The use of random allocation for the control of selection bias

Abstract

SUMMARY In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. In such situations, if the experiment calls for fixed treatment numbers, the experimenter can, using his knowledge of the number of treatments that have been assigned, bias the experiment by his selection of subjects. If we consider the method of assigning treatments as an experimental design, Blackwell & Hodges (1957) have shown that the minimax design is the truncated binomial. In this paper we show that random allocation is a restricted Bayes design within the class of Markov designs, and is in many senses preferable to the minimax design. In particular, it is possible for the random allocation design effectively to eliminate the bias asymptotically when the minimax design does not, and in no case will random allocation have a much worse performance than the minimax. In the problem of comparing the effectiveness of two treatments, it is common for the statistician to have the experimenter select 2n subjects suitable for treatment and treat n subjects with each treatment. Clearly, if the experimenter is aware of or guesses which treatment a subject will receive before he selects the subject, then he can, consciously or unconsciously, bias the experiment by his choice; therefore the assignment of treatments to subjects is usually done 'randomly'. Ideally, the 2n subjects will be selected in advance, and the treatment assignments made by random sampling without replacement. This is, however, not always possible. In many cases, suitable subjects arrive sequentially and must be treated immediately or not at all. Two examples which have been cited are clinical trials and cloud seeding experiments. In clinical trials a patient often must be treated as soon as the disease has been diagnosed; in cloud seeding it is physically impossible to collect the subjects (storm clouds) for simultaneous assignment of treatments. In some experiments it may be possible to eliminate this bias by having the selection of suitable subjects done by a third party who is ignorant of the past assignment of treatments, or by defining 'suitability' in a sufficiently objective manner to permit selection without an exercise of judgment. There remain many cases in which the best or only judges available are those involved in administering the treatments. For such situations it is reasonable to ask what strategy the statistician should adopt in the assignment of treatments in order to reduce the expected bias as much as possible. This problem was first considered by Blackwell & Hodges (1957), who have proposed the following model. In order to compare two treatments it is decided to treat 2n subjects, n with treatment A and n with treatment B. Candidates for treatment arrive sequentially, and as each arrives, the experimenter E decides whether or not it is a suitable subject. If it is