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Transporting Randomized Trial Effects to Real-World Populations via Riesz-Calibrated Optimal Transport

This paper introduces RICOT, a Riesz-calibrated Optimal Transport method that corrects inherent biases in standard transport approaches to achieve doubly robust, semiparametrically efficient estimation of randomized trial effects in real-world target populations, even under conditions of model misspecification and weak covariate overlap.

Original authors: Anik Burman, Margaret Gamalo, Promit Ghosal, Prosenjit Kundu

Published 2026-08-25

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CC BY 4.0

Original authors: Anik Burman, Margaret Gamalo, Promit Ghosal, Prosenjit Kundu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of medical research, the gold standard for proving a treatment works is the randomized controlled trial. In these studies, patients are randomly assigned to receive either a new medicine or a placebo, ensuring that any difference in outcomes is caused by the drug itself. However, the people who sign up for these trials are often very different from the patients doctors see in their daily practice. Trial participants might be younger, healthier, or live in specific geographic areas, while the real-world population includes people with complex medical histories, different ages, and diverse backgrounds. This gap creates a difficult problem: a drug might look perfect in the trial but behave differently when given to the general public. Scientists need a way to take the reliable results from the trial and "transport" them to the real world, adjusting for these differences so they can predict how the treatment will actually perform for everyone.

The challenge lies in figuring out how to weigh the trial data so that it looks like the real-world population. Traditional methods try to guess the rules that determine who gets into a trial, but if those guesses are wrong, the final prediction is wrong too. A team of researchers, including statisticians from Johns Hopkins University, the University of Chicago, and Pfizer, has developed a new approach called RiCOT to solve this. Instead of guessing the rules of participation, their method uses a mathematical technique known as optimal transport. Imagine two piles of sand, one representing the trial patients and the other representing the real-world patients. Optimal transport finds the most efficient way to move grains of sand from the trial pile to match the shape of the real-world pile. This creates a map that shows exactly how to adjust the trial data to fit the real world.

However, the researchers discovered that simply using this transport map introduces a subtle but persistent error. The mathematical smoothing used to make the calculation work creates a bias that does not disappear even if the study gets larger. To fix this, the team added a calibration step. They forced the transport map to satisfy specific balance conditions, ensuring that the adjusted trial data perfectly matches the real-world data on key characteristics. This calibration removes the error while keeping the benefits of the transport method. The result is a tool that can take treatment effects from a controlled experiment and accurately project them onto a real-world population, even when the two groups are very different.

The researchers tested their method through extensive computer simulations and by applying it to a real-world case involving a rare heart condition called transthyretin amyloid cardiomyopathy. In the simulations, they created scenarios where the trial and real-world populations were very different and where traditional methods failed. In these difficult cases, the new method remained accurate, while older techniques produced biased results that could lead to incorrect conclusions. When applied to the heart condition study, the method successfully transported the trial results to a registry of real-world patients who had received the treatment. It confirmed that the treatment reduced the risk of death over thirty months, providing a more reliable estimate of the drug's benefit for the general population than previous methods could offer.

The study also explored what happens when the data is messy or when the overlap between the trial and real-world groups is weak. The new method proved robust, maintaining its accuracy where other approaches broke down. It showed that by combining the geometric precision of transport with the balance of calibration, researchers can achieve a level of reliability that was previously difficult to reach. This work does not just offer a new calculation; it provides a more trustworthy way to bring the certainty of clinical trials into the complex reality of everyday medical care, ensuring that decisions about treatments are based on evidence that truly reflects the people who will use them.

Technical Summary: Transporting Randomized Trial Effects to Real-World Populations via Riesz-Calibrated Optimal Transport

Problem Statement
Randomized controlled trials (RCTs) provide internally valid causal estimates, but their generalizability to real-world populations is often limited by differences in covariate distributions between trial participants and the target population. Standard approaches to transportability, such as inverse probability of sampling weighting (IPSW) based on parametric propensity score models, are sensitive to model misspecification and weak overlap. While Optimal Transport (OT) offers a geometric alternative by aligning populations directly in covariate space, standard regularized OT (specifically entropic OT) induces a population-level bias in the resulting weights. This bias prevents the transport weights from consistently estimating the target-to-trial density ratio required for semiparametric efficiency, even as sample sizes grow, provided the regularization parameters remain fixed.

Methodology: RiCOT
The authors propose RiCOT (Riesz-calibrated Optimal Transport), a procedure that integrates Riesz calibration directly into a semi-unbalanced, entropic regularized OT framework.

