## Journal of Applied Probability

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

### Generalized Efron's biased coin design and its theoretical properties

#### 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