On restricted mean time in favor of treatment

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.

Use of win time for ordered composite endpoints in clinical trials

Consider the choice of outcome for overall treatment benefit in a clinical trial which measures the first time to each of several clinical events. We describe several new variants of the win ratio that incorporate the time spent in each clinical state over the common follow-up, where clinical state means the worst clinical event that has occurred by that time. One version allows restriction so that death during follow-up is most important, while time spent in other clinical states is still accounted for. Three other variants are described; one is based on the average pairwise win time, one creates a continuous outcome for each participant based on expected win times against a reference distribution and another that uses the estimated distributions of clinical state to compare the treatment arms. Finally, a combination testing approach is described to give robust power for detecting treatment benefit across a broad range of alternatives. These new methods are designed to be closer to the overall treatment benefit/harm from a patient's perspective, compared to the ordinary win ratio. The new methods are compared to the composite event approach and the ordinary win ratio. Simulations show that when overall treatment benefit on death is substantial, the variants based on either the participants' expected win times (EWTs) against a reference distribution or estimated clinical state distributions have substantially higher power than either the pairwise comparison or composite event methods. The methods are illustrated by re-analysis of the trial heart failure: a controlled trial investigating outcomes of exercise training.

Dissecting the restricted mean time in favor of treatment

The restricted mean time in favor (RMT-IF) summarizes the treatment effect on a hierarchical composite endpoint with mortality at the top. Its crude decomposition into "stage-wise effects," i.e., the net average time gained by the treatment prior to each component event, does not reveal the patient state in which the extra time is spent. To obtain this information, we break each stage-wise effect into subcomponents according to the specific state to which the reference condition is improved. After re-expressing the subcomponents as functionals of the marginal survival functions of outcome events, we estimate them conveniently by plugging in the Kaplan -- Meier estimators. Their robust variance matrices allow us to construct joint tests on the decomposed units, which are particularly powerful against component-wise differential treatment effects. By reanalyzing a cancer trial and a cardiovascular trial, we acquire new insights into the quality and composition of the extra survival times, as well as the extra time with fewer hospitalizations, gained by the treatment in question. The proposed methods are implemented in the rmt package freely available on the Comprehensive R Archive Network (CRAN).

Win-loss parameters for right-censored event data, with application to recurrent events

The win ratio has become a popular method for comparing multiple event data between two groups in clinical cohort studies. The win ratio compares the event data in prioritized order, where the first prioritized event is death and a typical example for the second prioritized event is hospitalization. Literature is sparse on inference for win and loss parameters, including the win ratio, for censored event data. Inference for two prioritized censored event times has been developed for independent right-censoring. Many clinical studies include recurrent event data such as hospitalizations. In this article, we suggest inference for win-loss parameters for death and a recurrent event outcome under independent right-censoring. The small sample properties of the proposed method are studied in a simulation study showing that the variance formula is accurate even for small samples. The method is applied on a data set from a randomized clinical trial.

Study design for restricted mean time analysis of recurrent events and death

The restricted mean time in favor (RMT-IF) of treatment has just been added to the analytic toolbox for composite endpoints of recurrent events and death. To help practitioners design new trials based on this method, we develop tools to calculate the sample size and power. Specifically, we formulate the outcomes as a multistate Markov process with a sequence of transient states for recurrent events and an absorbing state for death. The transition intensities, in this case the instantaneous risks of another nonfatal event or death, are assumed to be time-homogeneous but nonetheless allowed to depend on the number of past events. Using the properties of Coxian distributions, we derive the RMT-IF effect size under the alternative hypothesis as a function of the treatment-to-control intensity ratios along with the baseline intensities, the latter of which can be easily estimated from historical data. We also reduce the variance of the nonparametric RMT-IF estimator to calculable terms under a standard set-up for censoring. Simulation studies show that the resulting formulas provide accurate approximation to the sample size and power in realistic settings. For illustration, a past cardiovascular trial with recurrent-hospitalization and mortality outcomes is analyzed to generate the parameters needed to design a future trial. The procedures are incorporated into the rmt package along with the original methodology on the Comprehensive R Archive Network (CRAN).

