Forcing a Sequential Experiment to be Balanced

The object is to reduce imbalances in small-sized experiments, involving treatments and controls that arrive sequentially, while avoiding various forms of experimental bias characteristic of perfectly balanced plans. Complete randomization has the advantages of freedom from selection bias and from accidental bias due to the presence of nuisance factors. It also provides an inferential basis for calculating appropriate P-values. But in small samples it can lead to serious imbalances, degrading the power of the test. Efron offers a relatively simple alternative plan which gives the experimenter a slight ability for guessing which treatment will be next. This plan, biased coin design or BCD(p), consists of selecting the next subject to be treatment or control with probability p > 1/2 if there is an excess of subjects in the other group. If the trial is well balanced at the time, the next subject is chosen to be treatment with probability 0.5. Efron indicates a preference for p = 2/3 where r = p/(1 – p) = 2. The experimenter knows which choice is more likely to be next, but his knowledge is limited by chance.

The tendency for lack of balance using BCD(2/3) is much reduced from that of complete randomization, but the plan suffers from some tendencies to bias. These are evaluated and compared with those of Randomized Block Design, RB(b), which consists of complete randomization within a block of 2b subjects of which b must be selected to be treatments.