Stratified proportional win‐fractions regression analysis

The recently proposed proportional win‐fractions (PW) model extends the two‐sample win ratio analysis of prioritized composite endpoints to regression. Its proportionality assumption ensures that the covariate‐specific win ratios are invariant to the follow‐up time. However, this assumption is strong and may not be satisfied by every covariate in the model. We develop a stratified PW model that adjusts for certain prognostic factors without setting them as covariates, thus bypassing the proportionality requirement. We formulate the stratified model based on pairwise comparisons within each stratum, with a common win ratio across strata modeled as a multiplicative function of the covariates. Correspondingly, we construct an estimating function for the regression coefficients in the form of an incomplete U‐statistic consisting of within‐stratum pairs. Two types of asymptotic variance estimators are developed depending on the number of strata relative to the sample size. This in particular allows valid inference even when the strata are extremely small, such as with matched pairs. Simulation studies in realistic settings show that the stratified model outperforms the unstratified version in robustness and efficiency. Finally, real data from a major cardiovascular trial are analyzed to illustrate the potential benefits of stratification. The proposed methods are implemented in the R package WR, publicly available on the Comprehensive R Archive Network (CRAN).


Summary
This web appendix contains technical details, proofs and additional tables and figures referenced throughout the main text.

S1.1 Win function on recurrent events
For notational simplicity, consider only one counting process ( ) for recurrent nonfatal events. Let 1 < 2 < … denote the ordered times to the recurrent events counted by ( ). Following Mao et al. (2022) 1 , we can first compare on survival, and then the cumulative frequency of recurrent events, with ties further broken by time to the last occurrence. This leads to where and are on the th and th subject in the th stratum, respectively. Extending §2.4 of Mao et al. 1 slightly, we find that SPW( ) holds with regression parameter if where ( , 1 , 2 , …) is a nonparametric baseline joint survival function for ( , 1 , 2 , …). Like the counterexample offered in Mao and Wang (2021) 2 , model (S1) is sufficient but far from necessary for SPW( ).

S1.2 Efficiency of weights
Here we give a heuristic argument about the efficiency of weights. Consider only time-constant weights but allow weights to vary across strata, i.e.ĥ = for = 1, … , , where 's are some constants. Without loss of generality, assume ∑ =1 = 1 and for simplicity, assume is one dimensional. Then for each U-statistic estimating function in stratum , we have
• (C3) 0 , the true value of , lies in the interior of a compact subset  of ℝ for a fixed symmetric function ℎ ( ; ⋅, ⋅; ). The limit function ℎ ( ; ⋅, ⋅; ) is uniformly bounded with uniformly bounded total variation, is uniformly continuous in , and satisfies Let̂ be the estimator obtained from solving (5) of the main text. We first prove consistency. Define Therefore, which is globally negative definite under (C2), (C3) and (C5). Thus, ( ) is globally concave and̂ = arg max ( ). Because the parameter space  is compact, we may assume without loss of generality that̂ → * for some * ∈ .
Using Taylor expansion, the left hand side of (S2) is ⎞ ⎟ ⎟ ⎠ Using Hoeffding decomposition, the right hand side of (S2) is

S2 ADDITIONAL FIGURES AND TABLES FOR THE ACCORD STUDY DATASET
Figures S1 and S2 show the standardized score processes for the sex-and age-stratified analyses in Sections 5.1 and 5.2, respectively. Table S1 summarizes patient characteristics of the study cohort in Section 5.2.

S3 ANALYSIS OF THE HF-ACTION STUDY BY SPW( )
We conducted additional analyses on a cardiovascular trial to illustrate the applications of SPW models on recurrent events, using the win function  described in Section S1.1. The Heart Failure: A Controlled Trial Investigating Outcomes of Exercise Training (HF-ACTION) trial was a randomized controlled clinical trial conducted on a cohort of over 2,000 patients to evaluate the efficacy and safety of exercise training among patients with heart failure. 5 The primary endpoint was a composite of allcause mortality and all-cause hospitalizations. The primary analysis showed a moderate beneficial effect of exercise training in reducing the risk of the first composite event compared to usual care alone with a nonsignificant hazard ratio of 0.93 (P-value = 0.13).
We reanalyzed the HF-ACTION study by modeling all-cause death and repeated hospitalizations using SPW( ). The study cohort consists of 2,130 patients, with 1,060 randomized to exercise training and 1,070 to usual care. The mortality rate in the training arm is about 15.8%, with an average of 2.0 hospitalizations per patient; the mortality rate in the usual care arm is about 17.1%, with an average of 2.0 hospitalizations per patient. We included the treatment indicator, sex, age, and body mass index (BMI) in the SPW model stratified by heart failure etiology, i.e., nonischemic or ischemic. Table S2 summarizes the estimated win ratio and confidence intervals, constructed using the variance estimator under finite strata. Within each etiology group and adjusting for other predictors, patients going through exercise training are 6% more likely to have a favorable composite outcome