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Causal inference for calibrated scaling interventions on time-to-event processes

Helene Charlotte Wiese Rytgaard, Mark van der Laan

TL;DR

This paper tackles causal inference for time-to-event data by introducing α-scaled stochastic interventions that multiplicatively modify the intermediate event hazard $Λ^z$ through $α$, preserving individual heterogeneity. It adds calibrated interventions that tie $α$ to clinically meaningful targets and defines composite downstream effects, enabling interpretable causal questions about how changes in intermediate processes propagate to outcomes. The authors derive efficient influence curves for $α$-indexed, calibrated, and composite target parameters, establish double robustness, and sketch a TMLE framework that accommodates machine-learning nuisance estimation. Through simulation, they demonstrate finite-sample performance and discuss positivity issues when interventions extend beyond observed data support, highlighting practical guidance for applying these methods to time-to-event questions with mediators or time-varying exposures.

Abstract

This work develops a flexible inferential framework for nonparametric causal inference in time-to-event settings, based on stochastic interventions defined through multiplicative scaling of the intensity governing an intermediate event process. These interventions induce a family of estimands indexed by a scalar parameter α, representing effects of modifying event rates while preserving the temporal and covariate-dependent structure of the observed data generating mechanism. To enhance interpretability, we introduce calibrated interventions, where α is chosen to achieve a pre-specified goal, such as a desired level of cumulative risk of the intermediate event, and define corresponding composite target parameters capturing the downstream effects on the outcome process. This yields clinically meaningful contrasts while avoiding unrealistic deterministic intervention regimes. Under a nonparametric model, we derive efficient influence curves for α-indexed, calibrated, and composite target parameters and establish their double robustness properties. We further sketch a targeted maximum likelihood estimation (TMLE) strategy that accommodates flexible, machine learning based nuisance estimation. The proposed framework applies broadly to (causal) questions involving time-to-event treatments or mediators and is illustrated through different examples event-history settings. A simulation study demonstrates finite-sample inferential properties, and highlights the implications of practical positivity violations when interventions extend beyond observed data support.

Causal inference for calibrated scaling interventions on time-to-event processes

TL;DR

This paper tackles causal inference for time-to-event data by introducing α-scaled stochastic interventions that multiplicatively modify the intermediate event hazard through , preserving individual heterogeneity. It adds calibrated interventions that tie to clinically meaningful targets and defines composite downstream effects, enabling interpretable causal questions about how changes in intermediate processes propagate to outcomes. The authors derive efficient influence curves for -indexed, calibrated, and composite target parameters, establish double robustness, and sketch a TMLE framework that accommodates machine-learning nuisance estimation. Through simulation, they demonstrate finite-sample performance and discuss positivity issues when interventions extend beyond observed data support, highlighting practical guidance for applying these methods to time-to-event questions with mediators or time-varying exposures.

Abstract

This work develops a flexible inferential framework for nonparametric causal inference in time-to-event settings, based on stochastic interventions defined through multiplicative scaling of the intensity governing an intermediate event process. These interventions induce a family of estimands indexed by a scalar parameter α, representing effects of modifying event rates while preserving the temporal and covariate-dependent structure of the observed data generating mechanism. To enhance interpretability, we introduce calibrated interventions, where α is chosen to achieve a pre-specified goal, such as a desired level of cumulative risk of the intermediate event, and define corresponding composite target parameters capturing the downstream effects on the outcome process. This yields clinically meaningful contrasts while avoiding unrealistic deterministic intervention regimes. Under a nonparametric model, we derive efficient influence curves for α-indexed, calibrated, and composite target parameters and establish their double robustness properties. We further sketch a targeted maximum likelihood estimation (TMLE) strategy that accommodates flexible, machine learning based nuisance estimation. The proposed framework applies broadly to (causal) questions involving time-to-event treatments or mediators and is illustrated through different examples event-history settings. A simulation study demonstrates finite-sample inferential properties, and highlights the implications of practical positivity violations when interventions extend beyond observed data support.
Paper Structure (20 sections, 7 theorems, 33 equations, 6 figures)

This paper contains 20 sections, 7 theorems, 33 equations, 6 figures.

