A combined test for differences in scale based on the interquantile range

A class of tests due to Shoemaker (1999) for differences in scale which is valid for a variety of both skewed and symmetric distributions when location is known or unknown is considered. The class is based on the interquantile range and requires that the population variances are finite. In this paper, we firstly propose a permutation version of these tests that does not require the condition of finite variances and that is remarkably more powerful than the original one (which is based on asymptotic critical values). Secondly we solve the question of what quantile choose to compute the interquantile range by proposing, within the permutation framework, a combined interquantile test based on our permutation version of Shoemaker tests. Shoemaker showed that the more extreme interquantile range tests are more powerful than the less extreme ones, unless the underlying distributions are very highly skewed. Since in practice the researcher may not know if the underlying distributions are very highly skewed or not, the question arises. The combined interquantile test solves this question, is robust and more powerful than the stand alone tests. Thirdly, to evaluate the performance of the permutation version of the Shoemaker test and of the combined test we designed and conducted a much more detailed simulation study than that of Shoemaker (1999) that compared his tests to the F and the squared rank tests showing that his tests are better. Since it is well known that the F and the squared rank test are not good tests for differences in scale his results suffer of such a drawback, and for this reason in our study instead of considering the squared rank test we consider other tests following the suggestions of some authors: the Brown-Forsythe (1974) W50 test, Pan (1999) M50 and L50 tests, O’Brien (1979) R test and the modified Fligner-Killeen FK test due to Conover et al. (1981).

A class of tests due to Shoemaker (Commun Stat Simul Comput 28: 189-205, 1999) for differences in scale which is valid for a variety of both skewed and symmetric distributions when location is known or unknown is considered. The class is based on the interquantile range and requires that the population variances are finite. In this paper, we firstly propose a permutation version of it that does not require the condition of finite variances and is remarkably more powerful than the original one. Secondly we solve the question of what quantile choose by proposing a combined interquantile test based on our permutation version of Shoemaker tests. Shoemaker showed that the more extreme interquantile range tests are more powerful than the less extreme ones, unless the underlying distributions are very highly skewed. Since in practice you may not know if the underlying distributions are very highly skewed or not, the question arises. The combined interquantile test solves this question, is robust and more powerful than the stand alone tests. Thirdly we conducted a much more detailed simulation study than that of Shoemaker (1999) that compared his tests to the F and the squared rank tests showing that his tests are better. Since the F and the squared rank test are not good for differences in scale, his results suffer of such a drawback, and for this reason instead of considering the squared rank test we consider, following the suggestions of several authors, tests due to Brown-Forsythe (J Am Stat Assoc 69:364-367, 1974), Pan (J Stat Comput Simul 63:59-71, 1999), O'Brien (J Am Stat Assoc 74:877-880, 1979) and Conover et al. (Technometrics 23:351-361, 1981). © 2010 Springer-Verlag.