A Birth and Death Urn for Randomized Clinical Trials: Asymptotic Methods

Consider the situation in which subjects arrive sequentially in a clinical trial. There are K possibly unrelated treatments. Suppose balls in an urn are labeled with treatments. When a subject arrives, a ball is drawn randomly from the urn, the subject receives the treatment indicated on the ball and the ball is returned to the urn. If the treatment is successful, one ball of the same type is added to the urn. Otherwise, one ball of the same type is taken out from the urn. Under certain assumptions, the urn process can be embedded in a continuous time birth and death process, and associated distributional results can thereby be obtained. Since certain treatments can “die out,” we introduce a Poisson immigration process to replenish the urn. We derive the maximum likelihood estimators of the success probabilities, prove their consistency and obtain their limiting distributions using martingale theory. We then derive inference procedures for the comparison of the K treatments.