Journal of Applied Probability

Generalized Efron's biased coin design and its theoretical properties

Yanqing Hu

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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

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

First available in Project Euclid: 17 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

More than two treatments unequal allocation drift conditions Markov chain


Hu, Yanqing. Generalized Efron's biased coin design and its theoretical properties. J. Appl. Probab. 53 (2016), no. 2, 327--340.

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