Abstract

The treatment allocation mechanism in a randomized clinical trial can be optimized by maximizing the nonparametric efficiency bound for a specific measure of treatment effect. Optimal treatment allocations which may or may not depend on baseline covariates have been derived for a variety of effect measures focusing on the trial population, the patient population represented by the trial participants. Frequently, clinical trial data are used to estimate treatment effects in a target population that is related to but different from the trial population. This article provides optimal treatment allocations that account for the impact of such population differences. We consider three cases with different data configurations: transportation, generalization, and post-stratification. Our results indicate that, for general effect measures, optimal treatment allocations may depend on the covariate distribution in the target population but not on the configuration of data or information that describes the target covariate distribution. For estimating average treatment effects, there is a unique covariate-dependent allocation that achieves maximal efficiency regardless of the target covariate distribution and the associated data configuration.

Key words and phrases: covariate adjustment, covariate-dependent randomization, generalizability, optimal design, propensity score, transportability

Information

Preprint No.SS-2025-0330
Manuscript IDSS-2025-0330
Complete AuthorsWei Zhang, Zhiwei Zhang, Aiyi Liu
Corresponding AuthorsWei Zhang
Emailszhangwei@amss.ac.cn

References

  1. Bannick, M.S., Shao, J., Liu, J., Du, Y., Yi, Y., & Ye, T. (2025). A general form of covariate adjustment in randomized clinical trials. Biometrika, 112, asaf029.
  2. Bickel, P.J., Klaassen, C.A.J., Ritov, Y. & Wellner, J.A. (1993) Efficient and Adaptive Estimation for Semiparametric Models. Baltimore, MD: Johns Hopkins University Press.
  3. Breslow, N. & Day, N. (1987) Statistical methods in cancer research, volume II: The design and analysis of cohort studies World Health Organization.
  4. Cahn, P., Pozniak, A. L., Mingrone, H., Shuldyakov, A., Brites, C., Andrade-Villanueva,
  5. J. F. & et al. for the SAILING Study Team (2013). Dolutegravir versus raltegravir in antiretroviral-experienced, integrase-inhibitor-naive adults with HIV: Week 48 results from the randomised, double-blind, non-inferiority SAILING study. Lancet, 382, 700–708.
  6. Cappiello, L., Zhang, Z., Shen, C., Butala, N.M., Cui, X. & Yeh, R.W. (2021) Adjusting for population differences using machine learning methods. Journal of the Royal Statistical
  7. Society, Series C, 70, 750–769.
  8. Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., &
  9. Robins, J. (2018) Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal, 21: C1–C68.
  10. Cole, S.R. & Stuart, E.A. (2010) Generalizing evidence from randomized clinical trials to target populations: the ACTG 320 trial. American Journal of Epidemiology, 172, 107–115. Colnet B Mayer I Chen G Dieng A Li R
  11. Varoquaux G & et al (2024) Causal inference methods for combining randomized trials and observational studies: A review. Statistical Science, 39, 165–191.
  12. Cooper, D.A., Steigbigel, R.T., Gatell, J.M., Rockstroh, J.K., Katlama, C. & et al. for the
  13. BENCHMRK Study Teams. (2008). Subgroup and resistant analyses of Raltegravir for resistant HIV-1 infection. New England Journal of Medicine 359, 355–365.
  14. Dahabreh, I.J., Robertson, S.E., Tchetgen Tchetgen, E.J., Stuart, E.A. & Hernan, M.A.
  15. (2019) Generalizing causal inferences from individuals in randomized trials to all trialeligible individuals. Biometrics, 75, 685–694.
  16. Hastie, T., Tibshirani, R. & Friedman, J. (2009) The Elements of Statistical Learning: Data
  17. Mining, Inference, and Prediction, 2nd ed. New York, Springer-Verlag.
  18. Ingall, T.J., O’Fallon, W.M., Asplund, K., Goldfrank, L.R., Hertzberg, V.S., Louis, T.A. &
  19. et al. (2004) Findings from the reanalysis of the NINDS tissue plasminogen activator for acute ischemic stroke treatment trial. Stroke, 35, 2418–2424.
  20. Moore, K.L. & van der Laan, M.J. (2009) Covariate adjustment in randomized trials with binary outcomes: targeted maximum likelihood estimation. Statistics in Medicine, 28, 39–64.
  21. Neyman, J. (1934). On the two different aspects of the representative method: The method of stratified sampling and the method of purposive selection. Journal of the Royal Statistical Society, 97, 558–625.
