Nonparametric estimation of the total treatment effect with multiple outcomes in the presence of terminal events

As standards of care advance, patients are living longer and once-fatal diseases are becoming manageable. Clinical trials increasingly focus on reducing disease burden, which can be quantified by the timing and occurrence of multiple non-fatal clinical events. Most existing methods for the analysis of multiple event-time data require stringent modeling assumptions that can be difficult to verify empirically, leading to treatment efficacy estimates that forego interpretability when the underlying assumptions are not met. Moreover, most existing methods do not appropriately account for informative terminal events, such as premature treatment discontinuation or death, which prevent the occurrence of subsequent events. To address these limitations, we derive and validate estimation and inference procedures for the area under the mean cumulative function (AUMCF), an extension of the restricted mean survival time to the multiple event-time setting. The AUMCF is nonparametric, clinically interpretable, and properly accounts for terminal competing risks. To enable covariate adjustment, we also develop an augmentation estimator that provides efficiency at least equaling, and often exceeding, the unadjusted estimator. The utility and interpretability of the AUMCF are illustrated with extensive simulation studies and through an analysis of multiple heart-failure-related endpoints using data from the Beta-Blocker Evaluation of Survival Trial (BEST) clinical trial. Our open-source R package MCC makes conducting AUMCF analyses straightforward and accessible.

💡 Research Summary

The paper addresses a growing need in clinical trials to evaluate disease burden when patients experience multiple non‑fatal events over a prolonged follow‑up period, and when a terminal event such as death can pre‑empt further events. Existing methods for multiple‑event time data—non‑homogeneous Poisson models, marginal rate models, frailty models, and various regression approaches—typically rely on strong parametric or semi‑parametric assumptions and often fail to handle informative terminal events appropriately. To overcome these limitations, the authors introduce the Area Under the Mean Cumulative Function (AUMCF), an extension of the restricted mean survival time (RMST) to the setting of multiple recurrent or distinct events. The AUMCF is defined as the integral over a fixed horizon τ of the mean cumulative function (MCF), which itself is the expected number of events up to time t. When a terminal event is present, the MCF is weighted by the survival probability of the terminal event, ensuring that no events are counted after death. Consequently, the AUMCF represents the expected total “event‑free time lost” due to all non‑fatal events, providing a clinically intuitive summary of disease burden.

Estimation proceeds non‑parametrically: the survival function of the terminal event is estimated by Kaplan–Meier, the cumulative hazard of recurrent events by Nelson–Aalen, and these plug‑in estimators are integrated to obtain the AUMCF for each treatment arm. For a two‑arm randomized trial, the treatment effect is the difference Δ = θ₁ – θ₂ between the two AUMCFs. The authors derive the asymptotic normality of the estimator Δ̂, present an explicit variance formula Σ_Δ, and show how to construct Wald‑type confidence intervals and hypothesis tests. They compare this approach to a log‑rank‑type statistic previously proposed for recurrent events, noting that Δ̂ enjoys the desirable property that a significant Wald test automatically corresponds to a confidence interval excluding the null, a property the log‑rank statistic lacks under general censoring.

A major contribution is the development of a covariate‑adjusted, augmented estimator. Although randomization guarantees that the mean difference in baseline covariates between arms converges to zero, finite‑sample imbalances can reduce efficiency. The authors augment Δ̂ with a term βᵀ(W₁ – W₂), where W_j is the average covariate vector in arm j and β is chosen to minimize asymptotic variance. Closed‑form expressions for β, based on the covariance between the covariates and the influence functions of the unadjusted estimator, are provided, and the resulting estimator Δ̂_aug is shown to be consistent and at least as efficient as the unadjusted version.

The methodology is illustrated with data from the Beta‑Blocker Evaluation of Survival Trial (BEST). The trial collected three heart‑failure‑related secondary outcomes (hospitalization, myocardial infarction, heart transplant) alongside overall survival. Using the AUMCF, the authors demonstrate that bucindolol reduces the cumulative disease burden compared with placebo, with results that align with, but are more directly interpretable than, those from Cox proportional hazards, negative‑binomial regression, the LWYY marginal model, shared‑Gamma frailty, and win‑ratio approaches. Extensive simulation studies explore a range of censoring rates, terminal‑event frequencies, and event‑rate scenarios. Across all settings, the AUMCF estimator exhibits negligible bias, lower mean‑squared error, and robust coverage of confidence intervals relative to the competing methods, especially when terminal events are common.

Finally, the authors release an open‑source R package, MCC, which implements AUMCF estimation, covariate augmentation, variance estimation, and hypothesis testing, making the approach readily accessible to applied researchers. In summary, the paper provides a rigorous, non‑parametric framework for quantifying total treatment effect on multiple outcomes in the presence of informative terminal events, offering both theoretical guarantees and practical tools for modern clinical trial analysis.