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arxiv: 2604.25565 · v2 · submitted 2026-04-28 · 📊 stat.ME · math.ST· stat.TH

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CBARA: Covariate-Balanced-and-Adjusted Response-Adaptive Randomization

Hengjia Fang , Wei Ma
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Pith reviewed 2026-05-08 03:15 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords covariate-adaptive randomizationresponse-adaptive randomizationclinical trial designadaptive randomizationcovariate balanceallocation ratioasymptotic propertiesmodel robustness
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The pith

The CBARA procedure improves balance on both observed and unobserved covariates in clinical trials by updating allocation ratios to responses without requiring a correct outcome model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes the covariate-balanced-and-adjusted response-adaptive randomization procedure to combine the strengths of response-adaptive and covariate-adaptive designs in clinical trials. It updates the target allocation ratio dynamically based on patient responses and covariate profiles without assuming a correctly specified model for the responses. This approach aims to retain ethical benefits from adapting to responses while enhancing robustness and balance across covariates. A sympathetic reader would care because it offers a way to make treatment assignments fairer and more efficient when patient characteristics influence outcomes. The authors establish that the procedure achieves asymptotic improvements in balance and consistency of the allocation ratio through analysis of the adaptive process.

Core claim

The CBARA procedure integrates covariate-adjusted response-adaptive randomization with covariate-adaptive randomization by using a newly defined imbalance vector and three interrelated components: the allocation function, parameter estimation, and update mechanism. It updates the target allocation ratio according to observed responses and patient covariate profiles without requiring a correctly specified model. This retains CARA's ethical and efficiency considerations while improving robustness and extends the CAR principle from fixed target allocation ratios to covariate-adjusted adaptive target allocation ratios. The authors establish the asymptotic properties of covariate imbalance and of

What carries the argument

The imbalance vector together with the allocation function, parameter estimation, and update mechanism that together enable dynamic adjustment of target ratios while still pursuing balance in treatment allocation with respect to covariate features.

Load-bearing premise

The asymptotic guarantees rest on the assumption that a pseudo-Markov chain framework with a new discrepancy measure for transition kernels accurately captures the continuity and long-run behavior of the adaptive allocation process.

What would settle it

A simulation or trial dataset in which covariate imbalance fails to decrease or allocation ratios lose consistency after applying the CBARA updates under outcome model misspecification would show the central claims do not hold.

Figures

Figures reproduced from arXiv: 2604.25565 by Hengjia Fang, Wei Ma.

Figure 1
Figure 1. Figure 1: Dependency structure among assumptions on the allocation parameter sequence view at source ↗
read the original abstract

We propose the covariate-balanced-and-adjusted response-adaptive randomization (CBARA) procedure for adaptive design in clinical trials, which integrates the complementary strengths of covariate-adjusted response-adaptive randomization (CARA) and covariate-adaptive randomization (CAR). The CBARA procedure updates the target allocation ratio according to observed responses and patient covariate profiles without requiring a correctly specified model, thereby retaining CARA's ethical and efficiency considerations while improving robustness. In addition, the CBARA procedure extends the CAR principle from fixed target allocation ratios to covariate-adjusted adaptive target allocation ratios, yet still pursues balance in treatment allocation with respect to covariate features. This integration is enabled by a newly defined imbalance vector and three interrelated components: the allocation function, parameter estimation and update mechanism. We establish the asymptotic properties of covariate imbalance and the estimators under the CBARA procedure. The results demonstrate that the CBARA procedure can improve balance for both observed and unobserved covariates while preserving the consistency of the allocation ratio. The theoretical analysis is developed through a pseudo-Markov chain framework, where a new discrepancy measure for transition kernels is introduced to handle the continuity of Poisson equation solutions with respect to parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proposes the covariate-balanced-and-adjusted response-adaptive randomization (CBARA) procedure, which integrates covariate-adjusted response-adaptive randomization (CARA) and covariate-adaptive randomization (CAR). It updates the target allocation ratio based on observed responses and patient covariate profiles without requiring a correctly specified model. Asymptotic properties of the covariate imbalance and the estimators are established using a pseudo-Markov chain framework and a newly introduced discrepancy measure for transition kernels to ensure continuity of Poisson equation solutions with respect to parameters. The results claim that CBARA improves balance for both observed and unobserved covariates while preserving the consistency of the allocation ratio.

