Abstract

Covariate-adaptive randomization (CAR) is commonly used to enhance covariate

balance in randomized experiments, yet the development of statistical inference

procedures on quantile treatment effects (QTEs) under CAR is still limited, with

most existing work focusing on stratified CAR procedures.

In this paper, we

establish a unified theoretical framework for statistical inference on QTE under

CAR procedures, encompassing a wide range of commonly used CAR procedures. First, we introduce a simple regression estimator for QTE and derive its

asymptotic properties. To further enhance estimation efficiency, we propose an

imbalance vector adjusted estimator. Additionally, we develop methods for estimating the asymptotic variance, including the moving block estimator and the

covariate-adaptive randomized bootstrap. Simulation studies and a real data example demonstrate the effectiveness of the proposed methods, highlighting their

practical utility in various settings.

Key words and phrases: Bootstrap, covariate-adaptive randomization, quantile ∗Corresponding author. E-mail: stazlx@mail.zjgsu.edu.cn 1 treatment effect

Information

Preprint No.SS-2025-0214
Manuscript IDSS-2025-0214
Complete AuthorsYuhang Tao, Li-Xin Zhang
Corresponding AuthorsLi-Xin Zhang
Emailsstazlx@mail.zjgsu.edu.cn

References

  1. Anderson, S. J. and D. McKenzie (2022). Improving business practices and the boundary of the entrepreneur: A randomized experiment comparing training, consulting, insourcing, and outsourcing. Journal of Political Economy 130(1), 157–209.
  2. Atkinson, A. C. (1982). Optimum biased coin designs for sequential clinical trials with prognostic factors. Biometrika 69(1), 61–67.
  3. Baldi Antognini, A. and M. Zagoraiou (2011). The covariate-adaptive biased coin design for balancing clinical trials in the presence of prognostic factors. Biometrika 98(3), 519–535.
  4. Bugni, F. A., I. A. Canay, and A. M. Shaikh (2018). Inference under covariate-adaptive randomization. Journal of the American Statistical Association 113(524), 1784–1796.
  5. Bugni, F. A., I. A. Canay, and A. M. Shaikh (2019). Inference under covariate-adaptive randomization with multiple treatments. Quantitative Economics 10(4), 1747–1785.
  6. Burchardi, K. B., S. Gulesci, B. Lerva, and M. Sulaiman (2019). Moral hazard: Experimental evidence from tenancy contracts. The Quarterly Journal of Economics 134(1), 281–347.
  7. Chong, A., I. Cohen, E. Field, E. Nakasone, and M. Torero (2016). Iron deficiency and schooling attainment in peru. American Economic Journal: Applied Economics 8(4), 222–255.
  8. He, X., Y.-H. Hsu, and M. Hu (2010). Detection of treatment effects by covariate-adjusted expected shortfall. The Annals of Applied Statistics 4(4), 2114–2125.
  9. Hjort, N. L. and D. Pollard (2011). Asymptotics for minimisers of convex processes. arXiv preprint arXiv:1107.3806.
  10. Hu, F. and L.-X. Zhang (2020). On the theory of covariate-adaptive designs. arXiv preprint arXiv:2004.02994.
  11. Hu, Y. and F. Hu (2012). Asymptotic properties of covariate-adaptive randomization. The Annals of Statistics 40(3), 1794 – 1815.
  12. Jiang, L., P. C. Phillips, Y. Tao, and Y. Zhang (2023). Regression-adjusted estimation of quantile treatment effects under covariate-adaptive randomizations. Journal of Econometrics 234(2), 758–776.
  13. Knight, K. (1998). Limiting distributions for l1 regression estimators under general conditions. Annals of statistics 26(2), 755–770.
  14. Koenker, R. and G. Bassett Jr (1978). Regression quantiles. Econometrica 46(1), 33–50.
  15. Kremer, J. M., H. K. Genant, L. W. Moreland, A. S. Russell, P. Emery, C. Abud-Mendoza,
  16. J. Szechinski, T. Li, Z. Ge, J.-C. Becker, et al. (2006). Effects of abatacept in patients with methotrexate-resistant active rheumatoid arthritis: a randomized trial. Annals of internal medicine 144(12), 865–876.
  17. LaLonde, R. J. (1986). Evaluating the econometric evaluations of training programs with experimental data. The American economic review, 604–620.
  18. Lin, Y. and Z. Su (2012). Balancing continuous and categorical baseline covariates in sequential clinical trials using the area between empirical cumulative distribution functions. Statistics in medicine 31(18), 1961–1971.
  19. Lin, Y., M. Zhu, and Z. Su (2015). The pursuit of balance: an overview of covariate-adaptive randomization techniques in clinical trials. Contemporary clinical trials 45, 21–25.
  20. Liu, H., F. Tu, and W. Ma (2023). Lasso-adjusted treatment effect estimation under covariate38 adaptive randomization. Biometrika 110(2), 431–447.
