Conformal Selective Borrowing (CSB)

• CSB is a data-adaptive statistical framework that borrows only those external samples passing exchangeability tests to optimize the bias-variance trade-off.
• It integrates conformal exchangeability testing with doubly robust estimation and randomization-based inference to ensure finite-sample validity and strict type I error control.
• Applications include hybrid controlled trials, multi-regional trials, and causal discovery, where selective borrowing improves efficiency and inference precision.

Conformal Selective Borrowing (CSB) is a statistical framework for individualized, data-adaptive borrowing of information across heterogeneous datasets, with rigorous finite-sample guarantees. Fundamentally, CSB combines conformal inference for exchangeability testing with robust estimation (notably, doubly robust estimators) and exact randomization-based hypothesis testing. Its applications include hybrid controlled trials (HCTs) that combine randomized controlled trials (RCTs) with external controls (ECs), multi-regional clinical trials under covariate/outcome mismatch, and selective conformal prediction under interventions in causal discovery settings. The central objective of CSB is to optimize the bias-variance trade-off by adaptively borrowing only those external samples that are statistically indistinguishable—at the level of potential outcome distributions or calibration score behavior—from the target or primary population, while strictly controlling type I error and providing robust inference in finite samples (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026, Asiaee et al., 2 Mar 2026).

1. Conceptual Foundation and Motivation

CSB arises from the need to improve efficiency (statistical power and precision) in estimation and inference when primary data (e.g., from an RCT or target region) are limited, but additional, possibly biased, auxiliary or external data are available. Borrowing external data naively can introduce bias via unmeasured confounding, distributional shift, or intervention-induced outcome drift. CSB addresses two critical sources of mismatch:

• Covariate shift: Distributional differences in observed covariates between the auxiliary and target samples.
• Outcome incomparability (hidden bias/drift): Differences in the conditional outcome distributions not explained by measured covariates.

The CSB methodology provides a solution by carrying out unit-level, finite-sample-valid exchangeability tests, borrowing only auxiliary units that pass these tests. This paradigm achieves robustness to unmeasured confounding and drift, while maintaining (or often improving) efficiency over strictly target-only analyses (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026, Asiaee et al., 2 Mar 2026).

2. Algorithmic Structure of CSB

2.1 Conformal Exchangeability Testing

At the core of CSB is the computation of conformal (nonconformity) scores $s_j$ for candidate auxiliary units $j$. These scores quantify the degree to which each external observation aligns with the reference (target) distribution. In the HCT context, nearest-neighbor-based scores are widely used:

$$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$

where $\mathcal{C}$ is the calibration set of RCT controls, $X$ denotes covariates, and $Y$ is the (binary) outcome (Liu et al., 30 Apr 2025).

A label-conditional extension (LC-NN) compares these scores within strata of $Y$ to enforce conditional validity, crucial for binary outcomes (Liu et al., 30 Apr 2025). In broader settings, such as MRCT or regression, nonconformity scores based on regression residuals are adopted:

$s_j = \left| Y_j - \hat{f}_{-j}(X_j) \right|$

where $\hat{f}_{-j}$ is a predictor (e.g., regression model) trained on the reference sample (Zhu et al., 2024, Li et al., 2 Feb 2026).

2.2 Conformal $p$-Value and Selection

The conformal $j$0-value for each external observation quantifies its compatibility with the reference set:

$j$1

These $j$2 are finite-sample valid under exchangeability of $j$3 with the reference. Fixing a threshold $j$4 yields the selected set:

$j$5

Thus, $j$6 is the adaptively "borrowed" subset (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026).

2.3 Robust Estimation and Inference

CSB can be integrated with doubly robust (AIPW or DR) estimation of the parameter of interest. For example, in HCTs for binary outcomes, the CSB estimator has the general form:

$j$7

where the external sample is restricted to $j$8. For region-specific ATEs in MRCTs, similar DR forms with observed covariates are used (Liu et al., 30 Apr 2025, Li et al., 2 Feb 2026).

2.4 Threshold Optimization

Bias/variance tradeoffs are governed by the choice of $j$9. CSB uses a mean squared error (MSE) proxy, often estimated via bootstrap, to select the optimal threshold:

$$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$0

where

$$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$1

(Zhu et al., 2024, Liu et al., 30 Apr 2025, Li et al., 2 Feb 2026).

2.5 Randomization Inference

Final inference is performed via a Fisher randomization test, re-randomizing treatment assignments and re-selecting auxiliary units as per the CSB procedure at each permutation. This produces exact finite-sample type I error control, even after data-adaptive selection (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026).

