A nonparametric two-sample comparison for skewed data with unequal variances

Abstract

For practical situations, the pure shift model is probably notveryrealistic,anditisthusimportanttoassesstherobust- nessofdifferenttestsagainstdeviationsfromthismodel.Itis well established that heteroscedasticity (unequal variances) is at least as deleterious for the properties of the Wilcoxon- Mann-Whitney (WMW) test as for the t-test (1) and that the Welch test should replace the t-test when distributions are approximately normal and variances unequal. It is also worth remembering that the WMW test does not share the asymptoticrobustnesspropertiesofthet-test.Inaddition,un- equal variance is not the only problem frequently encoun- tered. Distributions can also be skewed, and the skewness of the two distributions may differ. Unfortunately, but not unexpectedly, even the Welch test is unable to maintain the nominalsignificancelevelwhendistributionsareskewed(1). Stochastic simulation is a valuable tool to compare test properties under different conditions. Based on simulation studies, Neuhauser (2) proposes that the modified or gener- alized Wilcoxon test by Brunner and Munzel (BM) (3) should be applied when it cannot be assumed that variances are equal and distributions symmetric. In such situations, Skovlund and Fenstad (1) have shown that the properties of the most commonly used classical two-sample tests (t-test, WMW, or Welch test) are not acceptable and thus suggested that transformations are necessary. It will in prac- tice of course often be difficult to come up with satisfactory transformations that will render an approximately symmet- ric distribution. It is thus of great value to identify relevant alternative tests and assess their properties. Based on simu- lation models with data from normal and gamma distribu- tions, Neuhauser concludes that the properties of the BM test are acceptable in situations with skewed distributions and heteroscedasticity. He also refers to similar findings from other simulation studies. Unfortunately, the literature is not unanimous on this issue. Fagerland and Sandvik (4) have examined the significance level of six different tests, including the BM test, and have shown that nonrobustness is a serious problem with all tests if distributions with different skewness are compared. The main message given by Neuhauser (2) is that the BM testcontrolstypeIerrorandshouldbeappliedwhennonsym- metrical distributions with unequal variance are compared. PreservationofthetypeIerrorisacrucialpropertyofasignif- icance test, and based on his findings, this seems not to be a bad idea. However, according to Fagerland and Sandvik (4), even the BM test fails to maintain the nominal signifi- cance level in many situations. For most settings studied, even small differences in variance are shown to lead to non- robustsignificancelevelsifthedegreeofskewnessislarge.In additiontoshapeandscale,animportantfactoriswhetherthe two samples have different size. When sample sizes are equal, their simulation studies show that the parametric tests are superior to the rank-based tests under the null hypothesis of equal means but not under the hypothesis of equality of medians. However, among the rank-based tests, the BM test is certainly better than the WMW and seems to be slightly better than other modifications.