  21. Liu, X., F. Hu, and W. Ma (2025). The properties of covariate-adaptive randomization procedures with possibly unequal allocation ratio. The Annals of Applied Statistics 19(2), 907–925.
  22. Liu, Y. and F. Hu (2023). The impacts of unobserved covariates on covariate-adaptive randomized experiments. The Annals of Statistics 51(5), 1895–1920.
  23. Liu, Y., L. Xia, and F. Hu (2025). Testing heterogeneous treatment effect with quantile regression under covariate-adaptive randomization. Journal of Econometrics 249, 105808.
  24. Liu, Y., Y. Zhou, P. Li, and F. Hu (2024). Cluster-adaptive network a/b testing: From randomization to estimation. Journal of Machine Learning Research 25(170), 1–48.
  25. Ma, W., F. Hu, and L. Zhang (2015). Testing hypotheses of covariate-adaptive randomized clinical trials. Journal of the American Statistical Association 110(510), 669–680.
  26. Ma, W., P. Li, L.-X. Zhang, and F. Hu (2024). A new and unified family of covariate adaptive randomization procedures and their properties. Journal of the American Statistical Association 119(545), 151–162.
  27. Ma, W., Y. Qin, Y. Li, and F. Hu (2020). Statistical inference for covariate-adaptive randomization procedures. Journal of the American Statistical Association 115(531), 1488–1497.
  28. Ma, W., F. Tu, and H. Liu (2022). Regression analysis for covariate-adaptive randomization: a robust and efficient inference perspective. Statistics in Medicine 41(29), 5645–5661.
  29. Ma, Z. and F. Hu (2013). Balancing continuous covariates based on kernel densities. Contemporary clinical trials 34(2), 262–269.
  30. Morgan, K. L. and D. B. Rubin (2012). Rerandomization to improve covariate balance in experiments. The Annals of Statistics 40(2), 1263 – 1282.
  31. Pocock, S. J. and R. Simon (1975). Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics, 103–115.
  32. Qin, Y., Y. Li, W. Ma, H. Yang, and F. Hu (2022). Adaptive randomization via mahalanobis distance. Statistica Sinica 1, 1–40.
  33. Rosenberger, W. F. and J. M. Lachin (2015). Randomization in clinical trials: theory and practice. John Wiley & Sons.
  34. Shao, J., X. Yu, and B. Zhong (2010). A theory for testing hypotheses under covariate-adaptive randomization. Biometrika 97(2), 347–360.
  35. Smith, R. L. (1984). Sequential treatment allocation using biased coin designs. Journal of the Royal Statistical Society Series B: Statistical Methodology 46(3), 519–543.
  36. Taves, D. R. (1974). Minimization: a new method of assigning patients to treatment and control groups. Clinical Pharmacology & Therapeutics 15(5), 443–453.
  37. Taves, D. R. (2010). The use of minimization in clinical trials. Contemporary clinical trials 31(2), 180–184. van der Heijde, D., L. Klareskog, V. Rodriguez-Valverde, C. Codreanu, H. Bolosiu, J. Melo40
  38. Gomes, J. Tornero-Molina, J. Wajdula, R. Pedersen, S. Fatenejad, et al. (2006). Comparison of etanercept and methotrexate, alone and combined, in the treatment of rheumatoid arthritis: two-year clinical and radiographic results from the tempo study, a double-blind, randomized trial. Arthritis & Rheumatism 54(4), 1063–1074.
  39. Ye, T., Y. Yi, and J. Shao (2022). Inference on the average treatment effect under minimization and other covariate-adaptive randomization methods. Biometrika 109(1), 33–47.
  40. Zelen, M. (1974). The randomization and stratification of patients to clinical trials. Journal of Chronic Diseases 27(7), 365–375.
  41. Zhang, L.-X. (2019). Theory on covariate-adaptive randomized clinical trials: Efficiency, selection bias and randomization methods. arXiv preprint arXiv:1912.03636.
  42. Zhang, L.-X. (2023). Asymptotic properties of multi-treatment covariate adaptive randomization procedures for balancing observed and unobserved covariates. arXiv preprint arXiv:2305.13842.
  43. Zhang, Y. and X. Zheng (2020). Quantile treatment effects and bootstrap inference under covariate-adaptive randomization. Quantitative Economics 11(3), 957–982.
  44. Zhou, Q., P. A. Ernst, K. L. Morgan, D. B. Rubin, and A. Zhang (2018). Sequential rerandomization. Biometrika 105(3), 745–752. Yuhang Tao

Acknowledgments

The authors are grateful for the detailed comments and suggestions of the

reviewers. This work was supported by grants from National Key R&D

Program of China (No.

2024YFA1013502), NSF of China (Grant Nos.

U23A2064 and 12031005) and the Summit Advancement Disciplines of Zhejiang Province (Zhejiang Gongshang University - Statistics).

Supplementary Materials

The supplementary materials contain proofs and additional numerical studies.

Supplementary materials are available for download.