Article
Farcical Results with Rank Procedures in Unbalanced Designs – Ranks and Pseudo-Ranks
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Authors
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Edgar Brunner
- Universitätsmedizin Göttingen Institue of Medical Statistics, Göttingen, Germany
| Published: | February 26, 2021 |
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Outline
Text
Background: If rank methods are used for d>2 samples then farcical results may be obtained in case of unequal sample sizes since the quantities pi = ∫H dFi on which rank procedures are based depend on the relative sample sizes through the definition of the weighted mean distribution function H in the experiment. This undesirable property applies to the well-known Kruskal-Wallis test [1], the trend test by Hettmansperger-Norton [2] as well as for the Akritas-Arnold-Brunner procedures [3] in factorial designs. In two-way layouts interactions may appear or disappear just by changing the ratios of the samples sizes while keeping fixed the underlying distributions. Such designs appear in the so-called sub-group analysis in clinical trials or in clinical epidemiology.
Methods: First some examples are presented and then the theoretical background of the farcical results is investigated by extending the well-known Mann-Whitney effect for two samples [4] to several samples and to factorial designs. It turns out that a nonparametric effect (a quantity not depending on relative samples sizes) could be based on comparisons of the distributions Fi for example, with the unweighted mean distribution G.
Results: The functionals ψi = ∫G dFi describe an effect of the distribution Fi relative to the mean distribution G. They can be estimated by simple plug-in estimators which can easily be computed by rank-like quantities, the so-called pseudo-ranks which have already been discussed in the literature [5]. The asymptotic distribution of these estimators and confidence intervals for ψi (see, e.g., [6] or [7]) are briefly discussed. For equal sample sizes and for two distributions ranks and pseudo-ranks are identical.
Conclusion: It is demonstrated that in case of unequal sample sizes procedures based on the nonparametric effects ψi do not have the farcical properties of rank-based procedures. An example from a register study in multiple sclerosis demonstrates the practical meaning.
The authors declare that they have no competing interests.
The authors declare that an ethics committee vote is not required.
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