Win statistics

Setup

Load the package hce and check the version:

library(hce)
packageVersion("hce")
#> [1] '0.6.5'

For citing the package, run citation("hce") (Samvel B. Gasparyan 2024).

Terminology

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

The concept of “win ratio” was introduced by Pocock et al. (2012). Unlike the win odds, the win ratio does not account for ties. The win odds, however, is the odds of winning, as described by 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 the Mann-Whitney odds by 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 considering win ratio, win odds, and net benefit together as win statistics.

The concept of winning for the active group is the same as concordance for the active group. For two variables, X and Y, a pair is concordant if the observation with the larger value of X also has the better value of Y (Agresti 2013). If X indicates the treatment group with the value 1 for the active group and 0 for the control group, and Y is the ordinal value for the analysis, then concordance means that a patient in the active treatment group has a better value than a patient in the control group. Discordance means the patient in the active group has a worse value than the control patient. Therefore, the win ratio is the total number of concordances divided by the total number of discordances. The win ratio can be obtained from the Goodman-Kruskal gamma (Kruskal and Goodman 1954), G, as follows:

WR = (1 + G)/(1 − G).

The net benefit is Somers’ D C/R (Somers 1962), while the win odds is the Mann-Whitney odds (Mann and Whitney 1947). Estimation of win statistics in the absence of censoring can be done using the theory of U-statistics (Hoeffding 1948).

Definitions

Two treatment groups are compared using an ordinal endpoint, and each comparison results in a win, loss, or 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 denoting the total wins for the active group, L the total losses, and T the total ties, so that O = W + L + T. The following quantities are called win statistics:

• Win Probability defined as $WP=\frac{W+0.5T}{O}$, which is the total number of wins, adding half of the total number of ties, divided by the overall number of comparisons.

• Number Needed to Treat defined as $NNT=\frac{1}{2WP-1}=\frac{O}{W-L}$ (rounded up to the nearest natural number for interpretation).

• 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 $NB=\frac{W - L}{O}=2WP-1=\frac{1}{NNT}$.

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:

The function propWINS() implements the formula above:

args("propWINS")
#> function (WO, WR, Overall = 1, alpha = NULL, N = NULL) 
#> NULL
propWINS(WO = 1.5, WR = 2)
#>   WIN LOSS TIE TOTAL WR  WO
#> 1   0    0   1     1  2 1.5

Suppose there are n₁ = 120 patients in the placebo group and n₂ = 150 in the active group. If the win ratio is 1.5 and the win odds is 1.25, then the number of wins and losses for the active group can be calculated using the Overall argument, which represents all possible comparisons.

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

This function call will provide the number of wins, losses, and ties for the active group based on the specified win odds and win ratio.