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

Jessica Gronsbell, Zachary R. McCaw, Isabelle-Emmanuella Nogues, Xiangshan Kong, Tianxi Cai, Lu Tian, LJ Wei

TL;DR

The paper introduces the area under the mean cumulative function (AUMCF) as a nonparametric, interpretable summary for total disease burden in settings with multiple recurrent events and terminal competing risks. It develops one- and two-sample estimators for the AUMCF, provides asymptotic inference, and adds a covariate-augmented estimator to improve efficiency while preserving consistency. Through BEST data and extensive simulations across independent risks, frailty, and time-varying treatment effects, the authors demonstrate that AUMCF provides robust type I error control and favorable power relative to several standard methods, with augmentation offering measurable efficiency gains when covariates are informative. The work includes open-source software (MCC) and replication resources, enabling practical application in clinical trial contexts where total event burden is a key outcome.

Abstract

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.

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

TL;DR

The paper introduces the area under the mean cumulative function (AUMCF) as a nonparametric, interpretable summary for total disease burden in settings with multiple recurrent events and terminal competing risks. It develops one- and two-sample estimators for the AUMCF, provides asymptotic inference, and adds a covariate-augmented estimator to improve efficiency while preserving consistency. Through BEST data and extensive simulations across independent risks, frailty, and time-varying treatment effects, the authors demonstrate that AUMCF provides robust type I error control and favorable power relative to several standard methods, with augmentation offering measurable efficiency gains when covariates are informative. The work includes open-source software (MCC) and replication resources, enabling practical application in clinical trial contexts where total event burden is a key outcome.

Abstract

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.

Paper Structure

This paper contains 14 sections, 2 theorems, 8 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Problem Setting and Notation
  4. The Area Under the Mean Cumulative Function (AUMCF)
  5. One sample setting
  6. Two sample setting
  7. Covariate adjustment
  8. Empirical studies
  9. Real data example
  10. Simulation studies
  11. Data generation
  12. Evaluation metrics
  13. Results
  14. Discussion

Key Result

theorem 1

Let $\Lambda_j^D(t)$ be the cumulative hazard function of the terminal event in the $j$th treatment arm, $N_{ij}^D(t) = \mathbb{I}( X_{ij} \le t, \delta_{ij} = 1)$, $M_{ij}^D(t) = N_{ij}^D(t) - \int_0^t Y_{ij}(u) d\Lambda_j^D(u)$, $M_{ij}(t) = N_{ij}(t) - \int_{0}^t Y_{ij}(u) dR_j(u)$, and $\pi_j =

Figures (2)

  • Figure 1: Overview of the area under the mean cumulative function (AUMCF). The AUMCF quantifies the total event-free time lost from all events and is illustrated in an example of a study with a 24 month follow-up period. The top panel shows three hypothetical observations while the bottom three panels illustrates the corresponding observation-level MCFs and AUMCFs. In observation 1, two non-terminal events (red circles) are experienced at months 6 and 12 and the process is censored at the end of follow-up (white circle). Therefore, 18 and 12 event-free months were lost and the AUMCF is equal to (24-6) + (24-12) = 30 event months. In observation 2, no events are experienced during 12 months before censoring occurs, which could represent late enrollment into the study. All that is known is that the AUMCF is positive since no event-free time is lost. In observation 3, one non-terminal event occurs at 6 months and the terminal event (black circle) occurs at 18 months of follow-up. The AUMCF is equal to (24-6) + (24-18) = 24 event months.
  • Figure 2: Mean cumulative functions (MCF) and area under the MCF (AUMCF) for patients from the Beta-Blocker Evaluation of Survival (BEST) Trial. The composite disease-burden endpoint consists of the times to hospitalization from any cause, myocardial infarction, heart transplantation, and death, considering all occurrences a patient experienced.

Theorems & Definitions (2)

  • theorem 1
  • theorem 2