Comparing medians

Many methods have been proposed for comparing the medians of J   independent groups. Generally, however, extant techniques require very restrictive assumptions or they are known to perform in an unsatisfactory manner in simulations. Included are many well-known rank-based methods plus certain types of bootstrap techniques. One goal here is to point out that two recently proposed methods also perform poorly when there are tied values. Another goal is to examine the small sample properties of several alternative methods that have not been previously studied. The main result is that a multiple comparison technique (called method R), is the only method to perform well in all the situations considered here. For an omnibus test with J>2J>2 groups and no tied values, two methods are found that control Type I error probabilities reasonably well, one of which is based in part on results in Liu and Singh [1997. Notions of limiting p-values based on data depth and bootstrap. J. Amer. Statist. Assoc. 92, 266–277]. With tied values, the second method is found to be more satisfactory, but even it can perform poorly.