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Generalized Efron's biased coin design and its theoretical properties

  • Yanqing Hu (a1)


In clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probability p > ½. Under this design the proportion of patients in any arm converges to ½, and the convergence rate is n-1, as opposed to n under some other popular designs. The generalization of Efron's design to K ≥ 2 arms and an unequal target allocation ratio (q1, . . ., qK) can be found in some papers, most of which determine the allocation probabilities ps in a heuristic way. Nonetheless, it has been noted that by using inappropriate ps, the proportion of patients in the K arms never converges to the target ratio. We develop a general theory to answer the question of what allocation probabilities ensure that the realized proportions under a generalized design still converge to the target ratio (q1, . . ., qK) with rate n-1.


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* Postal address: Department of Statistics, West Virginia University, PO Box 6330, Morgantown, WV 26506, USA. Email address:


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[1]Atkinson, A. C. (1982).Optimum biased coin designs for sequential clinical trials with prognostic factors.Biometrika 69, 6167.
[2]Baldi Antognini, A. and Giovagnoli, A. (2004).A new 'biased coin design' for the sequential allocation of two treatments.J. R. Statist. Soc. C 53, 651664.
[3]Burman, C. F. (1996).On sequential treatment allocations in clinical trials. Doctoral thesis. Chalmers University of Technology.
[4]Chen, Y.-P. (1999).Biased coin design with imbalance tolerance.Commun. Statist. Stoch. Models 15, 953975.
[5]Efron, B. (1971).Forcing a sequential experiment to be balanced.Biometrika 58, 403417.
[6]Eisele, J. R. and Woodroofe, M. B. (1995).Central limit theorems for doubly adaptive biased coin designs.Ann. Statist. 23, 234254.
[7]Han, B., Enas, N. and McEntegart, D. (2009).Randomization by minimization for unbalanced treatment allocation.Statist. Med. 28, 33293346.
[8]Han, B., Yu, M. and McEntegart, D. (2013).Weighted re-randomization tests for minimization with unbalanced allocation.Pharm. Statist. 12, 243253.
[9]Hu, F. and Rosenberger, W. F. (2006).The Theory of Response-Adaptive Randomization in Clinical Trials.John Wiley, Hoboken, NJ.
[10]Hu, F. and Zhang, L.-X. (2004).Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials.Ann. Statist. 32, 268301.
[11]Hu, F., Zhang, L.-X. and He, X. (2009).Efficient randomized-adaptive designs.Ann. Statist. 37, 25432560.
[12]Hu, F., Hu, Y., Ma, Z. and Rosenberger, W. F. (2014).Adaptive randomization for balancing over covariates.WIREs Comput. Statist. 6, 288303.
[13]Hu, al. (2015).Statistical inference of adaptive randomized clinical trials for personalized medicine.Clin. Investigation 5, 415425.
[14]Hu, J., Zhu, H. and Hu, F. (2015).A unified family of covariate-adjusted response-adaptive designs based on efficiency and ethics.J. Amer. Statist. Assoc. 110, 357367.
[15]Hu, Y. and Hu, F. (2012).Asymptotic properties of covariate-adaptive randomization.Ann. Statist. 40, 17941815.
[16]Kuznetsova, O. M. and Tymofyeyev, Y. (2012).Preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation.Statist. Med. 31, 701723.
[17]Kuznetsova, O. M. and Tymofyeyev, Y. (2013).Shift in re-randomization distribution with conditional randomization test.Pharm. Statist. 12, 8291.
[18]Ma, W., Hu, F. and Zhang, L. (2015).Testing hypotheses of covariate-adaptive randomized clinical trials.J. Amer. Statist. Assoc. 110, 669680.
[19]Markaryan, T. and Rosenberger, W. F. (2010).Exact properties of Efron's biased coin randomization procedure.Ann. Statist. 38, 15461567.
[20]Meyn, S. P. and Tweedie, R. L. (1993).Markov Chains and Stochastic Stability.Springer, London.
[21]Pocock, S. J. and Simon, R. (1975).Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial.Biometrics 31, 103115.
[22]Proschan, M., Brittain, E. and Kammerman, L. (2011).Minimize the use of minimization with unequal allocation.Biometrics 67, 11351141.
[23]Rosenberger, W. F. and Lachin, J. M. (2002).Randomization in Clinical Trials: Theory and Practice.John Wiley, New York.
[24]Rosenberger, W. F., Sverdlov, O. and Hu, F. (2012).Adaptive randomization for clinical trials.J. Biopharm. Statist. 22, 719736.
[25]Smith, R. L. (1984).Properties of biased coin designs in sequential clinical trials.Ann. Statist. 12, 10181034.
[26]Smith, R. L. (1984).Sequential treatment allocation using biased coin designs.J. R. Statist. Soc. B 46, 519543.
[27]Soares, J. F. and Wu, C.-F. J. (1983).Some restricted randomization rules in sequential designs.Commun. Statist. Theory Meth. 12, 20172034.
[28]Wei, L. J. (1978).An application of an urn model to the design of sequential controlled clinical trials.J. Amer. Statist. Assoc. 73, 559563.
[29]Wei, L. J. (1978).The adaptive biased coin design for sequential experiments.Ann. Statist. 6, 92100.
[30]Wei, L. J., Smythe, R. T. and Smith, R. L. (1986).K-treatment comparisons with restricted randomization rules in clinical trials..Ann. Statist. 14, 265274.
[31]Zhang, L.-X., Hu, F., Cheung, S. H. and Chan, W. S. (2007).Asymptotic properties of covariate-adjusted response-adaptive designs.Ann. Statist. 35, 11661182.


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