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 $n^{-1}$.