Multi-Study R-Learner for Estimating Heterogeneous Treatment Effects Across Studies Using Statistical Machine Learning

Estimating heterogeneous treatment effects (HTEs) is crucial for precision medicine. While multiple studies can improve the generalizability of results, leveraging them for estimation is statistically challenging. Existing approaches often assume identical HTEs across studies, but this may be violated due to various sources of between-study heterogeneity, including differences in study design, study populations, and data collection protocols, among others. To this end, we propose a framework for multi-study HTE estimation that accounts for between-study heterogeneity in the nuisance functions and treatment effects. Our approach, the multi-study R-learner, extends the R-learner to obtain principled statistical estimation with machine learning (ML) in the multi-study setting. It involves a data-adaptive objective function that links study-specific treatment effects with nuisance functions through membership probabilities, which enable information to be borrowed across potentially heterogeneous studies. The multi-study R-learner framework can combine data from randomized controlled trials, observational studies, or a combination of both. It’s easy to implement and flexible in its ability to incorporate ML for estimating HTEs, nuisance functions, and membership probabilities. In the series estimation framework, we show that the multi-study R-learner is asymptotically normal and more efficient than the R-learner when there is between-study heterogeneity in the propensity score model under homoscedasticity. We illustrate using cancer data that the proposed method performs favorably compared to existing approaches in the presence of between-study heterogeneity.

💡 Research Summary

The paper introduces the Multi‑Study R‑Learner, a novel framework for estimating heterogeneous treatment effects (HTEs) when data are pooled from multiple studies that may differ in design, populations, and data collection procedures. Traditional multi‑study methods often assume that the conditional average treatment effect is transportable across studies (i.e., identical HTEs), an assumption that is frequently violated in practice. The authors address this gap by extending the single‑study R‑learner (Nie & Wager, 2021) to a setting where both the propensity scores and outcome models can vary across studies, and where the overall HTE of interest is a weighted average of study‑specific HTEs.

Key methodological contributions include:

1. Multi‑Study Robinson Transformation – By generalizing Robinson’s (1988) orthogonalization, the authors derive a decomposition of the observed outcome that links the residualized outcome (Y‑m̂(X)) to a weighted sum of study‑specific treatment effect functions τ_k(X) multiplied by (A‑ê_k(X)) and the study membership probabilities p̂(k|X). This formulation naturally incorporates cross‑study borrowing through the estimated membership probabilities.
2. Objective Function (Multi‑Study R‑Loss) – The loss function L_n aggregates squared residuals across all observations, penalizing the deviation between the residualized outcome and the weighted combination of study‑specific treatment effects. A regularization term Λ(τ_k) controls complexity, allowing any off‑the‑shelf machine‑learning (ML) estimator (e.g., Lasso, kernel methods, neural nets) to be used for τ_k.
3. Estimation Procedure – The approach proceeds in three steps: (i) estimate study‑specific nuisance functions (propensity scores e_k(·) and outcome regressions m_k(·)) separately for each study; (ii) estimate the membership probabilities p(k|·) by pooling all data in a multi‑class classification model; (iii) solve the penalized multi‑study R‑loss using cross‑fitting to avoid over‑fitting of nuisance estimates. Cross‑fitting splits the data into Q folds, fits nuisance models on Q‑1 folds, and evaluates the loss on the held‑out fold, ensuring orthogonalization and robustness.
4. Theoretical Guarantees – Under a series‑estimation framework and homoscedastic error variance, the authors prove that the estimator of τ_k is asymptotically unbiased and √n‑consistent, converging to a normal distribution. In the special case of two studies, they analytically show that when propensity scores differ across studies, the multi‑study R‑learner attains a smaller asymptotic variance than the standard single‑study R‑learner, highlighting efficiency gains from borrowing information via membership probabilities.
5. Empirical Evaluation – Simulations vary the degree of between‑study heterogeneity in both propensity scores and treatment effects. Results demonstrate that as heterogeneity increases, the multi‑study R‑learner maintains lower mean‑squared error (MSE) and bias compared with meta‑analytic pooling, mixed‑effects models, causal forests, and the original R‑learner. An application to breast‑cancer data from several institutions illustrates practical benefits: the method yields more stable, study‑specific HTE estimates and a reliable overall personalized effect, whereas competing methods either over‑smooth across studies or ignore useful cross‑study information.
6. Practical Flexibility – Because the framework separates nuisance estimation from the final HTE regression, practitioners can plug in any modern ML algorithm (random forests, gradient boosting, deep nets) for each component. The membership model can be any probabilistic classifier, and the regularizer can be tuned via cross‑validation, making the method readily implementable in existing causal‑inference pipelines.

Overall, the Multi‑Study R‑Learner provides a principled, theoretically sound, and computationally tractable solution for heterogeneous treatment effect estimation in the presence of study‑level heterogeneity. By explicitly modeling study‑specific nuisance functions and using data‑driven membership probabilities to control information borrowing, it bridges the gap between fully pooled analyses (which assume transportability) and completely separate analyses (which ignore potential synergies). The paper’s blend of rigorous theory, extensive simulations, and real‑world validation positions the method as a valuable tool for multi‑institutional clinical research, precision‑medicine initiatives, and any setting where causal inference must reconcile heterogeneous data sources.