Power efficiency of Efron’s biased coin design

Efron’s biased coin design aims to both balance the experiment and preserve randomness. It has been noticed that, under the homoscedastic normal model, Efron’s design is uniformly more powerful than a perfect simple randomization. However, this optimality property does not hold for heteroscedastic models. For the latter, it is shown in this work that Efron’s biased coin provides more power than a perfect simple randomization for a large enough sample size. This is proved by studying the exponential rate at which the power converges to one, under the different designs, using large deviations theory. Specifically, we prove this power efficiency property for binary and normal responses, when the variances of the two treatments are different, and the probability of heads for the biased coin is equal to or greater than 2/3. A numerical study indicates that the power is larger even for small-sized experiments and the improvement can reach up to 4%.