  22. Nie, L., Zhang, Z., Rubin, D.B. & Chu, J. (2013) Calibration of treatment effect size through propensity score ratio reweighting, with application to clinical trials. Annals of Applied Statistics, 7, 1796–1813.
  23. NINDS rt-PA Stroke Study Group. (1995) Tissue plasminogen activator for acute ischemic stroke. New England Journal of Medicine, 333, 1581–1587.
  24. Polley, E.C., Rose, S. & van der Laan, M.J. (2011) Super learning. In Targeted Learning, pages 43–66. New York, Springer.
  25. Rosenblum, M. & van der Laan, M.J. (2010) Simple, efficient estimators of treatment effects in randomized trials using generalized linear models to leverage baseline variables. International Journal of Biostatistics, 6, article 13.
  26. Rudolph, K.E. & van der Laan, M.J. (2017) Robust estimation of encouragement design intervention effects transported across sites. Journal of the Royal Statistical Society, Series B, 79, 1509–1525.
  27. Steigbigel, R.T., Cooper, D.A., Kumar, P.N., Eron, J.E., Schechter, M., Markowitz, M. &
  28. et al. for the BENCHMRK Study Teams (2008) Raltegravir with optimized background therapy for resistant HIV-1 infection. New England Journal of Medicine 359, 339–354.
  29. Stuart, E.A., Cole, S.R., Bradshaw, C.P. & Leaf, P.J. (2011) The use of propensity scores to assess the generalizability of results from randomized trials. Journal of the Royal Statistical
  30. Society, Series A, 174, 369–386.
  31. Tian, L., Cai, T., Zhao, L. & Wei, L.J. (2012) On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial. Bi t ti ti 13 256 273
  32. Tsiatis, A.A. (2006) Semiparametric Theory and Missing Data. New York, Springer.
  33. Tsiatis, A.A., Davidian, M., Zhang, M. & Lu, X. (2008) Covariate adjustment for two-sample treatment comparisons in randomized clinical trials: a principled yet flexible approach. Statistics in Medicine, 27, 4658–4677.
  34. van der Laan, M.J. & Robins, J.M. (2003) Unified Methods for Censored Longitudinal Data and Causality. New York, Springer-Verlag.
  35. van der Laan, M.J. & Rose, S. (2011) Targeted Learning: Causal Inference for Observational and Experimental Data. New York, Springer.
  36. Ye, T., Shao, J., Yi, Y. & Zhao, Q. (2023) Toward better practice of covariate adjustment in analyzing randomized clinical trials. Journal of the American Statistical Association, 118, 2370–2382.
  37. Zhang, M., Tsiatis, A.A. & Davidian, M. (2008) Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics, 64, 707–715.
  38. Zhang, W., Zhang, Z. & Liu, A. (2023) Optimizing treatment allocation in randomized clinical trials by leveraging baseline covariates. Biometrics, 79, 2815–2829.
  39. Zhang, W., Zhang, Z. & Liu, A. (2025) An adaptive design for optimizing treatment assignment in randomized clinical trials. http://arxiv.org/abs/2509.00429.
  40. Zhang, Z. (2009) Covariate-adjusted putative placebo analysis in active-controlled clinical trials. Statistics in Biopharmaceutical Research, 1, 279–290.
  41. Zhang, Z. & Ma, S. (2019) Machine learning methods for leveraging baseline covariate information to improve the efficiency of clinical trials. Statistics in Medicine, 38, 1703–1714.
  42. Zhang, Z., Nie, L., Soon, G. & Hu, Z. (2016) New methods for treatment effect calibration, with applications to non-inferiority trials. Biometrics, 72, 20–29. Supporting Information The Supplementary Material includes proofs and details omitted in the main text. W2 = 1 W2 = 0 0.0 0.2 0.4 0.6 0.8 1.0 Propensity score −2 −1 Figure 1: The optimal CDR design for ATE estimation in the simulation study with a continuous outcome. popt(W1, 1) popt popt(W1, 0) popt 0.0 0.2 0.4 0.6 0.8 1.0 tr (W1, 1) popt tr (W1, 0) popt gen(W1, 1) popt gen(W1, 0) popt ps (W1, 1) ps (W1, 0) Propensity score −2 −1 Figure 2: Optimal CDR designs for LMR estimation in the simulation study with a count outcome.

Acknowledgments

The research of Wei Zhang was supported by the National Key R&D Program of China [grant number 2022YFA1004800]. The research of Aiyi Liu was supported by the Intramural Research Program of the National Institutes of Health (NIH). The contributions of the NIH author were made as part of their official duties as NIH federal employees, are in compliance with agency policy requirements, and are considered Works of the United States Government. However, the findings and conclusions presented in this paper are those of the author and do not necessarily reflect the views of the NIH or the U.S. Department of Health and Human Services.