Semi-Unbalanced Entropic OT: The method formulates a transport problem where the target marginal is fixed (representing the estimand's population) while the trial marginal is relaxed via a Kullback–Leibler divergence penalty. This "source-relaxed" formulation allows poorly aligned trial observations to receive reduced mass, adapting to weak overlap.
The Inferential Gap: The authors demonstrate that at fixed entropic ( $\epsilon$ ) and source-relaxation ( $\rho$ ) parameters, the induced trial-side weight $r_{\epsilon, \rho}$ converges to a biased limit that differs from the true density ratio $r^\dagger = dQ_X/dP_X$ . This bias is a population object that does not vanish with increasing sample size.
Riesz Calibration: To correct this, RiCOT imposes the Riesz moment conditions directly within the transport optimization. Specifically, it enforces that the weighted trial covariates match the target covariates on a growing sieve of basis functions $\mathcal{H}_J$ $H_{J}$ .
- The calibration is implemented as a multiplicative exponential tilt of the raw transport weight: $\hat{r}(x) = \exp(\gamma \hat{L}(x) + \theta^\top b_J(x))$ , where $\hat{L}$ is the transport offset and $\theta$ is a finite-dimensional parameter.
- This structure preserves the positivity and geometric properties of OT while ensuring the weights satisfy the balance conditions required for efficient causal inference.
Doubly Robust Estimation: The calibrated weights are combined with an outcome regression model (fitted via cross-fitting) to construct a one-step estimator for the average treatment effect in the treated target population ( $\tau_Q$ ). The estimator utilizes the efficient influence function, which features the density ratio as a Riesz representer.

Key Contributions

Characterization of Regularization Bias: The paper derives an exact fixed-point representation of the population trial-side weight in uncalibrated entropic OT. It provides a local expansion showing that the bias consists of a source-relaxation component (scaling with $\rho/\epsilon$ ) and an entropic smoothing component (scaling with $\epsilon$ ). Crucially, it proves that this bias persists at fixed regularization parameters, necessitating a correction beyond undersmoothing.
Riesz-Calibrated Transport: The authors introduce a calibration mechanism that removes the regularization-induced distortion without requiring the entropic or relaxation parameters to vanish. They show that with a growing calibration sieve, the calibrated weight consistently estimates the target-to-trial density ratio in $L^2(P_X)$ , even when regularization parameters remain fixed and positive.
Semiparametric Efficiency and Double Robustness: The resulting estimator is shown to be doubly robust: it is consistent if either the outcome regression or the calibrated weight is consistent. Under suitable rate conditions (specifically, that the product of the nuisance estimation errors is $o(N^{-1/2})$ ), the estimator achieves the semiparametric efficiency bound.
Variance Estimation: The paper proposes a direct variance estimator based on the influence function, which does not require resampling or repeated OT optimization, facilitating practical inference.

Results

Simulations: In simulation studies involving Gaussian shifts, covariance tilts, and mixture distributions, RiCOT demonstrated low bias and near-nominal coverage (95%) across various overlap and misspecification settings.
- When the sampling propensity score was misspecified, conventional augmented IPW (AIPW-PS) exhibited significant bias and poor coverage, whereas RiCOT maintained validity by relying on the consistency of the weighting branch.
- Under double misspecification (both outcome and weight models wrong), increasing the calibration dimension restored valid inference for RiCOT, though at a quantifiable cost to efficiency (variance remained slightly above the bound).
- The method remained stable under weakening overlap, whereas standard propensity score weights became unstable or ineffective.
Real-World Application: The method was applied to transport the treatment effect of tafamidis for transthyretin amyloid cardiomyopathy from a randomized trial to a real-world registry. The RiCOT estimate of the 30-month mortality risk difference ($-0.556$) was consistent with outcome-regression-based estimates but provided a more robust transport mechanism than parametric IPW, which yielded a more attenuated estimate ($-0.476$).

Significance and Claims
The paper positions RiCOT as a robust alternative to parametric sampling-propensity weighting, particularly in regulatory settings where the mechanism governing trial participation is unknown and analysis plans must be prespecified. The authors claim that their approach targets the density ratio through prespecified balance conditions rather than relying on the correct specification of a parametric model.

The significance lies in bridging the gap between the geometric flexibility of Optimal Transport and the rigorous requirements of semiparametric causal inference. By characterizing and correcting the specific bias introduced by regularization, the authors enable the use of stable, fixed-parameter OT solvers while still achieving the consistency and efficiency required for valid transportability. The method is presented not as uniformly superior to all existing methods (acknowledging that correctly specified parametric models can be more efficient), but as a necessary safeguard against model misspecification in real-world evidence generation.

The authors note limitations, including the assumption of a common baseline covariate space and the reliance on conditional transportability (which cannot be verified from data). They identify extensions to time-to-event outcomes and designs incorporating real-world controls as substantive next steps.

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