On the alternative hypotheses for the win ratio

Defining estimand for the win ratio: Separate the true effect from censoring

The win ratio has been increasingly used in trials with hierarchical composite endpoints. While the outcomes involved and the rule for their comparisons vary with the application, there is invariably little attention to the estimand of the resulting statistic, causing difficulties in interpretation and cross-trial comparison. We make the case for articulating the estimand as a first step to win ratio analysis and establish that the root cause for its elusiveness is its intrinsic dependency on the time frame of comparison, which, if left unspecified, is set haphazardly by trial-specific censoring. From the statistical literature, we summarize two general approaches to overcome this uncertainty-a nonparametric one that pre-specifies the time frame for all comparisons, and a semiparametric one that posits a constant win ratio across all times-each with publicly available software and real examples. Finally, we discuss unsolved challenges, such as estimand construction and inference in the presence of intercurrent events.

A win ratio approach for comparing crossing survival curves in clinical trials

Many clinical trials include time-to-event or survival data as an outcome. To compare two survival distributions, the log-rank test is often used to produce a P-value for a statistical test of the null hypothesis that the two survival curves are identical. However, such a P-value does not provide the magnitude of the difference between the curves regarding the treatment effect. As a result, the P-value is often accompanied by an estimate of the hazard ratio from the proportional hazards model or Cox model as a measurement of treatment difference. However, one of the most important assumptions for Cox model is that the hazard functions for the two treatment groups are proportional. When the hazard curves cross, the Cox model could lead to misleading results and the log-rank test could also perform poorly. To address the problem of crossing curves in survival analysis, we propose the use of the win ratio method put forward by Pocock et al. as an estimand for analysing such data. The subjects in the test and control treatment groups are formed into all possible pairs. For each pair, the test treatment subject is labelled a winner or a loser if it is known who had the event of interest such as death. The win ratio is the total number of winners divided by the total number of losers and its standard error can be estimated using Bebu and Lachin method. Using real trial datasets and Monte Carlo simulations, this study investigates the power and type I error and compares the win ratio method with the log-rank test and Cox model under various scenarios of crossing survival curves with different censoring rates and distribution parameters. The results show that the win ratio method has similar power as the log-rank test and Cox model to detect the treatment difference when the assumption of proportional hazards holds true, and that the win ratio method outperforms log-rank test and Cox model in terms of power to detect the treatment difference when the survival curves cross.

The win odds: statistical inference and regression

Generalized pairwise comparisons and win statistics (i.e., win ratio, win odds and net benefit) are advantageous in analyzing and interpreting a composite of multiple outcomes in clinical trials. An important limitation of these statistics is their inability to adjust for covariates other than by stratified analysis. Because the win ratio does not account for ties, the win odds, a modification that includes ties, has attracted attention. We review and combine information on the win odds to articulate the statistical inferences for the win odds. We also show alternative variance estimators based on the exact permutation and bootstrap as well as statistical inference via the probabilistic index. Finally, we extend multiple-covariate regression probabilistic index models to the win odds with a univariate outcome. As an illustration we apply the regression models to the data in the CHARM trial.

The stratified win statistics (win ratio, win odds, and net benefit)

The win odds and the net benefit are related directly to each other and indirectly, through ties, to the win ratio. These three win statistics test the same null hypothesis of equal win probabilities between two groups. They provide similar p-values and powers, because the Z-values of their statistical tests are approximately equal. Thus, they can complement one another to show the strength of a treatment effect. In this article, we show that the estimated variances of the win statistics are also directly related regardless of ties or indirectly related through ties. Since its introduction in 2018, the stratified win ratio has been applied in designs and analyses of clinical trials, including Phase III and Phase IV studies. This article generalizes the stratified method to the win odds and the net benefit. As a result, the relations of the three win statistics and the approximate equivalence of their statistical tests also hold for the stratified win statistics.

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 incompleteU $$ U $$ -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).

Power and Sample Size Calculations for the Restricted Mean Time Analysis of Prioritized Composite Endpoints

As a new way of reporting treatment effect, the restricted mean time in favor (RMT-IF) of treatment measures the net average time the treated have had a less serious outcome than the untreated over a specified time window. With multiple outcomes of differing severity, this offers a more interpretable and data-efficient alternative to the prototypical restricted mean (event-free) survival time. To facilitate its adoption in actual trials, we develop simple approaches to power and sample size calculations and implement them in user-friendly R programs. In doing so we model the bivariate outcomes of death and a nonfatal event using a Gumbel-Hougaard copula with component-wise proportional hazards structures, under which the RMT-IF estimand is derived in closed form. In a standard set-up for censoring, the variance of the nonparametric effect-size estimator is simplified and computed via a hybrid of numerical and Monte Carlo integrations, allowing us to compute the power and sample size as functions of component-wise hazard ratios. Simulation studies show that these formulas provide accurate approximations in realistic settings. To illustrate our methods, we consider designing a new trial to evaluate treatment effect on the composite outcomes of death and cancer relapse in lymph node-positive breast cancer patients, with baseline parameters calculated from a previous study.

