The 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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