Targeted learning of heterogeneous treatment effect curves for right censored or left truncated time-to-event data

1. Problem Statement

The paper addresses the challenge of estimating Heterogeneous Treatment Effects (HTE) for time-to-event outcomes in observational studies. Specifically, it targets the difference in conditional survival probabilities between two treatments ($A=1$ vs. $A=0$) for a given patient profile ($X$), denoted as $\theta(t|x) = P(T^1 > t | X=x) - P(T^0 > t | X=x)$.

The estimation is complicated by two common data issues in real-world clinical databases:

1. Right Censoring (RC): Patients are lost to follow-up before an event occurs.
2. Left Truncation (LT): Patients enter the study (e.g., a genomic database) only after a specific event (e.g., genomic testing), meaning those who die or experience the event before entry are unobserved.

Limitations of Existing Methods:

• Indirect Estimators (e.g., T-learners): Estimate survival curves for each treatment group separately and subtract them. This inherits regularization bias from the machine learning (ML) models used for each group, leading to poor finite-sample performance.
• Direct Estimators (e.g., Causal Survival Forests - CSFs, ltrc-R, ltrc-DR): While these target the contrast directly to reduce bias, they suffer from:
  • Non-smoothness: They estimate the effect at each time point independently, resulting in jagged, biologically implausible curves over time.
  • Instability: Some (like ltrc-DR) can produce unbounded estimates or negative weights.
  • Inefficiency: They fail to leverage the temporal structure of the data, treating time points as unrelated.

2. Methodology: surv-iTMLE

The authors propose surv-iTMLE (Survival Infinite-dimensional Targeted Minimum Loss-Based Estimation), a non-parametric framework designed to estimate smooth, bounded, and stable treatment effect curves under both RC and Left Truncated Right Censored (LTRC) settings.

The method operates in two main steps:

Step 1: Generating De-biased Pseudo-Outcomes (iTMLE)

Instead of estimating the survival curves directly, the algorithm constructs pseudo-outcomes that represent the difference in survival probabilities, corrected for bias.

• Loss Function: The method minimizes a loss function derived from the Mean Squared Error (MSE) of the target parameter.
• Efficient Influence Function (EIF): The authors derive the EIF for the LTRC setting. This EIF includes inverse probability weights (IPWs) for treatment assignment, right censoring, and left truncation.
• Sieve-based Targeting: Since the target parameter is infinite-dimensional (a function of time $t$ and covariates $X$), the method uses sieve-based targeting (specifically, a cosine polynomial basis).
  • It updates nuisance parameters (hazard functions) using a weighted regression against the sieve basis.
  • This step ensures that the sample average of the de-biasing term converges to zero, achieving "double robustness" and oracle efficiency.
  • To handle convergence issues with high-dimensional sieves, the authors suggest using penalized linear or log-linear regressions instead of logistic regression.

Step 2: Pooled Regression for Smooth Curves

To ensure the resulting treatment effect curve is smooth over time (unlike existing methods that estimate time points separately):

• Incremental Approach: The algorithm creates a long-format dataset where each patient contributes a row for every time point.
• Outcome Definition: The outcome is defined as the incremental change in the pseudo-outcome between consecutive time points ($\Delta Y_{iTMLE}(k)$).
• Pooled Regression: A regression model (e.g., Generalized Additive Model - GAM) is fitted to predict these incremental changes using covariates $X$ and time as a continuous predictor.
• Reconstruction: The final treatment effect curve $\hat{\theta}(t|X)$ is obtained by summing the predicted incremental changes from time 0 to $t$. This enforces smoothness and borrows strength across time points.

3. Key Contributions

1. Novel Estimator (surv-iTMLE): The first targeted learning procedure specifically designed to handle both left truncation and right censoring while estimating HTE curves.
2. Smoothness and Stability: By utilizing a pooled regression on incremental outcomes, the method produces biologically plausible, smooth treatment effect curves, addressing a major flaw in existing forest-based or point-wise estimators.
3. Theoretical Guarantees: The method provides bounded estimates and achieves oracle efficiency (performing as well as if nuisance functions were known) under standard causal assumptions (positivity, consistency, conditional exchangeability, no interference).
4. Practical Implementation: The authors provide a complete implementation pipeline, including guidance on handling nuisance function estimation (using Local Survival Stacking and SuperLearner) and sieve selection.

4. Results

Simulation Studies

The authors conducted extensive simulations across three Data Generating Processes (DGPs) with varying levels of left truncation and right censoring.

• Performance: surv-iTMLE consistently outperformed existing methods (T-learner, CSFs, ltrc-R, ltrc-DR) in terms of Root Mean Squared Error (RMSE).
• Bias Reduction: It showed significant improvements in bias, particularly in settings with high left truncation rates where CSFs struggled.
• Smoothness: Visual inspection of individual curves demonstrated that surv-iTMLE tracks the true underlying effect much more smoothly than the jagged estimates produced by CSFs and T-learners.
• LTRc Learners: The ltrc-R and ltrc-DR learners performed poorly in simulations, often due to the generation of negative weights and unbounded estimates.

Real-World Application: NSCLC Immunotherapy

The method was applied to the Flatiron Health-Foundation Medicine clinico-genomic database to evaluate immunotherapy vs. chemotherapy for Non-Small Cell Lung Cancer (NSCLC).

• Findings:
  • Temporal Patterns: surv-iTMLE revealed that immunotherapy offers limited benefit in the first 6 months but shows increasing effectiveness relative to chemotherapy over time.
  • Heterogeneity:
    • EGFR Mutations: Patients with EGFR mutations showed less benefit from immunotherapy compared to those without.
    • Age: Older patients generally benefited more, but effectiveness dropped significantly for patients aged 85+.
• Comparison: Existing estimators (CSF, T-learner) produced noisy, non-smooth curves that obscured these clinically meaningful temporal patterns.

5. Significance

• Personalized Medicine: The ability to generate smooth, patient-specific survival curves over time allows clinicians to better understand when a treatment is effective for specific subgroups, rather than just an average effect.
• Handling Complex Data: The framework is robust to the complexities of real-world data, specifically the combination of left truncation (common in retrospective genomic databases) and right censoring.
• Methodological Advancement: It bridges the gap between causal inference (targeted learning) and survival analysis, moving beyond point-wise estimation to functional estimation of treatment effects.
• Clinical Utility: In the NSCLC example, the method uncovered nuanced temporal dynamics (e.g., delayed onset of immunotherapy efficacy) that could inform treatment duration and monitoring strategies, which might be missed by standard methods.

In summary, surv-iTMLE represents a significant advancement in causal machine learning for time-to-event data, offering a robust, smooth, and efficient solution for estimating heterogeneous treatment effects in complex observational settings.