Asymptotic Relative Efficiencies of the Rank-Transformation Procedure in Randomized Complete Block Designs

原文摘要

This article provides insight into the powers of alternative tests for experimental effects in the presence of a blocking factor. Three rank tests and the usual F test are compared using asymptotic relative efficiencies (ARE's). Rank tests provide a useful alternative method of analysis when the assumptions of the F test are not met. Further, rank tests are usually more powerful than the F test when outliers are present or when the distribution of the data possesses heavy tails. The rank tests studied are Friedman's test, the aligned-ranks test, and the rank-transformation procedure. The rank-transformation test consists of ranking all of the data as one sample and then applying the usual F test to the ranks. Two main themes form the body of the article. The first theme is the development of the limiting noncentrality parameter of the rank-transformation statistic. The development uses the methods popularized by Hájek and Šidák (1967). A lemma is given that establishes the joint limiting distribution of individually asymptotically normal dependent random variables, if such a distribution exists. This lemma may have application outside the area studied here. The second theme is the evaluation of the ARE's among the four tests. Six distinct situations are analyzed. These situations are generated by selecting from several within-block densities and from several types of block effects that include location shifts and scale changes. The ARE's of the aligned-ranks test and Friedman's test relative to the usual F test are shown to be affected by the number of experimental factors, whereas the ARE of the rank-transformation procedure is not affected. Conversely, the ARE of the rank-transformation test is affected by block location shifts, whereas the other tests are unaffected. In the cases studied with normally distributed block location shifts, the ARE of the rank-transformation test to the normal-theory test falls to .866 as the variance of the block shifts becomes large relative to the within-block dispersion. For the cases studied with block shifts drawn from a uniform distribution, this ARE approaches 1 as the variance of the block shifts becomes large.

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