Significance. If the asymptotic guarantees hold, this work provides a valuable method for adaptive clinical trial designs that combines ethical and efficiency benefits of response-adaptive methods with improved covariate balance. The introduction of the imbalance vector and the discrepancy measure for transition kernels represents a novel contribution to the analysis of adaptive randomization procedures.

major comments (1)
  1. [Theoretical analysis] The new discrepancy measure for transition kernels is central to handling the continuity of Poisson equation solutions with respect to the running parameter estimates in the pseudo-Markov chain framework. The paper should provide explicit verification or bounds showing that this measure yields a uniform modulus of continuity along the actual adaptive trajectory, as the kernels are data-dependent; without this, the o(1) terms in the imbalance and asymptotic normality results may not vanish.
minor comments (1)
  1. The abstract mentions 'three interrelated components: the allocation function, parameter estimation and update mechanism' but the manuscript could clarify their definitions and interactions more explicitly in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of CBARA and for the constructive comment on the theoretical analysis. We address the point below and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: The new discrepancy measure for transition kernels is central to handling the continuity of Poisson equation solutions with respect to the running parameter estimates in the pseudo-Markov chain framework. The paper should provide explicit verification or bounds showing that this measure yields a uniform modulus of continuity along the actual adaptive trajectory, as the kernels are data-dependent; without this, the o(1) terms in the imbalance and asymptotic normality results may not vanish.

    Authors: We agree that an explicit uniform modulus of continuity along the data-dependent trajectory strengthens the argument. The manuscript introduces the discrepancy measure precisely to control the continuity of solutions to the Poisson equation with respect to the running parameter estimates, and the asymptotic results are derived under the almost-sure convergence of these estimates to their limits (which forces the transition kernels to approach a fixed limiting kernel). To make this step fully rigorous, the revised manuscript will add explicit bounds on the modulus of continuity that hold uniformly along the adaptive path, obtained by combining the Lipschitz continuity of the discrepancy with the rate of convergence of the parameter estimates established earlier in the proof. revision: yes

Circularity Check

0 steps flagged

No circularity: new discrepancy measure is independent analytical tool

full rationale

The CBARA paper introduces a pseudo-Markov chain framework together with a custom discrepancy measure on transition kernels specifically to obtain continuity of Poisson equation solutions with respect to the running parameter estimates. This construction is presented as an external mathematical device that bridges the adaptive updates to the required o(1) remainder control; it is not defined in terms of the target imbalance or allocation ratio, nor is any prediction shown to be a direct algebraic rearrangement of fitted quantities. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled from prior work, and the asymptotic claims on observed/unobserved balance and estimator consistency rest on the new measure rather than reducing to the procedure's own inputs by construction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Evaluation is limited to the abstract; no explicit free parameters, axioms, or invented entities are quantified, but the text implies reliance on the new framework and measure whose validity is unverified here.

axioms (1)
  • domain assumption The pseudo-Markov chain framework and new discrepancy measure for transition kernels correctly capture the continuity and asymptotic behavior of the adaptive allocation process.
    Invoked to establish asymptotic properties of imbalance and estimators.
invented entities (2)
  • imbalance vector no independent evidence
    purpose: To quantify and control covariate imbalance under adaptive target allocation ratios.
    Newly defined to enable integration of CAR and CARA principles.
  • discrepancy measure for transition kernels no independent evidence
    purpose: To handle continuity of Poisson equation solutions with respect to parameters in the theoretical analysis.
    Introduced specifically for the pseudo-Markov chain framework.

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Reference graph

Works this paper leans on

57 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    Rosenberger

    Giacomo Aletti, Andrea Ghiglietti, and William F. Rosenberger. Nonparametric covariate-adjusted response-adaptive design based on a functional urn model.The An- nals of Statistics, 46(6B):3838–3866, 2018

    2018

  2. [2]

    Baldi Antognini and M

    A. Baldi Antognini and M. Zagoraiou. The covariate-adaptive biased coin design for balancing clinical trials in the presence of prognostic factors.Biometrika, 98(3):519–535, 2011

    2011

  3. [3]

    The Efficient Covariate-Adaptive Design for high-order balancing of quantitative and qualitative covariates.Statistical Papers, 65(1):19–44, 2024

