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Stratification Trees for Adaptive Randomisation in Randomised Controlled Trials

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  • Max Tabord-Meehan

Abstract

This paper proposes an adaptive randomisation procedure for two-stage randomised controlled trials. The method uses data from a first-wave experiment in order to determine how to stratify in a second wave of the experiment, where the objective is to minimise the variance of an estimator for the average treatment effect. We consider selection from a class of stratified randomisation procedures which we call stratification trees: these are procedures whose strata can be represented as decision trees, with differing treatment assignment probabilities across strata. By using the first wave to estimate a stratification tree, we simultaneously select which covariates to use for stratification, how to stratify over these covariates, and the assignment probabilities within these strata. Our main result shows that using this randomisation procedure with an appropriate estimator results in an asymptotic variance which is minimal in the class of stratification trees. Moreover, our results are able to accommodate a large class of assignment mechanisms within strata, including stratified block randomisation. In a simulation study, we find that our method, paired with an appropriate cross-validation procedure, can improve on ad-hoc choices of stratification. We conclude by applying our method to the study in Karlan and Wood (2017, Journal of Behavioral and Experimental Economics, vol. 66, 1–8), where we estimate stratification trees using the first wave of their experiment.

Suggested Citation

  • Max Tabord-Meehan, 2023. "Stratification Trees for Adaptive Randomisation in Randomised Controlled Trials," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 90(5), pages 2646-2673.
    • Handle: RePEc:oup:restud:v:90:y:2023:i:5:p:2646-2673.
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    File URL: http://hdl.handle.net/10.1093/restud/rdac083
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    Cited by:

    1. Li Harrison H. & Owen Art B., 2024. "Double machine learning and design in batch adaptive experiments," Journal of Causal Inference, De Gruyter, vol. 12(1), pages 1-27.
    2. Bokelmann, Björn & Lessmann, Stefan, 2025. "Heteroscedasticity-aware stratified sampling to improve uplift modeling," European Journal of Operational Research, Elsevier, vol. 325(1), pages 118-131.
    3. Davide Viviano, 2020. "Experimental Design under Network Interference," Papers 2003.08421, arXiv.org, revised Nov 2025.
    4. Brian Quistorff & Gentry Johnson, 2020. "Machine Learning for Experimental Design: Methods for Improved Blocking," Papers 2010.15966, arXiv.org.
    5. Rosenman Evan T. R. & Owen Art B., 2021. "Designing experiments informed by observational studies," Journal of Causal Inference, De Gruyter, vol. 9(1), pages 147-171, January.
    6. Jiang, Liang & Li, Liyao & Miao, Ke & Zhang, Yichong, 2025. "Adjustments with many regressors under covariate-adaptive randomizations," Journal of Econometrics, Elsevier, vol. 249(PB).
    7. Zikai Li, 2025. "Bridging Stratification and Regression Adjustment: Batch-Adaptive Stratification with Post-Design Adjustment in Randomized Experiments," Papers 2510.22908, arXiv.org.
    8. Yuehao Bai, 2022. "Optimality of Matched-Pair Designs in Randomized Controlled Trials," American Economic Review, American Economic Association, vol. 112(12), pages 3911-3940, December.
    9. Masahiro Kato & Masaaki Imaizumi & Takuya Ishihara & Toru Kitagawa, 2023. "Asymptotically Optimal Fixed-Budget Best Arm Identification with Variance-Dependent Bounds," Papers 2302.02988, arXiv.org, revised Jul 2023.
    10. Ahnaf Rafi, 2023. "Efficient Semiparametric Estimation of Average Treatment Effects Under Covariate Adaptive Randomization," Papers 2305.08340, arXiv.org.
    11. Bugni, Federico A. & Gao, Mengsi, 2023. "Inference under covariate-adaptive randomization with imperfect compliance," Journal of Econometrics, Elsevier, vol. 237(1).
    12. Ruicheng Ao & Hongyu Chen & David Simchi-Levi, 2024. "Prediction-Guided Active Experiments," Papers 2411.12036, arXiv.org, revised Nov 2024.
    13. Federico A. Bugni & Ivan A. Canay & Azeem M. Shaikh, 2019. "Inference under covariate‐adaptive randomization with multiple treatments," Quantitative Economics, Econometric Society, vol. 10(4), pages 1747-1785, November.
    14. Davide Viviano & Jess Rudder, 2020. "Policy design in experiments with unknown interference," Papers 2011.08174, arXiv.org, revised May 2024.
    15. Harrison H. Li & Art B. Owen, 2023. "Double machine learning and design in batch adaptive experiments," Papers 2309.15297, arXiv.org.
    16. Masahiro Kato & Akihiro Oga & Wataru Komatsubara & Ryo Inokuchi, 2024. "Active Adaptive Experimental Design for Treatment Effect Estimation with Covariate Choices," Papers 2403.03589, arXiv.org, revised Jun 2024.
    17. Yichong Zhang & Xin Zheng, 2020. "Quantile treatment effects and bootstrap inference under covariate‐adaptive randomization," Quantitative Economics, Econometric Society, vol. 11(3), pages 957-982, July.
    18. Liang Jiang & Oliver B. Linton & Haihan Tang & Yichong Zhang, 2022. "Improving Estimation Efficiency via Regression-Adjustment in Covariate-Adaptive Randomizations with Imperfect Compliance," Papers 2201.13004, arXiv.org, revised Jun 2023.
    19. Jiang, Liang & Phillips, Peter C.B. & Tao, Yubo & Zhang, Yichong, 2023. "Regression-adjusted estimation of quantile treatment effects under covariate-adaptive randomizations," Journal of Econometrics, Elsevier, vol. 234(2), pages 758-776.
    20. Federico A. Bugni & Mengsi Gao, 2021. "Inference under Covariate-Adaptive Randomization with Imperfect Compliance," Papers 2102.03937, arXiv.org, revised Jul 2023.
    21. Liang Jiang & Xiaobin Liu & Peter C. B. Phillips & Yichong Zhang, 2024. "Bootstrap Inference for Quantile Treatment Effects in Randomized Experiments with Matched Pairs," The Review of Economics and Statistics, MIT Press, vol. 106(2), pages 542-556, March.
    22. Davide Viviano & Jelena Bradic, 2021. "Dynamic covariate balancing: estimating treatment effects over time with potential local projections," Papers 2103.01280, arXiv.org, revised Jan 2024.
    23. Masahiro Kato, 2023. "Worst-Case Optimal Multi-Armed Gaussian Best Arm Identification with a Fixed Budget," Papers 2310.19788, arXiv.org, revised Mar 2024.

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