# Adaptive Combination of Randomized and Observational Data

arXiv: https://arxiv.org/abs/2111.15012 · HTML: https://mlsanity.com/paper/2111.15012/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2021.

## Built on by (most cited first, 22 of 22)

- [Combining Observational and Randomized Data for Estimating Heterogeneous Treatment Effects](https://mlsanity.com/paper/2202.12891/) (2022)
- [Long-term Causal Inference Under Persistent Confounding via Data Combination](https://mlsanity.com/paper/2202.07234/) (2022)
- [Prediction-powered Generalization of Causal Inferences](https://mlsanity.com/paper/2406.02873/) (2024)
- [Falsification of Internal and External Validity in Observational Studies via Conditional Moment Restrictions](https://mlsanity.com/paper/2301.13133/) (2023)
- [Efficient Randomized Experiments Using Foundation Models](https://mlsanity.com/paper/2502.04262/) (2025)
- [Adaptive-TMLE for the Average Treatment Effect based on Randomized Controlled Trial Augmented with Real-World Data](https://mlsanity.com/paper/2405.07186/) (2024)
- [Combining Experimental and Historical Data for Policy Evaluation](https://mlsanity.com/paper/2406.00317/) (2024)
- [General Synthetic-Powered Inference](https://mlsanity.com/paper/2509.20345/) (2025)
- [Minimax Regret Estimation for Generalizing Heterogeneous Treatment Effects with Multisite Data](https://mlsanity.com/paper/2412.11136/) (2024)
- [Multi-Study R-Learner for Estimating Heterogeneous Treatment Effects Across Studies Using Statistical Machine Learning](https://mlsanity.com/paper/2306.01086/) (2023)
- [Collaborative Heterogeneous Causal Inference Beyond Meta-analysis](https://mlsanity.com/paper/2404.15746/) (2024)
- [Causal Effect Estimation from Observational and Interventional Data Through Matrix Weighted Linear Estimators](https://mlsanity.com/paper/2306.06002/) (2023)
- [Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data](https://mlsanity.com/paper/2406.06452/) (2024)
- [The Partial Testimony of Logs: Evaluation of Language Model Generation under Confounded Model Choice](https://mlsanity.com/paper/2605.01311/) (2026)
- [Estimating treatment effects from single-arm trials via latent-variable modeling](https://mlsanity.com/paper/2311.03002/) (2023)
- [Combining Incomplete Observational and Randomized Data for Heterogeneous Treatment Effects](https://mlsanity.com/paper/2410.21343/) (2024)
- [Improving RCT-Based Treatment Effect Estimation Under Covariate Mismatch via Calibrated Alignment](https://mlsanity.com/paper/2603.19186/) (2026)
- [Transportability without Graphs: A Bayesian Approach to Identifying s-Admissible Backdoor Sets](https://mlsanity.com/paper/2505.12801/) (2025)
- [Deconfounded Warm-Start Thompson Sampling with Applications to Precision Medicine](https://mlsanity.com/paper/2505.17283/) (2025)
- [Augmenting Limited and Biased RCTs through Pseudo-Sample Matching-Based Observational Data Fusion Method](https://mlsanity.com/paper/2509.18148/) (2025)
- [Testing Effect Homogeneity and Confounding in High-Dimensional Experimental and Observational Studies](https://mlsanity.com/paper/2602.19703/) (2026)
- [Prediction-Powered Data Fusion for Treatment Effect Estimation](https://mlsanity.com/paper/2610.12332/) (2026)

## Abstract (arXiv)

Data from both a randomized trial and an observational study are sometimes simultaneously available for evaluating the effect of an intervention. The randomized data typically allows for reliable estimation of average treatment effects but may be limited in sample size and patient heterogeneity for estimating conditional average treatment effects for a broad range of patients. Estimates from the observational study can potentially compensate for these limitations, but there may be concerns about whether confounding and treatment effect heterogeneity have been adequately addressed. We propose an approach for combining conditional treatment effect estimators from each source such that it aggressively weights toward the randomized estimator when bias in the observational estimator is detected. This allows the combination to be consistent for a conditional causal effect, regardless of whether assumptions required for consistent estimation in the observational study are satisfied. When the bias is negligible, the estimators from each source are combined for optimal efficiency. We show the problem can be formulated as a penalized least squares problem and consider its asymptotic properties. Simulations demonstrate the robustness and efficiency of the method in finite samples, in scenarios with bias or no bias in the observational estimator. We illustrate the method by estimating the effects of hormone replacement therapy on the risk of developing coronary heart disease in data from the Women's Health Initiative.