Rank Tests of Dispersion

Abstract

In a recent paper [1] Ansari and Bradley have shown the equivalence of two rank tests for comparing dispersion, one test due to Barton and David [2], the other to Ansari and Freund, and have provided tables of the exact distribution. They observe that Siegel and Tukey have proposed [11] a similar test which permits use of existing tables. They also exhibit the mean of the limiting normal distribution under the alternative hypothesis. Later Klotz [7] established the equivalence of all these tests. In the present paper it is shown that (1) Any of these tests is consistent against differences in dispersion if the two distributions have a common median and differ in a scale parameter, and under some less restrictive circumstances. But without such restrictions bizarre asymptotic behavior can arise--including good sensitivity against translation for some non-symmetric densities. One (not very natural) example is offered in which the test constructed for rejection if one of two scale parameters is the larger, actually turns out to be consistent against that parameter's being the smaller of the two. (2) No rank test (i.e., a test invariant under strictly increasing transformation of the scale) can hope to be a satisfactory test against dispersion alternatives without some sort of strong restrictions (e.g., equal or known medians) being placed on the class of admissible distribution pairs. (3) Box [3] has proposed testing equality of variances by applying the $t$ test to the logarithms of variances computed within small subgroups. He indicates how such tests should be robust (though not of exact size). Distribution free tests of exact size can be constructed by applying a rank test in place of the $t$ test. Wilcoxon's test applied to variances-within-triads has asymptotic efficiency .5 against normal alternatives. If the two samples each have 9 observations then the exact power is readily calculated and "efficiency" is again about .5.