Win odds: An adaptation of the win ratio to include ties

Abstract

The win ratio, a recently proposed measure for comparing the benefit of two treatment groups, allows ties in the data but ignores ties in the inference. In this article, we highlight some difficulties that this can lead to, and we propose to focus on the win odds instead, a modification of the win ratio which takes ties into account. We construct hypothesis tests and confidence intervals for the win odds, and we investigate their properties through simulations and in a case study. We conclude that the win odds should be preferred over the win ratio.

Power and sample size calculation for the win odds test: application to an ordinal endpoint in COVID-19 trials

“…The term "win ratio" was suggested by Pocock et al (2012) for an application in Heart Failure trials. Unlike the win odds, it does not account for ties, whereas the win odds is the odds of winning, following Dong et al (2020) (see also Peng 2020;Brunner et al 2021). The same statistic was named as Mann-Whitney odds in O' Brien and Castelloe (2006).…”

“…It should be noted that the function f ðxÞ is not range-preserving and hence may result in the lower bound of the CI (11) being less than 0. As suggested by Brunner et al (2021), a better approach is to exponentiate the limits of the CI for logð θ 1À θ Þ ¼ logðκÞ. For this purpose,…”

“…where SE is the standard error for θN (see ( 7)), is the approximate standard error for logð θN 1À θ N Þ ¼ logðκ N Þ from its linear Taylor series (or the δ-method, see Van der Vaart (2000)). These considerations lead to the CI in (12) for the win odds κN (Brunner et al 2021). Additionally, Kawaguchi et al (2011) indicated that consideration of logðκ N Þ and its approximate standard error SE θN ð1À θN Þ enabled better control of Type I error and coverage of confidence intervals for inferences about θ than direct consideration of θN , particularly when θ was not near 0.5 (see also Carr et al (1989) for other related properties of this transformation and for a simulation study).…”

“…To illustrate the coverage properties of the WO CI and the CI for the NNT, as well as to conduct a power analysis for this test, we will conduct a simulation study using categorization of variables from Beta distributions. The following approach was suggested by Brunner et al (2021). Suppose that the independent random variables � and η are ordinal and have 5 values that can be denoted by the numbers 1; 2; � � � ; 5 where a higher value indicates a worse outcome, meaning η wins against � if η < �.…”

“…Pocock et al (2012) introduced the concept of a win ratio (which does not account for ties) as a method of analyzing hierarchical composite endpoints in clinical trials where the endpoint components are not clinically equivalent (Packer 2016). The win odds statistic was introduced by Dong et al (2020) to account for the ties as half wins (see also Finkelstein and Schoenfeld (1999), for the review of recent methodology see Gasparyan et al (2021); Brunner et al (2021)). It estimates the treatment effect in the "Active" group against the "Placebo" group using the severity of the clinical outcomes in the composite as ordinal values.…”