A new distribution free inferential procedure for comparing paired observations, called the ‘rank difference test’, is presented. The new procedure is based on ranks, and is applicable as a robust alternative to the related f test in all situations where the Wilcoxon signed ranks test is applicable. In additions it may be applied to ordinal data or operational measures which do not meet the assumptions underlying the Wilcoxon. Since the rank difference statistic, D, can take on half‐integer, as well as integer, values, it has continuity advantages over the Wilcoxon for small samples. The exact null distribution of the rank difference test statistic, D, is derived and tabulated for samples sizes from 2 to 7 pairs. The null distribution of the rank difference test statistic is shown to be asymptotically the same as that of the Wilcoxon test statistic for large sample sizes. Permutation methods are used to derive the null distribution for sample sizes from 8 to 20 based on 400000 simulations for each sample size. The practical application of the rank difference test is evaluated by comparing its power and performance with that of the Wilcoxon, the sign test and the related t test, for several classic sets of data in the literature, and for several sets of simulated data.
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Diana Kornbrot (1990) studied this question.