Load the package hce and check the version
For citing the package run citation("hce") (Samvel B. Gasparyan 2024).
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 concept of winning for the active group is the same as concordance for the active group. For two variables, \(X\) and \(Y,\) this 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 the value 0 for the control group, and the variable \(Y\) is the ordinal value for the analysis, then concordance means that a patient with the active treatment has a better value than a patient with the control, while the 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).
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
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. \[\begin{align*} &L = O*\frac{2WP-1}{WR-1},\nonumber\\ &W = WR*L = WR*O*\frac{2WP-1}{WR-1},\nonumber\\ &T=O-W-L = O*\left[1 - (WR+1)\frac{2WP-1}{WR-1}\right]. \end{align*}\]
The 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 1.5Suppose there are \(n_1=120\)
patients in the placebo group and \(n_2=150\) in the active group. If 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 argument
Overall for all possible comparisons.