Efficacies of Rank-Transform Statistics in Two-Way Models with No Interaction

原文摘要

To test for treatment effects in a two-way model when the classical assumptions of normality of errors and constancy of variance cannot be verified, Hora and Conover (1984) proposed a rank test in which the entire data set is ranked, the ranks are scored, and then the classical analysis of variance F statistic is applied to the scored ranks. They showed that the limiting null distribution of this test statistic is a chi-squared distribution divided by its degrees of freedom. Simulation results suggest that this procedure, called the rank-transform procedure, has good power properties. This article determines the asymptotic relative efficiency of the rank-transform procedure relative to the classical F statistic. To do this, vectors of linear rank statistics are shown to have a limiting multivariate normal distribution under a sequence of Pitman alternatives. This work is based on the results of Hájek (1968). The rank-transform statistic is then expressed as a quadratic form in the vectors, divided by a consistent estimator of a variance component. It is then shown that the limiting distribution of this statistic is a noncentral chi-squared distribution divided by its degrees of freedom. The efficacy of the rank-transform procedure is the noncentrality parameter of this chi-squared distribution, which is shown to be close to the efficacy of the Kruskal—Wallis test. For the special case in which the data can be represented as a one-way layout, this efficacy coincides with the Kruskal—Wallis efficacy.

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