment arms. Under the null hypothesis, the t test assumes that the two samples arise from the same normally distributed population with unknown variance. Adequate control of the Type I error requires that the normality assumption holds, which is often examined by means of a preliminary Shapiro-Wilk test. The following two-stage procedure is widely accepted: If the preliminary test for normality is not significant, the t test is used; if the preliminary test rejects the null hypothesis of normality, a nonparametric test is applied in the main analysis.

Methods: 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.

Results: 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.

Conclusion: 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.

Figure 1

Estimated Type I error probability of the two-sample t  test at α   = .05 after both samples had passed the Shapiro-Wilk test for normality at α  _pre = .100, .050, .010, .005 (conditional), and without pretest (unconditional). Samples of equal size from the (A) exponential, (B) uniform, and (C) normal distribution.

Figure 2

Estimated Type I error probability of the two-sample t  test at α   = .05 after the residuals had passed the Shapiro-Wilk test for normality at α  _pre = .100, .050, .010, .005 (conditional), and without pretest (unconditional). Samples of equal size from the (A) exponential, (B) uniform, and (C) normal distribution.