    Alessandro Baldi Antognini, Rosamarie Frieri, Maroussa Zagoraiou, and Marco Novelli. The Efficient Covariate-Adaptive Design for high-order balancing of quantitative and qualitative covariates.Statistical Papers, 65(1):19–44, 2024

    2024

  4. [4]

    Ballou.A Response-adaptive Covariate-Balanced Randomization for Multi-Arm Clinical Trials

    Cassandra M. Ballou.A Response-adaptive Covariate-Balanced Randomization for Multi-Arm Clinical Trials. PhD thesis, Oregon Health & Science University, 2015. 135

    2015

  5. [5]

    Adaptive designs for normal responses with prognostic factors.Biometrika, 88(2):409–419, 2001

    Uttam Bandyopadhyay and Atanu Biswas. Adaptive designs for normal responses with prognostic factors.Biometrika, 88(2):409–419, 2001

    2001

  6. [6]

    On a Class of Optimal Type Covariate Adjusted Response Adaptive Allocations for Normal Treatment Responses.Austrian Journal of Statistics, 44(4):53–65, 2015

    Rahul Bhattacharya and Uttam Bandyopadhyay. On a Class of Optimal Type Covariate Adjusted Response Adaptive Allocations for Normal Treatment Responses.Austrian Journal of Statistics, 44(4):53–65, 2015

    2015

  7. [7]

    A class of Covariate-Adjusted Response- Adaptive Allocation Designs for Multitreatment Binary Response Trials.Journal of Biopharmaceutical Statistics, 28(5):809–823, 2018

    Atanu Biswas and Rahul Bhattacharya. A class of Covariate-Adjusted Response- Adaptive Allocation Designs for Multitreatment Binary Response Trials.Journal of Biopharmaceutical Statistics, 28(5):809–823, 2018

    2018

  8. [8]

    On a class of optimal covariate- adjusted response adaptive designs for survival outcomes.Statistical Methods in Medical Research, 25(6):2444–2456, 2016

    Atanu Biswas, Rahul Bhattacharya, and Eunsik Park. On a class of optimal covariate- adjusted response adaptive designs for survival outcomes.Statistical Methods in Medical Research, 25(6):2444–2456, 2016

    2016

  9. [9]

    Bugni, Ivan A

    Federico A. Bugni, Ivan A. Canay, and Azeem M. Shaikh. Inference under covariate-adaptive randomization.Journal of the American Statistical Association, 113(524):1784–1796, 2018

    2018

  10. [10]

    Van Der Laan

    Antoine Chambaz and Mark J. Van Der Laan. Inference in Targeted Group-Sequential Covariate-Adjusted Randomized Clinical Trials.Scandinavian Journal of Statistics, 41(1):104–140, 2014

    2014

  11. [11]

    Van Der Laan

    Antoine Chambaz, Wenjing Zheng, and Mark J. Van Der Laan. Targeted sequential design for targeted learning inference of the optimal treatment rule and its mean reward. The Annals of Statistics, 45(6):2537–2564, 2017

    2017

  12. [12]

    Covariate-adjusted response-adaptive designs for generalized linear models.Journal of Statistical Planning and Inference, 149:152–161, 2014

    Siu Hung Cheung, Li-Xin Zhang, Feifang Hu, and Wai Sum Chan. Covariate-adjusted response-adaptive designs for generalized linear models.Journal of Statistical Planning and Inference, 149:152–161, 2014

    2014

  13. [13]

    Air Markov Chain Monte Carlo.arXiv preprint arXiv:1801.09309, 2018

    Cyril Chimisov, Krzysztof Latuszynski, and Gareth Roberts. Air Markov Chain Monte Carlo.arXiv preprint arXiv:1801.09309, 2018

    work page arXiv 2018

  14. [14]

    A General (Non-Markovian) Framework for Covariate Adap- tive Randomization: Achieving Balance While Eliminating the Shift.arXiv preprint arXiv:2602.22648, 2026

    Hengjia Fang and Wei Ma. A General (Non-Markovian) Framework for Covariate Adap- tive Randomization: Achieving Balance While Eliminating the Shift.arXiv preprint arXiv:2602.22648, 2026

    work page arXiv 2026

  15. [15]

