Error-Controlled Borrowing from External Data Using Wasserstein Ambiguity Sets

TL;DR

BOND (Borrowing under Optimal Nonparametric Distributional robustness), a framework that formalizes data noncommensurability through Wasserstein ambiguity sets centered at the current-trial distribution, and demonstrates that many prominent borrowing methods can be reparameterized via an effective borrowing weight, rendering the calibration framework broadly applicable.

Abstract

Incorporating external data can improve the efficiency of clinical trials, but distributional mismatches between current and external populations threaten the validity of inference. While numerous dynamic borrowing methods exist, the calibration of their borrowing parameters relies mainly on ad hoc, simulation-based tuning. To overcome this, we propose BOND (Borrowing under Optimal Nonparametric Distributional robustness), a framework that formalizes data noncommensurability through Wasserstein ambiguity sets centered at the current-trial distribution. By deriving sharp, closed-form bounds on the worst-case mean drift for both continuous and binary outcomes, we construct a distributionally robust, bias-corrected Wald statistic that ensures asymptotic type I error control uniformly over the ambiguity set. Importantly, BOND determines the optimal borrowing strength by maximizing a worst-case power proxy, converting heuristic parameter tuning into a transparent, analytically tractable optimization problem. Furthermore, we demonstrate that many prominent borrowing methods can be reparameterized via an effective borrowing weight, rendering our calibration framework broadly applicable. Simulation studies and a real-world clinical trial application confirm that BOND preserves the nominal size under unmeasured heterogeneity while achieving efficiency gains over standard borrowing methods.

Figures (17)

• Figure 1: BOND-calibrated borrowing levels versus $\gamma$ for continuous outcomes under data-driven radii ($c=1.5$). Each panel plots the calibrated discount factors $(\lambda_0^\ast,\lambda_1^\ast)$ and the implied EBW $(w_0(\lambda_0^\ast),w_1(\lambda_1^\ast))$. Left: covariate shift + effect modification. Right: control drift with historical controls only (so $\lambda_1^\ast\equiv 0$).
• Figure 2: Empirical type I error (top) and power (bottom) versus $\gamma$ for continuous outcomes under data-driven radii ($c=1.5$). Left: covariate shift + effect modification. Right: control drift with historical controls only (so $\lambda_1^\ast\equiv 0$).
• Figure 3: Real-world sensitivity to $\rho_0$ for BOND vs. baselines. Left: Estimated treatment effect $\hat{\theta}$ with 95% robust confidence intervals (CIs) versus the tolerance radius $\rho_0$. Right: Effective borrowed historical control sample size ($n_{\mathrm{hist}}^{\mathrm{eff}}$) versus $\rho_0$.
• Figure 4: BOND calibrated borrowing levels versus $\gamma$ for continuous outcomes under Commensurate with $n_C=200$ and $n_H=500$. Each panel reports the optimizer $\lambda_a^\ast$ and the induced effective weight $w_a(\lambda_a^\ast)$ for arm $a\in\{0,1\}$.
• Figure 5: BOND calibrated borrowing levels versus $\gamma$ for continuous outcomes under Covariate shift + effect modification with $n_C=200$ and $n_H=500$. Each panel reports the optimizer $\lambda_a^\ast$ and the induced effective weight $w_a(\lambda_a^\ast)$ for arm $a\in\{0,1\}$.

Theorems & Definitions (47)

• Remark 2.1: Extensions beyond two arms and a single historical source
• Remark 2.2
• Proposition 3.1: Closed-form worst-case mean shifts
• Proposition 3.2: Closed-form of $b_+(\lambda)$
• Remark 3.3: Two-sided extension
• Proposition 3.6: Asymptotic normality
• Theorem 3.7: Asymptotic distributionally robust size control
• Proposition 3.8: Tightness and minimality of the robust correction
• Theorem 3.9: Asymptotic robust power and the robust noncentrality parameter
• Corollary 3.10: Robust-power optimal borrowing weight exists