Journal of Applied Probability

Generalized Efron's biased coin design and its theoretical properties

Yanqing Hu

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Abstract

In clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probability $p\gt\frac{1}{2}$. Under this design the proportion of patients in any arm converges to $\frac{1}{2}$, and the convergence rate is $n^{-1}$, as opposed to $n^{-1/2}$ under some other popular designs. The generalization of Efron's design to $K\geq2$ arms and an unequal target allocation ratio $(q_1, \ldots, q_K)$ can be found in some papers, most of which determine the allocation probabilities $p$s in a heuristic way. Nonetheless, it has been noted that by using inappropriate $p$s, 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 $(q_1, \ldots, q_K)$ with rate $n^{-1}$.

Article information

Source
J. Appl. Probab., Volume 53, Number 2 (2016), 327-340.

Dates
First available in Project Euclid: 17 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.jap/1466172857

Mathematical Reviews number (MathSciNet)
MR3514281

Zentralblatt MATH identifier
1343.60035

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 62P10: Applications to biology and medical sciences

Keywords
More than two treatments unequal allocation drift conditions Markov chain

Citation

Hu, Yanqing. Generalized Efron's biased coin design and its theoretical properties. J. Appl. Probab. 53 (2016), no. 2, 327--340. https://projecteuclid.org/euclid.jap/1466172857


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References

  • Atkinson, A. C. (1982). Optimum biased coin designs for sequential clinical trials with prognostic factors. Biometrika 69, 61–67.
  • 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, 651–664.
  • Burman, C. F. (1996). On sequential treatment allocations in clinical trials. Doctoral Thesis, Department of Mathematics, Chalmers University of Technology.
  • Chen, Y.-P. (1999). Biased coin design with imbalance tolerance. Commun. Statist. Stoch. Models 15, 953–975.
  • Efron, B. (1971). Forcing a sequential experiment to be balanced. Biometrika 58, 403–417.
  • Eisele, J. R. and Woodroofe, M. B. (1995). Central limit theorems for doubly adaptive biased coin designs. Ann. Statist. 23, 234–254.
  • Han, B., Enas, N. and McEntegart, D. (2009). Randomization by minimization for unbalanced treatment allocation. Statist. Med. 28, 3329–3346.
  • Han, B., Yu, M. and McEntegart, D. (2013). Weighted re-randomization tests for minimization with unbalanced allocation. Pharm. Statist. 12, 243–253.
  • Hu, F. and Rosenberger, W. F. (2006). The Theory of Response-Adaptive Randomization in Clinical Trials. John Wiley, Hoboken, NJ.
  • Hu, F. and Zhang, L.-X. (2004). Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials. Ann. Statist. 32, 268–301.
  • Hu, F., Zhang, L.-X. and He, X. (2009). Efficient randomized-adaptive designs. Ann. Statist. 37, 2543–2560.
  • Hu, F., Hu, Y., Ma, Z. and Rosenberger, W. F. (2014). Adaptive randomization for balancing over covariates. WIREs Comput. Statist. 6, 288–303.
  • Hu, F. \et (2015). Statistical inference of adaptive randomized clinical trials for personalized medicine. Clin. Investigation 5, 415–425.
  • 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, 357–367.
  • Hu, Y. and Hu, F. (2012). Asymptotic properties of covariate-adaptive randomization. Ann. Statist. 40, 1794–1815.
  • 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, 701–723.
  • Kuznetsova, O. M. and Tymofyeyev, Y. (2013). Shift in re-randomization distribution with conditional randomization test. Pharm. Statist. 12, 82–91.
  • Ma, W., Hu, F. and Zhang, L. (2015). Testing hypotheses of covariate-adaptive randomized clinical trials. J. Amer. Statist. Assoc. 110, 669–680.
  • Markaryan, T. and Rosenberger, W. F. (2010). Exact properties of Efron's biased coin randomization procedure. Ann. Statist. 38, 1546–1567.
  • Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Pocock, S. J. and Simon, R. (1975). Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics 31, 103–115.
  • Proschan, M., Brittain, E. and Kammerman, L. (2011). Minimize the use of minimization with unequal allocation. Biometrics 67, 1135–1141.
  • Rosenberger, W. F. and Lachin, J. M. (2002). Randomization in Clinical Trials: Theory and Practice. John Wiley, New York.
  • Rosenberger, W. F., Sverdlov, O. and Hu, F. (2012). Adaptive randomization for clinical trials. J. Biopharm. Statist. 22, 719–736.
  • Smith, R. L. (1984). Properties of biased coin designs in sequential clinical trials. Ann. Statist. 12, 1018–1034.
  • Smith, R. L. (1984). Sequential treatment allocation using biased coin designs. J. R. Statist. Soc. B 46, 519–543.
  • Soares, J. F. and Wu, C.-F. J. (1983). Some restricted randomization rules in sequential designs. Commun. Statist. Theory Meth. 12, 2017–2034.
  • Wei, L. J. (1978). An application of an urn model to the design of sequential controlled clinical trials. J. Amer. Statist. Assoc. 73, 559–563.
  • Wei, L. J. (1978). The adaptive biased coin design for sequential experiments. Ann. Statist. 6, 92–100.
  • Wei, L. J., Smythe, R. T. and Smith, R. L. (1986). $K$-treatment comparisons with restricted randomization rules in clinical trials. Ann. Statist. 14, 265–274.
  • Zhang, L.-X., Hu, F., Cheung, S. H. and Chan, W. S. (2007). Asymptotic properties of covariate-adjusted response-adaptive designs. Ann. Statist. 35, 1166–1182.