    CBARA: Covariate-Balanced-and-Adjusted Response-Adaptive Randomization

    Hengjia Fang and Wei Ma. Supplement to “CBARA: Covariate-Balanced-and-Adjusted Response-Adaptive Randomization”. 2026. 136

    2026

  16. [16]

    G. B. Folland.Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics. Wiley, New York, 2nd edition, 1999

    1999

  17. [17]

    G. Fort, E. Moulines, and P. Priouret. Convergence of adaptive and interacting Markov chain Monte Carlo algorithms.The Annals of Statistics, 39(6):3262–3289, 2011

    2011

  18. [18]

    G. Fort, E. Moulines, P. Priouret, and P. Vandekerkhove. A central limit theorem for adaptive and interacting Markov chains.Bernoulli, 20(2):457–485, 2014

    2014

  19. [19]

    Response-Adaptive Randomization Procedure in Clinical Trials with Surrogate Endpoints.Statistics in Medicine, 43(30):5911–5921, 2024

    Jingya Gao, Feifang Hu, and Wei Ma. Response-Adaptive Randomization Procedure in Clinical Trials with Surrogate Endpoints.Statistics in Medicine, 43(30):5911–5921, 2024

    2024

  20. [20]

    Probability And Mathematical Statistics [Unnumbered]

    Peter Hall and Christopher Charles Heyde.Martingale Limit Theory and Its Appli- cation. Probability And Mathematical Statistics [Unnumbered]. Academic Press, New York, 1980

    1980

  21. [21]

    Multi-Arm Covariate-Adaptive Random- ization.Science China Mathematics, 66(1):163–190, 2023

    Feifang Hu, Xiaoqing Ye, and Li-Xin Zhang. Multi-Arm Covariate-Adaptive Random- ization.Science China Mathematics, 66(1):163–190, 2023

    2023

  22. [22]

    On the theory of covariate-adaptive designs.arXiv preprint arXiv:2004.02994, 2020

    Feifang Hu and Li-Xin Zhang. On the theory of covariate-adaptive designs.arXiv preprint arXiv:2004.02994, 2020

    work page arXiv 2004

  23. [23]

    Cheung, and Wai S

    Feifang Hu, Li-Xin Zhang, Siu H. Cheung, and Wai S. Chan. Doubly adaptive biased coin designs with delayed responses.Canadian Journal of Statistics, 36(4):541–559, 2008

    2008

  24. [24]

    A Unified Family of Covariate-Adjusted Response-Adaptive Designs Based on Efficiency and Ethics.Journal of the American Statistical Association, 110(509):357–367, 2015

    Jianhua Hu, Hongjian Zhu, and Feifang Hu. A Unified Family of Covariate-Adjusted Response-Adaptive Designs Based on Efficiency and Ethics.Journal of the American Statistical Association, 110(509):357–367, 2015

    2015

  25. [25]

    Asymptotic properties of covariate-adaptive randomiza- tion.The Annals of Statistics, 40(3):1794–1815, 2012

    Yanqing Hu and Feifang Hu. Asymptotic properties of covariate-adaptive randomiza- tion.The Annals of Statistics, 40(3):1794–1815, 2012

    2012

  26. [26]

    Covariate Balancing Propensity Score.Journal of the Royal Statistical Society Series B: Statistical Methodology, 76(1):243–263, 2014

    Kosuke Imai and Marc Ratkovic. Covariate Balancing Propensity Score.Journal of the Royal Statistical Society Series B: Statistical Methodology, 76(1):243–263, 2014

    2014

  27. [27]

    Large deviations for martingales.Stochastic Processes and their Applications, 96(1):143–159, 2001

    Emmanuel Lesigne and Dalibor Voln´ y. Large deviations for martingales.Stochastic Processes and their Applications, 96(1):143–159, 2001. 137

    2001

  28. [28]

    The properties of covariate-adaptive randomization procedures with possibly unequal allocation ratio.The Annals of Applied Statistics, 19(2):907–925, 2025

    Xiao Liu, Feifang Hu, and Wei Ma. The properties of covariate-adaptive randomization procedures with possibly unequal allocation ratio.The Annals of Applied Statistics, 19(2):907–925, 2025

    2025

  29. [29]

    Balancing Unobserved Covariates With Covariate- Adaptive Randomized Experiments.Journal of the American Statistical Association, 117(538):875–886, 2022

