352 Introduction to Robust Estimation and Hypothesis Testing type I error, when comparing groups having identical distributions. But when the distributions differ, under general conditions an incorrect estimate of the standard errors is being used, which might adversely affect power. A method aimed at improving the Kruskall­Wallis test was derived by Rust and Fligner (1984), assuming that tied values occur with probability zero. The explicit goal stated by Rust and Fligner is to test the hypothesis that J groups have a common median, but under general conditions it fails to do this in a satisfactory manner. Letting p jk = P(X i j < X ik ), their technique is appropriate for testing the hypothesis that for all J groups, p jk = .5. However, their method is based on the assumption that the distributions of the J groups differ in location only. If this assumption is violated, in essence the Rust­Fligner method is testing the hypothesis that the groups have identical distributions. A possible appeal of their method is that it is asymptotically distribution free under weaker conditions than the Kruskall­Wallis test. A rank-based method that can handle tied values was derived by Brunner, Dette, and Munk (1997). Extensive comparisons with the Rust­Fligner method, when ties occur with probability zero, have not been made. With small sample sizes, the choice of method might make a practical difference, but a detailed study of when this is the case has yet to be performed. Here it is merely remarked that situations can be constructed where, with a common sample size of 50, the choice of method makes a practical difference. For example,