Technical Summary: Assessing the Robustness of Heterogeneous Treatment Effects in Survival Analysis under Informative Censoring

1. Problem Statement

In clinical studies, particularly in oncology, patient dropout is prevalent, with rates reaching up to 53% in certain cancers. When dropout is informative—meaning the censoring time $C$ depends on the unobserved survival time $T$—it introduces systematic censoring bias. Standard survival analysis methods (e.g., Kaplan-Meier, Cox models) and existing heterogeneous treatment effect (HTE) estimators typically rely on the assumption of non-informative censoring ($T \perp C \mid X, A$). When this assumption is violated, unbiased point estimation of the Conditional Average Treatment Effect (CATE) becomes impossible, as the expected survival time for censored patients cannot be identified from observed data alone.

Current literature lacks practical tools to assess the robustness of CATE estimates when informative censoring is present. Existing methods for partial identification in survival analysis focus on bounding survival functions rather than causal effects, while causal inference methods for informative censoring generally target the Average Treatment Effect (ATE) rather than the CATE.

2. Methodology

The paper proposes an assumption-lean framework that shifts the goal from point estimation to partial identification. Instead of estimating a single CATE value, the framework constructs informative lower and upper bounds on the CATE, explicitly accounting for censoring bias.

2.1 Partial Identification of CATE

The authors derive bounds for the CATE, $\tau_{a_1, a_2}(x) = E[T(a_1) - T(a_2) \mid X=x]$, by first bounding the Conditional Average Potential Outcome (CAPO), $\mu(x, a) = E[T(a) \mid X=x]$.

• Lower Bound ($\mu^-$): Derived by assuming that censored patients experience the event immediately after their censoring time (worst-case scenario). This yields a lower bound based on observed data:
  $$\mu^-(x, a) = E[\tilde{T} \mid X=x, A=a]$$
  where $\tilde{T} = \min(T, C)$.

• Upper Bound ($\mu^+$): Derived using a sensitivity function $\gamma(x, a)$, which represents the maximum expected remaining survival time after dropout. The upper bound incorporates domain knowledge (e.g., clinical benchmarks for post-dropout survival) or a conservative theoretical maximum ($t_{max}$):
  $$\mu^+(x, a) = \nu(0, x, a)[1 - \xi(x, a)] + \nu(1, x, a)\xi(x, a) + \gamma(x, a)\xi(x, a)$$
  Here, $\xi(x, a)$ is the censoring strength (probability of being censored), and $\nu(\delta, x, a)$ are expected conditional survival times.

• CATE Bounds: The CATE bounds are constructed by combining the CAPO bounds:
  $$\tau^-_{a_1, a_2}(x) = \mu^-(x, a_1) - \mu^+(x, a_2) \leq \tau_{a_1, a_2}(x) \leq \mu^+(x, a_1) - \mu^-(x, a_2) = \tau^+_{a_1, a_2}(x)$$

The width of these bounds is determined by the censoring strength $\xi(x, a)$ and the sensitivity function $\gamma(x, a)$. The bounds become informative (narrow) when censoring is low or when strong domain knowledge constrains $\gamma(x, a)$.

2.2 SurvB-learner: A Model-Agnostic Meta-Learner

To estimate these bounds efficiently and without plug-in bias, the authors introduce SurvB-learner, a two-stage meta-learner compatible with arbitrary machine learning models (e.g., random forests, neural networks).

• Stage 1 (Nuisance Estimation): Uses cross-fitting to estimate nuisance functions: propensity scores $\hat{\pi}_a(x)$, censoring strengths $\hat{\xi}(x, a)$, and conditional survival times $\hat{\nu}(\delta, x, a)$.
• Stage 2 (Pseudo-Outcome Regression): Constructs pseudo-outcomes $\hat{\phi}^\pm(x, a)$ based on the efficient influence functions of the bounds. These pseudo-outcomes are regressed on covariates $X$ to estimate the final bounds $\hat{\mu}^\pm(x, a)$.

2.3 Theoretical Properties

The SurvB-learner is proven to possess:

1. Consistency: Converges to the true bounds as sample size increases.
2. Double Robustness: The estimator remains consistent if either the propensity score model is correctly specified OR the combination of the survival and censoring models is correctly specified.
3. Quasi-Oracle Efficiency: Achieves the same asymptotic convergence rate as an oracle estimator that knows the true nuisance functions, provided the nuisance estimators converge at sufficient rates.

The framework is also extended to continuous treatments (using kernel smoothing) and observational data with hidden confounding (using Generalized Marginal Sensitivity Models).

3. Key Contributions

1. Assumption-Lean Framework: A novel approach to audit censoring bias in CATE estimates by replacing the non-informative censoring assumption with sensitivity functions to form informative bounds.
2. SurvB-learner: A model-agnostic meta-learner designed to estimate these bounds efficiently, correcting for plug-in bias and offering double robustness.
3. Theoretical Guarantees: Formal proofs of consistency, double robustness, and quasi-oracle efficiency for the proposed estimator.
4. Empirical Validation: Demonstration of effectiveness using synthetic data (showing superior performance over plug-in learners) and a real-world clinical trial (ADJUVANT dataset).

4. Experimental Results

• Synthetic Data: Experiments across various functional forms (Exponential, Sin, Logistic-Sin) and censoring strengths (0.2, 0.4, 0.6) show that SurvB-learner significantly outperforms the naive plug-in learner. SurvB-learner achieves lower Root Mean Squared Error (RMSE) against oracle bounds and lower variance. The plug-in learner is shown to be unstable and often produces bounds that are too narrow (unfaithful).
• Real-World Data (ADJUVANT Trial): Applied to a non-small cell lung cancer (NSCLC) trial with high censoring rates (37% for DFS, 63% for OS). The framework successfully identified patient subgroups (e.g., specific genetic co-alterations like CDK4+MYC) where the lower bound of the treatment effect remained strictly positive, indicating robust treatment benefits despite informative censoring. This provided insights that standard point estimators could not reliably offer in the presence of such bias.

5. Significance and Claims

The paper claims to be the first to provide a partial identification tool specifically for CATE estimation under informative censoring. Its significance lies in:

• Clinical Utility: Enabling clinicians to identify patient subgroups that reliably benefit from treatment even when dropout mechanisms are biased, without relying on potentially invalid point estimates.
• Robustness: Offering a "safety net" for causal inference in survival analysis where standard assumptions fail. The framework degrades gracefully to point estimation when censoring is weak but remains valid when censoring is strong.
• Flexibility: The model-agnostic nature allows integration with state-of-the-art machine learning models, making it applicable to complex, high-dimensional medical data.

The authors conclude that their framework provides valid and interpretable results, helping to mitigate the risks of biased treatment effect estimates in survival studies. Future work is suggested to focus on learning sensitivity functions from external data and extending inference tools.