Table of Contents

  1. Introduction
  2. General setting and notation
  3. Intervention framework and target parameters
  4. $\alpha$-scaling stochastic interventions
  5. Post-interventional distribution
  6. Intervention-specific target parameters
  7. Calibrated interventions and composite target parameters
  8. Causal reading and feasibility of calibration
  9. Illustration of application
  10. Efficiency theory for estimation
  11. Efficiency theorems for targeted substitution estimators
  12. Targeted maximum likelihood estimation
  13. Initial estimation
  14. Estimation of clever covariates
  15. Estimation of clever weights
  16. ...and 5 more sections

Key Result

Lemma 3.1

The function $\alpha \mapsto \Psi^{a,\alpha}_z(P)$ is increasing and concave in $\alpha$, and strictly so when $L^a(P):=P (\Lambda^z(\tau \mid \mathcal{F}^{a}_{\tau-})>0)>0$, where $\mathcal{F}^{a}_{t}$ is the filtration generated by the observed data but evaluated in $A_0=a$. Moreover, $\lim_{\alph

Figures (6)

  • Figure 1: True values of the intervention-specific parameters for the operation example. The left plot shows $\alpha \mapsto \Psi_1^{\alpha}(P)$; the right plot shows $\alpha \mapsto \Psi_z^{\alpha}(P)$. In the left plot, the difference between the curve and the horizontal red dashed line constitutes the contrast comparing the outcome risk when modifying the rate of operations by factor $\alpha$ to the outcome risk with no modification of the rate of operation. Comparing the curves on the left and the right plots reveals the trade-off between how frequently operations occur and their downstream impact on the outcome.
  • Figure 2: True values of the intervention-specific parameters for the drop-in example. The left plot shows the intervention-specific absolute risks of death as function of $\alpha$, $\alpha \mapsto \Psi_1^{1,\alpha}(P)$ and $\alpha \mapsto \Psi_1^{0,\alpha}(P)$; the right plot shows the intervention-specific absolute probability of drop-in initiation as function of $\alpha$, $\alpha \mapsto \Psi_z^{1,\alpha}(P)$ and $\alpha \mapsto \Psi_z^{0,\alpha}(P)$. On the left plot the colored horizontal segments illustrate the indirect component $\Psi_1^{0,1}(P)-\Psi_1^{0,\alpha^{ 1}(P)}(P) = -0.0584$ (the length of the horizontal blue segment) and the direct component $\Psi_1^{0,\alpha^{ 1}(P)}(P)-\Psi_1^{1,1}(P) = 0.1192$ (the length of the horizontal black segment), which sum up to the total average treatment effect $\Psi_1^{0,1}(P)-\Psi_1^{1,1}(P) = 0.0608$.
  • Figure 3: Estimated $\alpha$-specific auxiliary ($\Psi_z^{\alpha}$, upper panels) and target ($\Psi_1^{\alpha}$, lower panels) parameters across a grid of values of $\alpha$. Left column: correctly specified nuisance models. Right column: misspecified outcome and covariate intensities. Red dashed lines indicate initial plug-in estimates; solid lines are TMLE updates. For small $\alpha$ (strongly down-weighting operation occurrence) estimation quality deteriorates because a few individuals receive very large inverse probability weights when we attempt to enforce an intervention that is unrealistic for them; this produces inflated variance and undercoverage of confidence intervals.
  • Figure 4: Quantiles (97.5%, 99%, 100%) of inverse probability weights across Monte Carlo replicates for each $\alpha$ (vertical axis on a log scale). Extremely large weights appear for small $\alpha$, signaling severe practical positivity violations in those intervention regimes. Investigation shows these extreme weights arise from a small subset of individuals who, according to their covariate/history profile, are very unlikely never to undergo the operation; forcing $\alpha$ near zero effectively requires unlikely counterfactual behaviour for these subjects and produces unstable estimates. These diagnostics could be consulted before interpreting contrasts: regimes with extreme quantiles indicate estimands that rely on extrapolation which should be avoided.
  • Figure 5: Estimates of calibrated (left) and composite (right) parameters $\alpha^{\theta}(P)$ and $\Psi^{\theta}_1(P)$. Lower $\theta$ produce smaller calibrated $\alpha$ and are associated with greater numerical instability.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Lemma 3.1
  • Theorem 1: Efficient influence curve for the $\alpha$-indexed parameters
  • Lemma 4.1: General form of the efficient influence curves for calibration parameters.
  • Theorem 2: Efficient influence curve of composite parameter.
  • Theorem 3: Asymptotically linear estimation of the $\alpha$-fixed parameters.
  • Lemma 4.2: Asymptotically linear estimation of calibrated $\alpha(P)$.
  • Theorem 4: Asymptotically linear estimation of composite parameter.