Tipping Point Sensitivity Analysis for Missing Data in Time-to-Event Endpoints: Model-Based and Ad hoc Approaches

Technical Summary: Tipping Point Sensitivity Analysis for Missing Data in Time-to-Event Endpoints

Problem Statement
In confirmatory clinical trials, the treatment policy estimand is frequently favored by regulatory authorities as it evaluates the effect of treatment assignment regardless of post-randomization intercurrent events (e.g., treatment discontinuation). While standard survival analysis methods (e.g., Kaplan–Meier, Cox proportional hazards) implicitly target this estimand, they rely on the independent censoring assumption (Censoring At Random, CAR). This assumption posits that, conditional on observed covariates, the event time process is independent of the censoring time process.

However, study discontinuation often coincides with intercurrent events (such as adverse events or perceived lack of efficacy), potentially violating the CAR assumption. When censoring depends on unobserved outcomes (Censoring Not At Random, CNAR), standard analyses may yield biased estimates of the treatment effect. Regulatory guidance (e.g., ICH E9 (R1)) emphasizes the need for sensitivity analyses to assess the robustness of conclusions to such departures. Tipping point analysis provides a structured framework to quantify how strongly assumptions about the censoring mechanism must be altered to change (or "tip") the study conclusions. Despite methodological extensions to time-to-event data, practical guidance linking these constructions to the primary estimand and analysis model remains limited.

Methodology
The paper describes and contrasts three specific approaches for conducting tipping point analyses in time-to-event settings, utilizing a multiple imputation framework. The methods are categorized into model-based and ad hoc approaches:

1. Model-Based Imputation (Delta-Adjustment):

   • Mechanism: A parametric survival model (e.g., Weibull) is fitted to the observed data under the assumption of independent censoring. A sensitivity parameter, $\delta$, is introduced to modify the hazard function for censored patients after their censoring time ($t > c_i$).
   • Implementation: For the experimental arm, the hazard is inflated ($\delta > 1$) to simulate worse outcomes; for the control arm, it is deflated ($\delta < 1$). The method employs proper multiple imputation, sampling from the posterior predictive distribution to account for parameter uncertainty.
   • Constraint: Imputed times are truncated at the maximum follow-up time to mimic administrative censoring.

2. Ad Hoc Approach 1: Landmark Sampling:

   • Mechanism: This heuristic method does not rely on a parametric model. A subset ($\kappa$) of censored patients is deterministically assigned an extreme outcome.
   • Implementation: In the experimental arm, a random subset of censored patients is imputed to have an event exactly at the time of censoring (worst-case scenario). In the control arm, a subset is imputed to remain event-free until the data cut-off (best-case scenario).
   • Equivalence: This approach corresponds to the model-based approach where $\delta \to \infty$ (all events) or $\delta \to 0$ (all censored).

3. Ad Hoc Approach 2: Percentile Sampling:

   • Mechanism: Another heuristic method where censored patients are imputed by randomly sampling observed event times from a specific percentile of the pooled data.
   • Implementation: For the experimental arm, event times are sampled from the worst $\theta$-percentile of observed times (shortest survival). For the control arm, times are sampled from the best $\theta$-percentile (longest survival).
   • Assumption: This assumes censored patients behave similarly to the specific subpopulation from which they are sampled.

Analysis Framework
The tipping point is identified by varying the sensitivity parameter ($\delta$, $\kappa$, or $\theta$) across a predefined range. For each value, multiple imputed datasets are generated, and treatment effects (e.g., Hazard Ratios) are estimated and combined using Rubin's rules. The analysis determines the parameter value at which the results "tip"—defined as the loss of nominal significance or the crossing of the confidence interval over the null value (HR = 1).

The paper emphasizes that these methods do not redefine the estimand but rather assess the robustness of inference for the treatment policy estimand under alternative assumptions about unobserved data. To ensure clinical plausibility, the authors propose anchoring sensitivity parameters to external data (e.g., "jump-to-reference" scenarios) and visualizing the impact via Kaplan–Meier (KM) curves before and after imputation.

Case Studies and Results
The authors illustrate these methods using re-constructed data from two clinical trials:

• CodeBreaK200 (NSCLC): Characterized by higher early discontinuation in the control arm.
  • Findings: The FDA's model-based analysis suggested a 55% hazard deflation in the control arm would tip the results. The authors' re-analysis confirmed this, noting that the resulting Hazard Ratio (0.68) between early discontinuers and the experimental arm seemed clinically implausible given the lack of alternative effective therapies, suggesting the original results were not robust to extreme informative censoring.
• ExteNET (Breast Cancer): Characterized by higher early discontinuation in the experimental (neratinib) arm.
  • Findings: The model-based approach required a 120% hazard inflation to tip the results. The ad hoc landmark sampling approach (imputing events at discontinuation) was more conservative, tipping the results when only 15 patients were imputed as events. The authors noted that a 120% inflation implies a hazard rate nearly three times higher than the control arm for discontinuers, which appears highly improbable given the drug's favorable benefit-risk profile.

Key Contributions and Claims
The paper does not claim to introduce novel methodology but rather aims to summarize and clarify existing approaches used in practice. Its primary contributions include:

1. Structured Comparison: Providing a clear taxonomy and step-by-step algorithms for model-based versus ad hoc tipping point approaches, highlighting their underlying assumptions and computational differences.
2. Interpretability Guidance: Emphasizing that sensitivity parameters operate on different scales and cannot be directly compared numerically. Instead, the authors advocate for interpreting results through clinical plausibility assessments (e.g., comparing imputed KM curves to external data or clinical expectations).
3. Visual Tools: Demonstrating the utility of plotting pooled Kaplan–Meier curves from imputed datasets to visualize the magnitude of shifts caused by different imputation assumptions.
4. Software Implementation: The authors provide an open-source R package, tipse, to facilitate the implementation of these methods.

Significance
The paper argues that while tipping point analyses are increasingly requested by regulators, they are often applied inconsistently. By linking these analyses explicitly to the treatment policy estimand and providing a framework for assessing clinical plausibility, the authors aim to improve the reliability of sensitivity analyses in regulatory settings. The work underscores that the "tipping point" is not merely a statistical threshold but a scenario that must be evaluated against clinical reality to determine the robustness of a trial's conclusions.