Randomized interventional effects in semicompeting risks

In clinical studies, the risk of the primary (terminal) event may be modified by intermediate events, resulting in semicompeting risks. To study the treatment effect on the terminal event mediated by the intermediate event, researchers wish to decompose the total effect into direct and indirect effects. In this article, we extend the randomized interventional approach to time-to-event outcomes, where both intermediate and terminal events are subject to right censoring. We envision a random draw for the intermediate event process from a reference distribution, either marginally over time-varying confounders or conditionally given the observed history. We present the identification formula for interventional effects. We also discuss some variants of the identification assumptions. We estimate the treatment effects using nonparametric maximum likelihood estimation and propose a sensitivity analysis. We study the effect of matched unrelated donor versus haploidentical donor on death mediated by relapse in a hematopoietic cell transplantation study with graft-versus-host disease (GVHD) as the time-varying confounder. We find that matched unrelated donor transplantation is preferable in terms of survival rates under the use of post-transplantation PTCy GVHD prophylaxis for lymphoma patients.

💡 Research Summary

The paper addresses causal mediation analysis in the context of semi‑competing risks, where an intermediate event (e.g., disease relapse) may occur before a terminal event (e.g., death) and the occurrence of the terminal event censors further observation. Traditional mediation frameworks—natural direct/indirect effects or separable effects—require strong sequential ignorability or partial isolation assumptions that are difficult to verify when time‑varying confounders (such as graft‑versus‑host disease, GVHD) are present. To overcome these limitations, the authors extend the randomized interventional effects framework to continuous‑time survival data with right censoring.

Key contributions are:

1. Potential‑outcome formulation: The authors define counting processes for the intermediate event (N_1(t;z)) and the terminal event (N_2(t;z,N_1(t))) under binary treatment (Z). The cumulative incidence function (CIF) for the terminal event, (F(t;z)=P{N_2(t;z)=1}), serves as the primary estimand.

2. Identification assumptions: Standard causal assumptions are adapted to the semi‑competing risk setting: (i) treatment ignorability given baseline covariates (X) and the full history of time‑varying confounders (L(t)); (ii) random censoring conditional on (X, L(t)) and the intermediate event status; (iii) positivity for all treatment‑covariate histories; and (iv) consistency linking observed times to potential outcomes.

3. Hazard representation: The authors express the hazard of the intermediate event ((d\Lambda^*(t;z,X,L(t)))) and the cause‑specific hazards of the terminal event before and after the intermediate event ((d\Lambda_0(t;z,X,L(t))) and (d\Lambda_1(t;z,r,X,L(t)))). These hazards are shown to be identifiable from observed data under the stated assumptions.

4. Randomized intervention design: The central methodological innovation is to imagine drawing the entire intermediate‑event trajectory from an external reference distribution (G(\cdot;z_1)) that mimics the distribution of the intermediate event under treatment level (z_1). Two counterfactual worlds are constructed:

   • Interventional direct effect – set treatment to (z_2) while keeping the intermediate‑event process fixed at (G(\cdot;z_1)). This blocks the pathway through the mediator but retains any effect of treatment that operates directly on the terminal event or via time‑varying confounders.
   • Interventional indirect effect – keep treatment at (z_1) and replace the intermediate‑event distribution with that under (z_2). This isolates the portion of the treatment effect that operates through the mediator.

   Because the reference distribution can be defined marginally over (L(t)) or conditionally on the full history, the framework accommodates different assumptions about how the mediator interacts with time‑varying confounders.

5. Estimation via non‑parametric maximum likelihood: The authors propose a non‑parametric likelihood for the observed counting processes, treating the hazard functions as step functions. Maximizing this likelihood yields estimators of the cumulative hazards, which are then integrated to obtain the CIFs under each interventional regime. Plug‑in estimators provide direct and indirect effects without needing parametric modeling of the hazard functions.

6. Sensitivity analysis for unmeasured confounding: To assess robustness, a latent variable (U) that may affect both the intermediate and terminal events is introduced. By varying sensitivity parameters governing the strength of (U)’s influence, researchers can examine how estimates of direct and indirect effects change, thereby quantifying the impact of potential hidden bias.

7. Application to hematopoietic cell transplantation: The methodology is applied to a cohort of lymphoma patients undergoing either matched unrelated donor (MUD) transplantation or haploidentical donor transplantation. GVHD is treated as a time‑varying confounder, relapse as the mediator, and death as the terminal outcome. Using the proposed framework, the authors find that under post‑transplant cyclophosphamide (PTCy) GVHD prophylaxis, MUD transplantation yields a lower cumulative incidence of death both through a direct pathway and via reduced relapse‑mediated mortality. The overall effect favors MUD over haploidentical transplantation.

8. Discussion of assumptions and extensions: The paper acknowledges that the ignorability assumption relies on randomization or thorough adjustment for observed covariates; violations could bias results. The handling of time‑varying confounders assumes they are fully observed, which may not hold in practice; discretization or interpolation strategies are suggested. The authors note that the framework can be generalized to multiple mediators, multiple competing terminal events, or fully competing risks settings.

In summary, this work provides a rigorous, flexible approach for decomposing treatment effects into direct and indirect components when dealing with semi‑competing risks and time‑varying confounding. By leveraging a randomized interventional perspective, it avoids the stringent sequential ignorability required by natural‑effects methods, while still offering identifiable estimands and practical estimation procedures. The application to real‑world transplant data demonstrates the method’s relevance for informing clinical decisions where mediation mechanisms are of substantive interest.