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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

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  • Olga V. Demler
  • Ilona A. Demler

Abstract

The win ratio (WR) is a novel statistic used in randomized controlled trials that can account for hierarchies within event outcomes. In this paper we report and study the long-run non-transitive behavior of the win ratio and the closely related Area Under the Receiver Operating Characteristics Curve (AUC) and argue that their transitivity cannot be taken for granted. Crucially, traditional within-group statistics (i.e., comparison of means) are always transitive, while the WR can detect non-transitivity. Non-transitivity provides valuable information on the stochastic relationship between two treatment groups, which should be tested and reported. We specify the necessary conditions for transitivity, the sufficient conditions for non-transitivity, and demonstrate non-transitivity in a real-life large randomized controlled trial for the WR of time-to-death. Our results can be used to rule out or evaluate the possibility of non-transitivity and show the importance of studying the strength of stochastic relationships.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2309.01791
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    References listed on IDEAS

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    1. Gillen, Daniel L. & Emerson, Scott S., 2007. "Nontransitivity in a class of weighted logrank statistics under nonproportional hazards," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 123-130, January.
    2. Lumley, Thomas & Gillen, Daniel L., 2016. "Characterising transitive two-sample tests," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 118-123.
    3. 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.
    4. Gorbunova, A.V. & Lebedev, A.V., 2022. "Nontransitivity of tuples of random variables with polynomial density and its effects in Bayesian models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 181-192.
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