Sample size formula for general win ratio analysis

Originally proposed for the analysis of prioritized composite endpoints, the win ratio has now expanded into a broad class of methodology based on general pairwise comparisons. Complicated by the non-i.i.d. structure of the test statistic, however, sample size estimation for the win ratio has lagged behind. In this article, we develop general and easy-to-use formulas to calculate sample size for win ratio analysis of different outcome types. In a nonparametric setting, the null variance of the test statistic is derived using U-statistic theory in terms of a dispersion parameter called the standard rank deviation, an intrinsic characteristic of the null outcome distribution and the user-defined rule of comparison. The effect size can be hypothesized either on the original scale of the population win ratio, or on the scale of a "usual" effect size suited to the outcome type. The latter approach allows one to measure the effect size by, for example, odds/continuation ratio for totally/partially ordered outcomes and hazard ratios for composite time-to-event outcomes. Simulation studies show that the derived formulas provide accurate estimates for the required sample size across different settings. As illustration, real data from two clinical studies of hepatic and cardiovascular diseases are used as pilot data to calculate sample sizes for future trials.

On recurrent-event win ratio

The win ratio approach proposed by Pocock et al. (2012) has become a popular tool for analyzing composite endpoints of death and non-fatal events like hospitalization. Its standard version, however, draws on the non-fatal event only through the first occurrence. For statistical efficiency and clinical interpretability, we construct and compare different win ratio variants that make fuller use of recurrent events. We pay special attention to a variant called last-event-assisted win ratio, which compares two patients on the cumulative frequency of the non-fatal event, with ties broken by the time of its latest episode. It is shown that last-event-assisted win ratio uses more data than the standard win ratio does but reduces to the latter when the non-fatal event occurs at most once. We further prove that last-event-assisted win ratio rejects the null hypothesis with large probability if the treatment stochastically delays all events. Simulations under realistic settings show that the last-event-assisted win ratio test consistently enjoys higher power than the standard win ratio and other competitors. Analysis of a real cardiovascular trial provides further evidence for the practical advantages of the last-event-assisted win ratio. Finally, we discuss future work to develop meaningful effect size estimands based on the extended rules of comparison. The R-code for the proposed methods is included in the package WR openly available on the Comprehensive R Archive Network.

Event-specific win ratios for inference with terminal and non-terminal events

For semi-competing risks data involving a non-terminal event and a terminal event we derive the asymptotic distributions of the event-specific win ratios under proportional hazards (PH) assumptions for the relevant cause-specific hazard functions of the non-terminal and terminal event, respectively. The win ratios converge to the respective hazard ratios under the PH assumptions and therefore are censoring-free, whether or not the censoring distributions in the two treatment arms are the same. With the asymptotic bivariate normal distributions of the win ratios, confidence intervals and testing procedures are obtained. Through extensive simulation studies and data analysis, we identified proper transformations of the win ratios that yield good control of the type one error rate for various testing procedures while maintaining competitive power. The confidence intervals also have good coverage probabilities. Furthermore, a test for the PH assumptions and a test of equal hazard ratios are developed. The new procedures are illustrated in the clinical trial Aldosterone Antagonist Therapy for Adults With Heart Failure and Preserved Systolic Function, which evaluated the effects of spironolactone in patients with heart failure and a preserved left ventricular ejection fraction.

Statistical models for composite endpoints of death and non-fatal events: a review

The proper analysis of composite endpoints consisting of both death and non-fatal events is an intriguing and sometimes contentious topic. The current practice of analyzing time to the first event often draws criticisms for ignoring the unequal importance between component events and for leaving recurrent-event data unused. Novel methods that address these limitations have recently been proposed. To compare the novel versus traditional approaches, we review three typical models for composite endpoints based on time to the first event, composite event process, and pairwise hierarchical comparisons. The pros and cons of these models are discussed with reference to the relevant regulatory guidelines, such as the recently released ICH-E9(R1) Addendum "Estimands and Sensitivity Analysis in Clinical Trials". We also discuss the impact of censoring when the model assumptions are violated and explore sensitivity analysis strategies. Simulation studies are conducted to assess the performance of the reviewed methods under different settings. As demonstration, we use publicly available R-packages to analyze real data from a major cardiovascular trial.

