Annals of Statistics

Exact properties of Efron’s biased coin randomization procedure

Tigran Markaryan and William F. Rosenberger

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Efron [Biometrika 58 (1971) 403–417] developed a restricted randomization procedure to promote balance between two treatment groups in a sequential clinical trial. He called this the biased coin design. He also introduced the concept of accidental bias, and investigated properties of the procedure with respect to both accidental and selection bias, balance, and randomization-based inference using the steady-state properties of the induced Markov chain. In this paper we revisit this procedure, and derive closed-form expressions for the exact properties of the measures derived asymptotically in Efron’s paper. In particular, we derive the exact distribution of the treatment imbalance and the variance-covariance matrix of the treatment assignments. These results have application in the design and analysis of clinical trials, by providing exact formulas to determine the role of the coin’s bias probability in the context of selection and accidental bias, balancing properties and randomization-based inference.

Article information

Ann. Statist., Volume 38, Number 3 (2010), 1546-1567.

First available in Project Euclid: 24 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E15: Exact distribution theory 62K99: None of the above, but in this section
Secondary: 62L05: Sequential design 62J10: Analysis of variance and covariance

Accidental bias exact distribution theory randomization test restricted randomization selection bias


Markaryan, Tigran; Rosenberger, William F. Exact properties of Efron’s biased coin randomization procedure. Ann. Statist. 38 (2010), no. 3, 1546--1567. doi:10.1214/09-AOS758.

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  • Baldi Antognini, A. (2008). A theoretical analysis of the power of biased coin designs. J. Statist. Plann. Inference 138 1792–1798.
  • Baldi Antognini, A. and Giovagnoli, A. (2004). A new “Biased coin design” for the sequential allocation of two treatments. J. Roy. Statist. Soc. Ser. C 53 651–664.
  • Blackwell, D. and Hodges, J. L. (1957). Design for the control of selection bias. Ann. Math. Statist. 28 449–460.
  • Chen, Y-P. (1999). Biased coin design with imbalance intolerance. Comm. Statist. Stochastic Models 15 953–975.
  • Efron, B. (1971). Forcing a sequential experiment to be balanced. Biometrika 58 403–417.
  • Eisele, J. R. (1994). The doubly adaptive biased coin design for sequential clinical trials. J. Statist. Plann. Inference 38 249–261.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications. I. Wiley, New York.
  • Hollander, M. and Peña, E. (1988). Nonparametric tests under restricted treatment-assignment rules. J. Amer. Statist. Assoc. 83 1141–1151.
  • Hu, F. and Zhang, L.-X. (2004). Asymptotic properties of doubly adaptive biased coin designs for multi-treatment clinical trials. Ann. Statist. 32 268–301.
  • Markaryan, T. (2009). Exact distributional properties of Efron’s biased coin design with applications to clinical trials. Ph.D. dissertation, George Mason Univ., Fairfax, VA.
  • Rosenberger, W. F. and Lachin, J. L. (2002). Randomization in Clinical Trials: Theory and Practice. Wiley, New York.
  • Smith, R. L. (1984). Sequential treatment allocation using biased coin designs. J. Roy. Statist. Soc. Ser. B 46 519–543.
  • Smythe, R. T. and Wei, L. J. (1983). Significance tests with restricted randomization. Biometrika 70 496–500.
  • Soares, J. F. and Wu, C. F. J. (1982). Some restricted randomization rules in sequential designs. Comm. Statist. A—Theory Methods 12 2017–2034.
  • Steele, J. M. (1980). Efron’s conjecture on vulnerability to bias in a method for balancing sequential trials. Biometrika 67 503–504.
  • Wei, L. J. (1978). The adaptive biased coin design for sequential experiments. Ann. Statist. 6 92–100.