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Ranks and Pseudo‐ranks—Surprising Results of Certain Rank Tests in Unbalanced Designs

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  • Edgar Brunner
  • Frank Konietschke
  • Arne C. Bathke
  • Markus Pauly

Abstract

Rank‐based inference methods are applied in various disciplines, typically when procedures relying on standard normal theory are not justifiable. Various specific rank‐based methods have been developed for two and more samples and also for general factorial designs (e.g. Kruskal–Wallis test or Akritas–Arnold–Brunner test). It is the aim of the present paper (1) to demonstrate that traditional rank procedures for several samples or general factorial designs may lead to surprising results in case of unequal sample sizes as compared with equal sample sizes, (2) to explain why this is the case and (3) to provide a way to overcome these disadvantages. Theoretical investigations show that the surprising results can be explained by considering the non‐centralities of the test statistics, which may be non‐zero for the usual rank‐based procedures in case of unequal sample sizes, while they may be equal to 0 in case of equal sample sizes. A simple solution is to consider unweighted relative effects instead of weighted relative effects. The former effects are estimated by means of the so‐called pseudo‐ranks, while the usual ranks naturally lead to the latter effects. A real data example illustrates the practical meaning of the theoretical discussions.

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  • Edgar Brunner & Frank Konietschke & Arne C. Bathke & Markus Pauly, 2021. "Ranks and Pseudo‐ranks—Surprising Results of Certain Rank Tests in Unbalanced Designs," International Statistical Review, International Statistical Institute, vol. 89(2), pages 349-366, August.
    • Handle: RePEc:bla:istatr:v:89:y:2021:i:2:p:349-366
    DOI: 10.1111/insr.12418
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    References listed on IDEAS

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    1. Gao, Xin & Alvo, Mayer, 2005. "A Unified Nonparametric Approach for Unbalanced Factorial Designs," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 926-941, September.
    2. Umlauft, Maria & Placzek, Marius & Konietschke, Frank & Pauly, Markus, 2019. "Wild bootstrapping rank-based procedures: Multiple testing in nonparametric factorial repeated measures designs," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 176-192.
    3. Olivier Thas & Jan De Neve & Lieven Clement & Jean-Pierre Ottoy, 2012. "Probabilistic index models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(4), pages 623-671, September.
    4. Dennis Dobler & Sarah Friedrich & Markus Pauly, 2020. "Nonparametric MANOVA in meaningful effects," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 997-1022, August.
    5. Edgar Brunner & Frank Konietschke & Markus Pauly & Madan L. Puri, 2017. "Rank-based procedures in factorial designs: hypotheses about non-parametric treatment effects," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1463-1485, November.
    6. Edgar Brunner & Madan Puri, 2001. "Nonparametric methods in factorial designs," Statistical Papers, Springer, vol. 42(1), pages 1-52, January.
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    1. Olga V. Demler & Ilona A. Demler, 2023. "Non-Transitivity of the Win Ratio and the Area Under the Receiver Operating Characteristics Curve (AUC): a case for evaluating the strength of stochastic comparisons," Papers 2309.01791, arXiv.org, revised Sep 2023.

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