Conditioning Efron’s biased coin design to ensure final balance

Abstract

This paper derives a new randomization procedure by conditioning Efron’s (1971) biased coin design to a prespecified final balance. The new procedure remains a function of the original bias parameter which now controls the probability of intermediate balance rather than final balance. As the sample size increases, the design’s selection bias and intermediate balance are similar to those of the original biased coin, but unlike the original biased coin it always guarantees final balance. It is also shown that the permuted block design for equal allocation is a special case of the new procedure when used in blocks. The latter can substitute the permuted blocks with the added benefit of reducing the expected number of deterministic assignments. The new design is also noteworthy since it shows that a randomization procedure with new properties can be obtained by conditioning an existing one to a subset in its allocation space. New relationships among existing designs can be established in the process, further elucidating the protean nature of randomization.