Doubly adaptive biased coin designs with delayed responses

In clinical studies, patients are usually accrued sequentially. Response- adaptive designs are then useful tools for assigning treatments to incoming patients as a function of the treatment responses observed thus far. In this regard, doubly adaptive biased coin designs have advantageous properties under the assump- tion that their responses can be obtained immediately after testing. However, it is a common occurrence that responses are observed only after a certain period of time. The authors examine the effect of delayed responses on doubly adaptive biased coin designs and derive some of their asymptotic properties. It turns out that these designs are relatively insensitive to delayed responses u nder widely satisfied conditions. This is illustrated with a simulation study. In most clinical trials, patients are accrued sequentially . Response-adaptive designs are valu- able and ethical randomization schemes that formulate treatment allocation as a function of the previous responses of patients. A major intention of such designs is to skew the probability of treatment allocation to increase the chance that more patients receive better treatment. The early important research work on adaptive designs can be traced back to Thompson (1933) and Robbins (1952). In the literature, response-adaptive designs can be catego rized into two classes. The first is the target-driven response-adaptive design, which is constructed using an optimal (or desired) allocation target in which a specific criterion is optimized based on a population response model. The second class is the design-driven response-adaptive design. The allocation rule for a member of this class is typically established with an intuitive mot ivation, but not with an optimal crite- rion in the formal sense (Rosenberger & Lachin 2002). A well-studied family of design-driven response-adaptive designs is based on urn models. The early influential developments in this area include the play-the-winner rule (for the comparison of two treatments with binary responses) of Zelen (1969) and the randomized play-the-winner rule of Wei & Durham (1978) and Wei (1979). An extensive review of the properties of urn models is provided in Rosenberger (2002), and some of the general asymptotic properties of urn models can be found in Janson (2004) and Bai & Hu (1999, 2005).

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