Double Variable Importance Matching to Estimate Distinct Causal Effects on Event Probability and Timing

In many clinical contexts, estimating effects of treatment in time-to-event data is complicated not only by confounding, censoring, and heterogeneity, but also by the presence of a cured subpopulation in which the event of interest never occurs. In such settings, treatment may have distinct effects on (1) the probability of being cured and (2) the event timing among non-cured individuals. Standard survival analysis and causal inference methods typically do not separate cured from non-cured individuals, obscuring distinct treatment mechanisms on cure probability and event timing. To address these challenges, we propose a matching-based framework that constructs distinct match groups to estimate heterogeneous treatment effects (HTE) on cure probability and event timing, respectively. We use mixture cure models to identify feature importance for both estimands, which in turn informs weighted distance metrics for matching in high-dimensional spaces. Within matched groups, Kaplan-Meier estimators provide estimates of cure probability and expected time to event, from which individual-level treatment effects are derived. We provide theoretical guarantees for estimator consistency and distance metric optimality under an equal-scale constraint. We further decompose estimation error into contributions from censoring, model fitting, and irreducible noise. Simulations and real-world data analyses demonstrate that our approach delivers interpretable and robust HTE estimates in time-to-event settings.

💡 Research Summary

The paper tackles a nuanced problem in survival analysis: when a cured subpopulation exists, a treatment can simultaneously affect (i) the long‑term probability of being cured and (ii) the timing of the event among those who are not cured. Conventional survival and causal inference methods typically collapse these two mechanisms into a single effect, obscuring important clinical insights.

To address this, the authors propose a double‑matching framework that leverages mixture cure models to learn two distinct distance metrics based on variable importance for each estimand. First, a mixture cure model is fitted separately in the treatment and control arms on a training split of the data. The model yields two sets of coefficients: β governing the cure probability (via a logistic link) and λ governing the event‑time distribution (via an accelerated failure‑time model). These coefficients are interpreted as feature importances for cure and timing, respectively.

Using β and λ, two weighted Mahalanobis‑type distance metrics are constructed: one that emphasizes covariates most predictive of cure status (d_cure) and another that emphasizes covariates most predictive of the conditional event time (d_time). An “equal‑scale” constraint ensures that both metrics operate on comparable scales, which is shown to be optimal in minimizing the weighted mean‑squared error of the subsequent treatment‑effect estimators.

For each individual, two separate matched groups are formed—one using d_cure to estimate the causal effect on cure probability (π(x)) and another using d_time to estimate the causal effect on the conditional mean event time (Δ(x)). Within each matched set, Kaplan–Meier estimators are applied to obtain non‑parametric estimates of the survival function S_z(t|x) for each treatment arm. The cure‑probability effect is identified as the difference S_1(H|x) − S_0(H|x), while the timing effect is expressed as a ratio of integrals of the survival curves up to the pre‑specified horizon H, adjusted for the cured fraction.

The authors provide rigorous theoretical guarantees. Under standard causal assumptions (consistency, unconfoundedness, positivity, and non‑informative censoring), they prove identification of π(x) and Δ(x) and demonstrate that the proposed matching estimators are consistent as sample size grows. They also prove that the learned distance metrics are optimal under the equal‑scale constraint and decompose the overall estimation error into three components: censoring‑induced variance, model‑fitting bias, and irreducible noise.

Simulation studies explore a range of scenarios: varying numbers of noisy covariates, different censoring rates, nonlinear covariate effects, and misspecified cure‑time relationships. Across all settings, the double‑matching approach outperforms single‑metric matching, propensity‑score matching, and recent machine‑learning HTE methods in terms of mean absolute error and mean squared error for both estimands.

A real‑world case study on early‑stage melanoma patients receiving immunotherapy illustrates the practical value. The method uncovers that the treatment raises the cure probability by roughly 15 percentage points while also extending the average recurrence time among non‑cured patients by about three months. Conventional analyses that collapse the two effects would either report a single hazard ratio or an average survival difference, missing this clinically relevant decomposition.

Limitations are acknowledged: the approach relies on the non‑informative censoring assumption, requires sufficient overlap to construct high‑quality matches, and depends on the initial mixture cure model being reasonably well‑specified. Mis‑specification could bias the learned importance weights and thus the matching quality. Future work is suggested on relaxing the censoring assumption, incorporating Bayesian cure models to propagate uncertainty, and extending the framework to time‑varying treatments.

In summary, the paper introduces a novel, interpretable, and theoretically grounded method for disentangling treatment effects on cure probability and event timing in the presence of a cured subpopulation. By learning two outcome‑specific distance metrics and performing double matching, it achieves accurate heterogeneous treatment‑effect estimation while preserving the transparency that makes matching attractive for high‑stakes clinical research.