Stopping Rules in Balanced Allocation Problems: Exact and Asymptotic Distributions

摘要

The properties of balanced randomization procedures for several treatments under a general (nonuniform) allocation are reviewed. Under one of such schemes, the truncated multinomial randomization design, the initial assignment probabilities are employed until a treatment fulfills its quota of subjects, after which these probabilities become these of the conditional distribution for the remaining available treatments, and so on. Another design, the random allocation rule, chooses at random one of possible balanced assignments of the given number of subjects per treatment. These two schemes turn out to be quite different: the target fulfillment instants for the truncated multinomial randomization design occur much earlier than for the random allocation rule. Under the truncated multinomial randomization design, the limiting behavior of these instants is determined by normal order statistics with different variances; for the random allocation rule this role is played by the sums of independent but heterogeneous geometric random variables. The formulas for the selection bias and the accidental bias of both procedures are given.