| The problem of testing the equality of the medians of several populations is considered. Standard nonparametric procedures for this problem require that the populations have the same shapes in order to preserve their nominal significance level under the null hypothesis. This study presents several modifications of the Kruskal-Wallis test that may be appropriate for handling the problems of unequal variances or even different shapes in the one-way classification model. A desirable property of the proposed tests is that they are asymptotically distribution-free when the populations are symmetric with equal medians or means, while they retain all desirable properties of the original test under the usual assumption of identical populations.;In a Monte Carlo study, the proposed tests were compared with other commonly advocated alternatives, including the classical F test, Welch's approximation method, the Kruskal-Wallis test, and Rust-Fligner's modification of the Kruskal-Wallis test. One of our modifications was proved to be the test of choice when the populations are symmetric/heavy-tailed and the variances are unequal. As expected, the F test and the Kruskal-Wallis tests failed to perform properly, regardless of distribution type, when the variances were unequal. The Type I error rates of the Welch test were systematically affected by type of distribution. To make matters worse, the power of the Welch test was considerably inferior to that of our modification when data were from symmetric/heavy-tailed distributions. The loss in power was more serious as the tail weights increased. On the other hand, the Type I error performances of both the Rust-Fligner and our modification were very consistent over different types of the symmetric distributions. But the Rust-Fligner test frequently failed to perform properly, with a tendency to become liberal. Finally, it should be noted that none of the parametric and nonparametric procedures examined here exhibited robustness with respect to variance heterogeneity when the populations were asymmetric. These results clearly suggest the need for developing a robust procedure for these conditions. |