    Yang Liu and Feifang Hu. Balancing Unobserved Covariates With Covariate- Adaptive Randomized Experiments.Journal of the American Statistical Association, 117(538):875–886, 2022

    2022

  30. [30]

    The impacts of unobserved covariates on covariate-adaptive randomized experiments.The Annals of Statistics, 51(5):1895–1920, 2023

    Yang Liu and Feifang Hu. The impacts of unobserved covariates on covariate-adaptive randomized experiments.The Annals of Statistics, 51(5):1895–1920, 2023

    1920

  31. [31]

    Testing Hypotheses of Covariate-Adaptive Ran- domized Clinical Trials.Journal of the American Statistical Association, 110(510):669– 680, 2015

    Wei Ma, Feifang Hu, and Li-Xin Zhang. Testing Hypotheses of Covariate-Adaptive Ran- domized Clinical Trials.Journal of the American Statistical Association, 110(510):669– 680, 2015

    2015

  32. [32]

    A New and Unified Family of Covariate Adaptive Randomization Procedures and Their Properties.Journal of the American Statistical Association, 119(545):151–162, 2024

    Wei Ma, Ping Li, Li-Xin Zhang, and Feifang Hu. A New and Unified Family of Covariate Adaptive Randomization Procedures and Their Properties.Journal of the American Statistical Association, 119(545):151–162, 2024

    2024

  33. [33]

    Statistical Inference for Covariate- Adaptive Randomization Procedures.Journal of the American Statistical Association, 115(531):1488–1497, 2020

    Wei Ma, Yichen Qin, Yang Li, and Feifang Hu. Statistical Inference for Covariate- Adaptive Randomization Procedures.Journal of the American Statistical Association, 115(531):1488–1497, 2020

    2020

  34. [34]

    Meurer, Jason T

    William J. Meurer, Jason T. Connor, and Jeffrey Glassberg. Simulation of various randomization strategies for a clinical trial in sickle cell disease.Hematology, 21(4):241– 247, 2016

    2016

  35. [35]

    S. P. Meyn and R. L. Tweedie.Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Cambridge University Press, Cambridge ; New York, 2nd edition, 2009

    2009

  36. [36]

    Wright.Numerical Optimization

    Jorge Nocedal and Stephen J. Wright.Numerical Optimization. Springer Series in Operations Research. Springer, New York, 2nd edition, 2006

    2006

  37. [37]

    Some quantitative results on Lipschitz inverse and implicit functions the- orems.arXiv preprint arXiv:1204.4916, 2012

    Phan Phien. Some quantitative results on Lipschitz inverse and implicit functions the- orems.arXiv preprint arXiv:1204.4916, 2012

    work page arXiv 2012

  38. [38]

    Pocock and Richard Simon

    Stuart J. Pocock and Richard Simon. Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial.Biometrics, 31(1):103–115, 1975. 138

    1975

  39. [39]

    Rosenberger and John M

    William F. Rosenberger and John M. Lachin.Randomization in Clinical Trials: Theory and Practice. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc, Hoboken, New Jersey, second edition, 2016

    2016

  40. [40]

    Rosenberger and Oleksandr Sverdlov

    William F. Rosenberger and Oleksandr Sverdlov. Handling Covariates in the Design of Clinical Trials.Statistical Science, 23(3):404–419, 2008

    2008

  41. [41]

    Rosenberger, A

    William F. Rosenberger, A. N. Vidyashankar, and Deepak K. Agarwal. Covariate- Adjusted Response-Adaptive Designs for Binary Response.Journal of Biopharmaceu- tical Statistics, 11(4):227–236, 2001

    2001

  42. [42]

    J. Shao, X. Yu, and B. Zhong. A theory for testing hypotheses under covariate-adaptive randomization.Biometrika, 97(2):347–360, 2010

    2010

  43. [43]

    CRC Press, Taylor & Francis Group, Boca Raton, FL, 2016

    Oleksan Sverdlov, editor.Modern Adaptive Randomized Clinical Trials: Statistical and Practical Aspects. CRC Press, Taylor & Francis Group, Boca Raton, FL, 2016

    2016

  44. [44]

    Donald R. Taves. Minimization: A new method of assigning patients to treatment and control groups.Clinical Pharmacology & Therapeutics, 15(5):443–453, 1974