Adjusting win statistics for dependent censoring

For composite outcomes whose components can be prioritized on clinical importance, the win ratio, the net benefit and the win odds apply that order in comparing patients pairwise to produce wins and subsequently win proportions. Because these three statistics are derived using the same win proportions and they test the same hypothesis of equal win probabilities in the two treatment groups, we refer to them as win statistics. These methods, particularly the win ratio and the net benefit, have received increasing attention in methodological research and in design and analysis of clinical trials. For time-to-event outcomes, however, censoring may introduce bias. Previous work has shown that inverse-probability-of-censoring weighting (IPCW) can correct the win ratio for bias from independent censoring. The present article uses the IPCW approach to adjust win statistics for dependent censoring that can be predicted by baseline covariates and/or time-dependent covariates (producing the CovIPCW-adjusted win statistics). Theoretically and with examples and simulations, we show that the CovIPCW-adjusted win statistics are unbiased estimators of treatment effect in the presence of dependent censoring.

Event-specific win ratios and testing with terminal and non-terminal events

BACKGROUND/AIMS

In clinical trials, the primary outcome is often a composite endpoint defined as time to the first occurrence of either death or certain non-fatal events. Thus, a portion of available data would be omitted. In the win ratio approach, priorities are given to the clinically more important events, and more data are used. However, its power may be low if the treatment effect is predominantly on the non-terminal event.

METHODS

We propose event-specific win ratios obtained separately on the terminal and non-terminal events. They can then be used to form global tests such as a linear combination test, the maximum test, or a χ2 test.

RESULTS

In simulations, these tests often improve the power of the original win ratio test. Furthermore, when the terminal and non-terminal events experience differential treatment effects, the new tests are often more powerful than the log-rank test for the composite outcome. Whether the treatment effect is primarily on the terminal events or not, the new tests based on the event-specific win ratios can be useful when different types of events are present. The new tests can reject the null hypothesis of no difference in the event distributions in the two treatment arms with the terminal event showing detrimental effect and the non-terminal event showing beneficial effect. The maximum test and the χ2 test do not have test-estimation coherency, but the maximum test has the coherency that the global null is rejected if and only if the null for one of the event types is rejected. When applied to data from the trial Aldosterone Antagonist Therapy for Adults With Heart Failure and Preserved Systolic Function (TOPCAT), the new tests all reject the null hypothesis of no treatment effect while both the log-rank test used in TOPCAT and the original win ratio approach show non-significant p-values.

CONCLUSION

Whether the treatment effect is primarily on the terminal events or the non-terminal events, the maximum test based on the event-specific win ratios can be a useful alternative for testing treatment effect in clinical trials with time-to-event outcomes when different types of events are present.

A class of proportional win-fractions regression models for composite outcomes

The win ratio is gaining traction as a simple and intuitive approach to analysis of prioritized composite endpoints in clinical trials. To extend it from two-sample comparison to regression, we propose a novel class of semiparametric models that includes as special cases both the two-sample win ratio and the traditional Cox proportional hazards model on time to the first event. Under the assumption that the covariate-specific win and loss fractions are proportional over time, the regression coefficient is unrelated to the censoring distribution and can be interpreted as the log win ratio resulting from one-unit increase in the covariate. U-statistic estimating functions, in the form of an arbitrary covariate-specific weight process integrated by a pairwise residual process, are constructed to obtain consistent estimators for the regression parameter. The asymptotic properties of the estimators are derived using uniform weak convergence theory for U-processes. Visual inspection of a "score" process provides useful clues as to the plausibility of the proportionality assumption. Extensive numerical studies using both simulated and real data from a major cardiovascular trial show that the regression methods provide valid inference on covariate effects and outperform the two-sample win ratio in both efficiency and robustness. The proposed methodology is implemented in the R-package WR, publicly available from the Comprehensive R Archive Network (CRAN).

The inverse-probability-of-censoring weighting (IPCW) adjusted win ratio statistic: an unbiased estimator in the presence of independent censoring

The win ratio method has received much attention in methodological research, ad hoc analyses, and designs of prospective studies. As the primary analysis, it supported the approval of tafamidis for the treatment of cardiomyopathy to reduce cardiovascular mortality and cardiovascular-related hospitalization. However, its dependence on censoring is a potential shortcoming. In this article, we propose the inverse-probability-of-censoring weighting (IPCW) adjusted win ratio statistic (i.e., the IPCW-adjusted win ratio statistic) to overcome censoring issues. We consider independent censoring, common censoring across endpoints, and right censoring. We develop an asymptotic variance estimator for the logarithm of the IPCW-adjusted win ratio statistic and evaluate it via simulation. Our simulation studies show that, as the amount of censoring increases, the unadjusted win proportions may decrease greatly. Consequently, the bias of the unadjusted win ratio estimate may increase greatly, producing either an overestimate or an underestimate. We demonstrate theoretically and through simulation that the IPCW-adjusted win ratio statistic gives an unbiased estimate of treatment effect.