Generalized Delayed Response in Randomized Play-the-Winner Rule

Abstract

In clinical trials, a particular patient's response may be delayed in the sense that it is not obtained before the entrance of the next patient. Delayed responses are particularly important in the context of adaptive trials where the allocations are done on the basis of the available responses. Bandyopadhyay and Biswas (Bandyopadhyay, U., Biswas, A. (1996). Delayed response in randomized play-the-winner rule: a decision theoretic outlook. Calcutta Statistical Association Bulletin 46:69–88) and Biswas (Biswas, A. (1999). Delayed response in randomized play-the-winner rule revisited. Comm. Statist. Simul. Comput. 28:715–731) have studied a simple delayed response model in the context of the randomized play-the-winner rule (RPW), a popular adaptive design, originally developed by Wei and Durham (Wei, L. J., Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Am. Statist. Assoc. 73:838–843) and Wei (Wei, L. J. (1979). The generalized Polya's urn for sequential medical trials. Ann. Statist. 7:291–296). The present article provides a generalized model for the delayed response in the RPW rule. The model is then theoretically developed and the exact and limiting proportion of patients allocated to the two treatments are obtained. The exact expression of the variance of the proportion of allocation is also obtained.