Should we always choose a nonparametric test when comparing two apparently nonnormal distributions?

Abstract 

When clinical data are subjected to statistical analysis, a common question is how to choose an appropriate significance test. Comparing two independent groups with observations measured on a continuous scale, the question is typically whether to choose the two-sample-t test or the Wilcoxon–Mann–Whitney test (WMW test). Similar results are often obtained, but which conclusion can be drawn if significance tests give highly different P-values? The t test is optimal for normally distributed observations with common variance and robust to deviations from normality if sample sizes are not very small. The WMW test makes no distributional assumptions, but depends heavily on equal shape and variance of the two distributions (homoscedasticity). We have compared the properties of the traditional two-sample t test, a modified t test allowing unequal variance, and the WMW test by stochastic simulation. All show acceptable behaviour when the two distributions have similar variance. When variances differ, the modified t test is superior to the other two.

Keywords:  P-Values, Two-sample t test, Welch's test, Wilcoxon–Mann–Whitney test, Heteroscedasticity, Stochastic simulation