Covariate-Adaptive Sample Size Re-estimation for Population-Standardized Historical Control Designs in Single-Arm Trials

Technical Summary: Covariate-Adaptive Sample Size Re-estimation for Population-Standardized Historical Control Designs in Single-Arm Trials

Problem Statement
Externally controlled single-arm trials are increasingly utilized when randomized controls are infeasible, particularly in rare diseases or settings where concurrent control allocation is ethically or practically difficult. However, a fundamental challenge arises from baseline imbalances between the active-arm trial and historical control data (e.g., from registries or previous trials). These differences in patient characteristics, calendar time, or measurement procedures can cause simple differences in outcome means to reflect population or data-source disparities rather than treatment effects.

While methods such as propensity scores, G-computation, and inverse probability of treatment weighting (IPTW) exist to adjust for confounding, sample size planning in these contexts remains problematic. Traditional approaches often fix the sample size based on planning assumptions regarding the historical control mean and variance. However, when the target estimand is defined for the actually enrolled active-arm population, the standardized control mean and its variance become explicit functions of the enrolled population's covariate distribution. If the actual enrolled population deviates from the planning assumptions, fixed designs may suffer from a loss of power or inflated type I error. Furthermore, standard sample size re-estimation (SSR) methods typically rely on nuisance parameters (like variance) or conditional power based on interim outcomes, which are not applicable here because the active-arm outcomes are not available during the enrollment phase for re-estimation.

Methodology
The authors propose a unified framework for population-standardized estimands combined with outcome-blinded, covariate-adaptive sample size re-estimation (SSR).

1. Population-Standardized Estimand:

   • The target population is defined as the subjects actually enrolled in the single-arm trial.
   • A pre-specified balancing score $Z = s(X)$ (e.g., a trial participation score) summarizes the distributional differences between the active arm and historical controls.
   • Historical control outcomes are standardized to the active-arm score distribution $F_{Z,A}$ using IPTW. The control parameter $\eta_0(F_{Z,A})$ is defined as the expected outcome if the enrolled active-arm population had received control treatment.
   • The treatment effect estimand is $\Delta(F_{Z,A}) = \eta_1 - \eta_0(F_{Z,A})$, where $\eta_1$ is the active-arm outcome.

2. Sample Size Design and Re-estimation:

   • Initial Design: An expected active-arm score distribution $F^{plan}_{Z,A}$ is pre-specified. The required sample size $N(F)$ is calculated based on the variance of the treatment effect estimator, which includes the variance of the standardized control parameter $\tau^2_0(F)$.
   • Outcome-Blinded SSR: During the trial, the method updates the required sample size using only the accumulating baseline covariates of enrolled subjects, without using active-arm outcomes.
   • At pre-specified review times, the empirical score distribution $\hat{F}_{Z,A,n}$ is calculated. The standardized control parameter $\hat{\eta}_0(\hat{F}_{Z,A,n})$ and the required sample size $N_{re}(n)$ are recalculated.
   • Two versions are proposed:
     • Point-estimate SSR: Uses the recalculated sample size directly.
     • Conservative SSR: Uses an upper bound of an uncertainty interval for the sample size to account for finite-sample estimation error of the score distribution.
   • Enrollment stops when the current sample size $n$ reaches the updated target $N_{target}(n)$ or hits a pre-specified maximum.

3. Theoretical Properties:

   • The paper establishes conditions (A1–A5) under which the procedure maintains approximate type I error control and conditional power.
   • Theorem 1: Demonstrates that if re-estimation depends only on the score process (not outcomes), the final test controls type I error up to a discrepancy term induced by the empirical score distribution.
   • Theorem 2: Shows that conditional on the final observed score distribution, securing the recalculated sample size yields power at least at the target level (up to normal approximation error).

Key Contributions

• Definition of Estimand: The paper formally defines a population-standardized estimand where historical controls are standardized to the actual enrolled population via a balancing score, making the control parameter a function of the target distribution.
• Covariate-Adaptive SSR: It develops a novel SSR procedure that updates the sample size based solely on baseline covariate distributions. This addresses the specific challenge that the variance and effect size of the standardized control depend on the enrolled population's characteristics.
• Theoretical Guarantees: It provides sufficient conditions for approximate type I error control and conditional power preservation in repeated blinded SSR settings, distinguishing it from methods that rely on unblinded outcome data.
• Integration of Planning and Execution: The framework unifies fixed designs based on planning distributions with adaptive designs based on observed enrollment distributions.

Results

• Illustrative Application (ADCS Data): Using data from Alzheimer's Disease Cooperative Study trials, the authors demonstrated that the standardized control parameter and required sample size vary significantly depending on the covariate adjustment set and the evolving enrolled population profile. The proposed SSR yielded different final sample sizes across scenarios, reflecting the actual covariate profiles observed during enrollment, whereas fixed designs yielded static sample sizes.
• Simulation Studies:
  • Scenario: Simulations were conducted where the true active-arm covariate distribution differed from the planning assumption (distributional shift).
  • Power: Fixed designs based solely on planning assumptions lost power as the true distribution deviated from the plan. In contrast, the proposed covariate-adaptive SSR maintained power near the target level (0.80) by adjusting the sample size to the observed distribution.
  • Comparison: The proposed SSR performed similarly to an "oracle" design (which knows the true distribution at the planning stage) and outperformed a "max-scenario" fixed design in terms of sample size efficiency while maintaining power.
  • Type I Error: The procedure maintained approximate type I error control (near 0.05) across scenarios, confirming that outcome-blinded re-estimation does not inflate false positive rates.
  • Unadjusted Designs: Unadjusted fixed designs showed high rejection rates in some settings, but the authors note this reflects misspecification of the control parameter due to imbalance rather than true power.

Significance and Claims
The paper claims to provide a practical design strategy for externally controlled single-arm trials that explicitly targets population-standardized treatment effects. By linking the definition of the target population, the standardization of external controls, and sample size re-estimation into a single framework, the method addresses the uncertainty of the target population at the design stage rather than relying solely on post-hoc covariate adjustment.

The authors emphasize that their approach allows for the adaptation of sample size to the actual enrolled population without unblinding treatment outcomes, thereby preserving the integrity of the trial while ensuring adequate power. The study concludes that this covariate-adaptive, outcome-blinded SSR is a viable strategy to mitigate the risks of power loss associated with distributional shifts between planning assumptions and reality in trials utilizing historical controls.

Limitations
The authors acknowledge several limitations:

• The framework assumes observed baseline covariates sufficiently account for comparability; unmeasured confounding or differences in measurement/definitions can still bias results.
• Model misspecification of the balancing score or conditional control parameters can affect the standardized control parameter and variance estimates.
• The method relies on normal approximations and assumes the active-arm design value is independent of the score distribution. Extensions for non-normal outcomes, small samples, or complex censoring are noted as future work.
• The current focus is on single-arm designs with purely historical controls; hybrid designs with concurrent controls require further development.