Open Access
June 2010 Exact properties of Efron’s biased coin randomization procedure
Tigran Markaryan, William F. Rosenberger
Ann. Statist. 38(3): 1546-1567 (June 2010). DOI: 10.1214/09-AOS758


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.


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Tigran Markaryan. William F. Rosenberger. "Exact properties of Efron’s biased coin randomization procedure." Ann. Statist. 38 (3) 1546 - 1567, June 2010.


Published: June 2010
First available in Project Euclid: 24 March 2010

zbMATH: 1189.62130
MathSciNet: MR2662351
Digital Object Identifier: 10.1214/09-AOS758

Primary: 62E15 , 62K99
Secondary: 62J10 , 62L05

Keywords: Accidental bias , exact distribution theory , Randomization test , restricted randomization , selection bias

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 3 • June 2010
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