Design of paediatric trials with benefit-risk endpoints using a composite score of adverse events of interest (AEI) and win-statistics

A fundamental problem in the regulatory evaluation of a therapy is assessing whether the benefit outweighs the associated risks. This work proposes designing a trial that assesses a composite endpoint consisting of benefit and risk, hence, making the core of the design of the study, to assess benefit and risk. The proposed benefit risk measure consists of efficacy measure(s) and a risk measure that is based on a composite score obtained from pre-defined adverse events of interest (AEI). This composite score incorporates full aspects of adverse events of interest (i.e. the incidence, severity, and duration of the events). We call this newly proposed score the AEI composite score. After specifying the priorities between the components of the composite endpoint, a win-statistic (i.e. win ratio, win odds, or net benefit) is used to assess the difference between treatments in this composite endpoint. The power and sample size requirements of such a trial design are explored via simulation. Finally, using Dupixent published adult study results, we show how we can design a paediatric trial where the primary outcome is a composite of prioritized outcomes consisting of efficacy endpoints and the AEI composite score endpoint. The resulting trial design can potentially substantially reduce sample size compared to a trial designed to assess the co-primary efficacy endpoints, therefore it may address the challenge of slow enrollment and patient availability for paediatric studies.

Nonparametric inference of general while-alive estimands for recurrent events

Measuring the treatment effect on recurrent events like hospitalization in the presence of death has long challenged statisticians and clinicians alike. Traditional inference on the cumulative frequency unjustly penalizes survivorship as longer survivors also tend to experience more adverse events. Expanding a recently suggested idea of the "while-alive" event rate, we consider a general class of such estimands that adjust for the length of survival without losing causal interpretation. Given a user-specified loss function that allows for arbitrary weighting, we define as estimand the average loss experienced per unit time alive within a target period and use the ratio of this loss rate to measure the effect size. Scaling the loss rate by the width of the corresponding time window gives us an alternative, and sometimes more photogenic, way of showing the data. To make inferences, we construct a nonparametric estimator for the loss rate through the cumulative loss and the restricted mean survival time and derive its influence function in closed form for variance estimation and testing. As simulations and analysis of real data from a heart failure trial both show, the while-alive approach corrects for the false attenuation of treatment effect due to patients living longer under treatment, with increased statistical power as a result. The proposed methods are implemented in the R-package WA, which is publicly available from 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.

Quantifying Treatment Effects in Trials with Multiple Event-Time Outcomes

BACKGROUND

Data on the occurrence times of multiple outcomes, reflecting the temporal profile of disease burden/progression, have been used to estimate treatment effects in various recent randomized trials. Most procedures for analyzing these data require specific model assumptions. When the assumptions are not met, the results may be misleading. Robust, model-free procedures for study design and analysis that enable clinically meaningful interpretations are warranted.

METHODS

For each treatment group, we constructed and summarized the estimated mean cumulative count of events over time by the area under the curve (AUC), which can be interpreted as the mean total event-free time lost from multiple undesirable outcomes. A higher curve, and resulting larger AUC, implies a worse treatment. The treatment effect is quantified by the ratio and/or difference of AUCs. The timing and occurrence of recurrent heart failure hospitalizations (HFHs) and cardiovascular (CV) death from Prospective Comparison of ARNI with ARB Global Outcomes in HF with Preserved Ejection Fraction (PARAGON-HF), comparing sacubitril/valsartan with valsartan, are presented for illustration. We also discuss the design of future studies on the basis of the proposed method.

RESULTS

With 48 months of follow-up, estimated AUCs, representing the total event-free time lost to HFHs and CV death, were 11.3 and 13.1 event-months for sacubitril/valsartan and valsartan, respectively. The ratio of these AUCs was 0.86 (95% confidence interval, 0.75 to 1.00; P=0.049), a 14% reduction of disease burden favoring combination therapy. A future study, similar to PARAGON-HF, designed using the new proposal would require fewer patients would than a conventional time-to-first-event analysis.

CONCLUSIONS

The proposed method is robust and model-free and provides a clinically interpretable, time-scale summary of the treatment effect. (Funded by National Institutes of Health.).

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.