3. Theoretical Guarantees

CSB provides the following guarantees:

• Finite-Sample Validity: The conformal $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$2 satisfy $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$3 under exchangeability, with label-conditional validity in the LC-NN extension (Liu et al., 30 Apr 2025, Zhu et al., 2024).
• Exact Type I Error Control: The Fisher randomization test with a selection-aware CSB statistic yields $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$4 for any fixed threshold $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$5, even after threshold selection (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026).
• Adaptive Power–Robustness Trade-Off: When no hidden bias is present, CSB closely matches the power and bias of full-borrowing; when bias is present, CSB adapts by discarding contaminated units, maintaining small bias and lowered MSE relative to both no-borrow and full-borrow strategies (Zhu et al., 2024, Liu et al., 30 Apr 2025, Li et al., 2 Feb 2026).
• Contamination-Robust Coverage: In interventions with possibly contaminated calibration sets, CSB allows explicit finite-sample lower bounds on coverage:

$$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$6

where $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$7 and $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$8 is the contamination fraction. A coverage correction is possible by inflating the nominal level (Asiaee et al., 2 Mar 2026).

4. Methodological Extensions and Applications

4.1 Hybrid and Multi-Regional Controlled Trials

In HCTs, CSB combines nearest-neighbor conformal tests with doubly robust AIPW estimators for risk difference, risk ratio, or odds ratio for binary outcomes (Liu et al., 30 Apr 2025). In MRCTs, CSB enables selective borrowing for region-specific estimands, combining small-sample covariate-rich estimators with large-sample restricted estimators and using conformal inference to select borrowable auxiliary-region patients adaptively (Li et al., 2 Feb 2026).

4.2 Selective Conformal Inference under Interventions

In interventional regimes, such as genomics experiments, CSB restricts calibration scores to interventions estimated (via partial causal learning) to leave the test variable unaffected. This leads to (i) tighter intervals, (ii) explicit coverage loss formulas under contamination, and (iii) algorithms such as perturbation-intersection and local invariant causal prediction to identify eligible calibration sets (Asiaee et al., 2 Mar 2026).

4.3 Finite-Sample Inference and Post-Selection Validity

CSB's selection step is always "replayed" at each permutation of the randomization test, preserving exact type I error control even after selection. This is critical for valid post-selection inference in small samples or high-dimensional settings (Liu et al., 30 Apr 2025, Zhu et al., 2024).

5. Empirical Results

Simulations and real-data analyses across diverse applications consistently support the efficiency and robustness of CSB. Key findings include:

• Simulation studies: Under no bias (e.g., $$s_j = \min\left\{ d(X_j, X_i) : i\in\mathcal{C}, Y_i = Y_j \right\}$$9), full-borrowing and CSB lead to ≥20% reductions in MSE versus no-borrowing. As hidden bias is introduced, CSB adapts, maintaining small bias and controlling type I error, while full-borrowing becomes severely biased (Liu et al., 30 Apr 2025, Zhu et al., 2024).
• Lung cancer hybrid trial: CSB-FRT strictly controlled the type I error, selected 264 of 335 matched ECs, and yielded estimates between those of no-borrowing and full-borrowing approaches (Liu et al., 30 Apr 2025, Zhu et al., 2024).
• Multi-regional trial: CSB produced 10–50% reductions in mean squared error, sharper confidence intervals, and higher power than both NB-AllCov and full-borrowing estimators (Li et al., 2 Feb 2026).
• Intervention studies: In both synthetic and real CRISPRi data, CSB significantly tightened conformal prediction intervals when support sets could be identified, with predicted coverage losses matching theoretical lower bounds (Asiaee et al., 2 Mar 2026).

6. Practical Implementation Considerations

CSB is computationally feasible for moderate sample sizes and is compatible with parallel computation. Tuning parameters include the choice of distance metric (Euclidean on standardized covariates is typical), number of folds (CV+ with $\mathcal{C}$0 is recommended), and threshold grid. Bootstrapping is used for threshold optimization. In practice, the Fisher randomization test with $\mathcal{C}$1 in the range 2000–10000 permutations typically suffices (Liu et al., 30 Apr 2025, Zhu et al., 2024).

Limitations include potential over-borrowing when bias is extremely subtle (no uniform power gain under strict type I error), and inability to remove bias arising from unmeasured confounders or covariates not present in both datasets (Liu et al., 30 Apr 2025, Li et al., 2 Feb 2026). For weak null hypotheses, asymptotic validity requires studentized statistics (Liu et al., 30 Apr 2025).

7. Summary and Outlook

CSB constitutes a unified, finite-sample-valid framework for robust, individualized information borrowing across heterogeneous data regimes, combining conformal inference, robust estimation, and randomization-based inference. The method is scalable, generic, and offers substantial efficiency gains without compromising validity. The growing suite of applications—in hybrid and multi-regional clinical trials, causal discovery, and statistical prediction under interventions—demonstrates its adaptability and relevance for modern statistical challenges where heterogeneity and small-sample validity concerns are paramount (Liu et al., 30 Apr 2025, Zhu et al., 2024, Li et al., 2 Feb 2026, Asiaee et al., 2 Mar 2026).