ABSTRACT
Department of Mathematics, Zhejiang University and Zhejiang University City College
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 10.3 Urn Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
10.3.1 Generalized Po´lya Urn (GPU) . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.3.2 Drop-the-Loser Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 10.3.3 Generalized Drop-the-Loser and Immigrated Urn Model 229 10.3.4 Randomly Reinforced Urn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.4 Optimal and Efficient RAR Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 10.4.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 10.4.2 Target-Driven Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 10.4.3 Variability and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 10.4.4 Efficient RAR Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10.5 Selection Bias and Lack of Randomness . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.6 Survival and Delayed Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 10.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Response-adaptive treatment allocation procedures are sequentially adaptive schemes that use past treatment assignments and patients’ responses to select future treatment assignments. Historically, response-adaptive treatment allocation procedures were developed for the purpose of assigning more patients to
the empirically better treatment. Early important work on response-adaptive designs can be traced back to Thompson [37] and Robbins [26]. Since then, many response-adaptive designs have been proposed in the literature [32]. A history of the subject is discussed in Rosenberger and Lachin [29] and Hu and Rosenberger [11]. The most famous non-randomized response-adaptive treatment allocation procedure is the play-the-winner (PW) rule proposed by Zelen [42], in which a success on a treatment results in the next patient’s assignment to the same treatment, and a failure on the treatment results in the next patient’s assignment to the opposite treatment. Wei and Durham [40] introduced the randomized play-the-winner (RPW) rule for which a patient’s treatment assignment is determined by randomly drawing a ball from the urn and the urn composition is updated based on the outcomes from the previous patients such that the balls corresponding to the more successful treatment are selected more frequently. The RPW rule was used occasionally in practice [27]. In particular, it was applied in a pediatric trial of extracorporeal membrane oxygenation (ECMO; Bartlett et al. [4]), which compared the ECMO therapy versus the conventional therapy. Unfortunately, the ECMO trial did not provide conclusive results (the trial stopped after enrolling 12 infants, of whom one infant was randomized to the conventional therapy and died and 11 infants were randomized to ECMO treatment and all survived). According to the modern theory on response-adaptive randomization designs, the failure of the ECMO trial can be explained mainly by the trial’s small sample size and the poor operating characteristics of the RPW rule, in particular, the rule’s high variability and dependence on the initial composition of the balls in the urn. In the past two decades, research on response-adaptive randomization has advanced substantially-many new methods have been developed and statistical properties of existing methods have been established.
