Win odds, win ratio, and net benefit

Load the package hce and check the version

library(hce)
packageVersion("hce")
#> [1] '0.5.0'

For citing the package run citation("hce") (Samvel B. Gasparyan 2022).

Two treatment groups are compared using an ordinal endpoint and each comparison results in a win, loss, or a tie for the patient in the active group compared to a patient in the placebo group. All possible (overall) combinations are denoted by \(O\), with \(W\) denoted the total wins for the active group, \(L\) total losses, and \(T\) the total ties, so that \(O=W+L+T.\) Then the following quantities are called win statistics

  • Win proportion defined as \(WP=\frac{W+0.5T}{O}\), that is, the total number of wins, added half of the total number of ties, divided by the overall number of comparisons.
  • Number needed to treat defined as \(NNT=\frac{1}{2WP-1}\) rounded up to the nearest natural number.
  • Win ratio defined as \(WR=\frac{W}{L}\).
  • Win odds defined as \(WO=\frac{W+0.5T}{L+0.5T} = \frac{WP}{1-WP}\).
  • Net Benefit defined as \(NB=\frac{W - L}{O}\).

Given the overall number of comparisons \(O,\) the win proportion \(WP\) and the win ratio \(WR\), it is possible to find the total number of wins and losses. Indeed, first the win odds can be found \(WO=\frac{WP}{WP+1}\) and

\[\begin{align*} &L = O*\frac{2WP-1}{WR-1},\nonumber\\ &W = WR*L = WR*O*\frac{2WP-1}{WR-1},\nonumber\\ &T=O-W-L = O*\left[1 - (WR+1)\frac{2WP-1}{WR-1}\right]. \end{align*}\]

The concept of win probability for the binary and continuous outcomes has been described in the paper by Buyse (2010) as “proportion in favor of treatment” (see also Rauch et al. (2014)), while in Verbeeck et al. (2021) it is called “probabilistic index”.

The concept of “win ratio” was introduced in Pocock et al. (2012), which, unlike the win odds, does not account for ties, whereas the win odds is the odds of winning, following G. Dong et al. (2020) (see also Peng (2020); Brunner, Vandemeulebroecke, and Mütze (2021); Samvel B. Gasparyan, Kowalewski, et al. (2021)). The same statistic was named as Mann-Whitney odds in O’Brien and Castelloe (2006). In Samvel B. Gasparyan, Folkvaljon, et al. (2021) the “win ratio” was used as a general term and included ties in the definition. Gaohong Dong et al. (2022) suggested to consider win ratio, win odds, and net benefit together as win statistics.

The function propWINS() implements the formula above

args("propWINS")
#> function (WO, WR, Overall = 1) 
#> NULL
propWINS(WO = 1.5, WR = 2)
#>   WIN LOSS TIE TOTAL
#> 1 0.4  0.2 0.4     1

In case we have \(n_1=120\) patients in the placebo group and \(n_2=150\) in the active group and need to know given the win ratio and win odds above how many wins and losses we will have for the active group then the argument Overall can be used to specify the number of all comparisons

propWINS(WO = 1.25, WR = 1.5, Overall = 120*150)
#>    WIN LOSS  TIE TOTAL
#> 1 6000 4000 8000 18000

References

Brunner, Edgar, Marc Vandemeulebroecke, and Tobias Mütze. 2021. “Win Odds: An Adaptation of the Win Ratio to Include Ties.” Statistics in Medicine. https://doi.org/10.1002/sim.8967.
Buyse, Marc. 2010. “Generalized Pairwise Comparisons of Prioritized Outcomes in the Two-Sample Problem.” Statistics in Medicine 29 (30): 3245–57.
Dong, Gaohong, Bo Huang, Johan Verbeeck, Ying Cui, James Song, Margaret Gamalo-Siebers, Duolao Wang, et al. 2022. “Win Statistics (Win Ratio, Win Odds, and Net Benefit) Can Complement One Another to Show the Strength of the Treatment Effect on Time-to-Event Outcomes.” Pharmaceutical Statistics. https://doi.org/10.1002/pst.2251.
Dong, G, DC Hoaglin, J Qiu, RA Matsouaka YW Chang, J Wang, and M Vandemeulebroecke. 2020. “The Win Ratio: On Interpretation and Handling of Ties.” Statistics in Biopharmaceutical Research 12 (1): 99–106. https://doi.org/10.1080/19466315.2019.1575279.
Gasparyan, Samvel B. 2022. hce: Design and Analysis of Hierarchical Composite Endpoints. CRAN: The Comprehensive R Archive Network, R Package, Version 0.5.0. https://CRAN.R-project.org/package=hce.
Gasparyan, Samvel B, Folke Folkvaljon, Olof Bengtsson, Joan Buenconsejo, and Gary G Koch. 2021. “Adjusted Win Ratio with Stratification: Calculation Methods and Interpretation.” Statistical Methods in Medical Research 30 (2): 580–611. https://doi.org/10.1177/0962280220942558.
Gasparyan, Samvel B, Elaine K Kowalewski, Folke Folkvaljon, Olof Bengtsson, Joan Buenconsejo, John Adler, and Gary G Koch. 2021. “Power and Sample Size Calculation for the Win Odds Test: Application to an Ordinal Endpoint in COVID-19 Trials.” Journal of Biopharmaceutical Statistics 31 (6): 765–87. https://doi.org/10.1080/10543406.2021.1968893.
O’Brien, RG, and JM Castelloe. 2006. Exploiting the Link Between the Wilcoxon–Mann–Whitney Test and a Simple Odds Statistic. Cary, NC: SAS Institute Inc. https://support.sas.com/resources/papers/proceedings/proceedings/sugi31/209-31.pdf.
Peng, Lei. 2020. “The Use of the Win Odds in the Design of Non-Inferiority Clinical Trials.” Journal of Biopharmaceutical Statistics 30 (5): 941–46. https://doi.org/10.1080/10543406.2020.1757690.
Pocock, SJ, CA Ariti, TJ Collier, and D Wang. 2012. “The Win Ratio: A New Approach to the Analysis of Composite Endpoints in Clinical Trials Based on Clinical Priorities.” European Heart Journal 33 (2): 176–82. https://doi.org/10.1093/eurheartj/ehr352.
Rauch, Geraldine, Antje Jahn-Eimermacher, Werner Brannath, and Meinhard Kieser. 2014. “Opportunities and Challenges of Combined Effect Measures Based on Prioritized Outcomes.” Statistics in Medicine 33 (7): 1104–20. https://doi.org/10.1002/sim.3923.
Verbeeck, Johan, Vaiva Deltuvaite-Thomas, Ben Berckmoes, Tomasz Burzykowski, Marc Aerts, Olivier Thas, Marc Buyse, and Geert Molenberghs. 2021. “Unbiasedness and Efficiency of Non-Parametric and UMVUE Estimators of the Probabilistic Index and Related Statistics.” Statistical Methods in Medical Research 30 (3): 747–68. https://doi.org/10.1177/0962280220966629.