    1974

  45. [45]

    van der Vaart.Asymptotic Statistics

    Aad W. van der Vaart.Asymptotic Statistics. Number 3 in Cambridge Series on Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge New York Port Melbourne New Delhi Singapore, 8th edition, 2007

    2007

  46. [46]

    Number 338 in Grundlehren Der Mathematischen Wissenschaften

    C´ edric Villani.Optimal Transport: Old and New. Number 338 in Grundlehren Der Mathematischen Wissenschaften. Springer, Berlin, 2009

    2009

  47. [47]

    L. J. Wei. An application of an urn model to the design of sequential controlled clinical trials.Journal of the American Statistical Association, 73(363):559–563, 1978

    1978

  48. [48]

    Sequential covariate-adjusted ran- domization via hierarchically minimizing Mahalanobis distance and marginal imbalance

    Haoyu Yang, Yichen Qin, Yang Li, and Feifang Hu. Sequential covariate-adjusted ran- domization via hierarchically minimizing Mahalanobis distance and marginal imbalance. Biometrics, 80(2):ujae047, 2024

    2024

  49. [49]

    A Bayesian response-adaptive covariate- balanced randomization design with application to a leukemia clinical trial.Statistics in Medicine, 30(11):1218–1229, 2011

    Ying Yuan, Xuelin Huang, and Suyu Liu. A Bayesian response-adaptive covariate- balanced randomization design with application to a leukemia clinical trial.Statistics in Medicine, 30(11):1218–1229, 2011

    2011

  50. [50]

    M. Zelen. The randomization and stratification of patients to clinical trials.Journal of Chronic Diseases, 27(7-8):365–375, 1974. 139

    1974

  51. [51]

    Statistical Inference with M-Estimators on Adaptively Collected Data

    Kelly Zhang, Lucas Janson, and Susan Murphy. Statistical Inference with M-Estimators on Adaptively Collected Data. InAdvances in Neural Information Processing Systems, volume 34, pages 7460–7471. Curran Associates, Inc., 2021

    2021

  52. [52]

    Asymptotic properties of multi-treatment covariate adaptive random- ization procedures for balancing observed and unobserved covariates.arXiv preprint arXiv:2305.13842, 2023

    Li-Xin Zhang. Asymptotic properties of multi-treatment covariate adaptive random- ization procedures for balancing observed and unobserved covariates.arXiv preprint arXiv:2305.13842, 2023

    work page arXiv 2023

  53. [53]

    A New Family of Covariate-Adjusted Response Adaptive Designs and Their Properties.Applied Mathematics-A Journal of Chinese Universities, 24(1):1–13, 2009

    Li-Xin Zhang and Fei-fang Hu. A New Family of Covariate-Adjusted Response Adaptive Designs and Their Properties.Applied Mathematics-A Journal of Chinese Universities, 24(1):1–13, 2009

    2009

  54. [54]

    Asymptotic properties of covariate-adjusted response-adaptive designs.The Annals of Statistics, 35(3):1166–1182, 2007

    Li-Xin Zhang, Feifang Hu, Siu Hung Cheung, and Wai Sum Chan. Asymptotic properties of covariate-adjusted response-adaptive designs.The Annals of Statistics, 35(3):1166–1182, 2007

    2007

  55. [55]

    Wenxin Zhang, Aaron Hudson, Maya Petersen, and Mark van der Laan. An Online Meta-Level Adaptive-Design Framework with Targeted Learning Inference: Applica- tions to Evaluating and Utilizing Surrogate Outcomes in Adaptive Designs.arXiv preprint arXiv:2408.02667, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  56. [56]

    Incorporating covariates infor- mation in adaptive clinical trials for precision medicine.Pharmaceutical Statistics, 21(1):176–195, 2022

    Wanying Zhao, Wei Ma, Fan Wang, and Feifang Hu. Incorporating covariates infor- mation in adaptive clinical trials for precision medicine.Pharmaceutical Statistics, 21(1):176–195, 2022

    2022

  57. [57]

    Covariate-adjusted response adaptive designs incorporating covariates with and without treatment interactions.Canadian Journal of Statistics, 43(4):534–553, 2015

    Hongjian Zhu. Covariate-adjusted response adaptive designs incorporating covariates with and without treatment interactions.Canadian Journal of Statistics, 43(4):534–553, 2015. 140

    2015