Resumen
We derive a lower bound on the asymptotic variance of the allocation proportions from response-adaptive randomization procedures when the allocation proportions are asymptotically normal. A procedure that attains this lower bound is defined to be asymptotically best. We then compare the asymptotic variances of five procedures, for which allocation proportions converge, to the lower bound. We find that a procedure by Zelen and a procedure by Ivanova attain the lower bound and a procedure by Eisele and its extension to K>2 treatments can attain the lower bound but are, in general, not asymptotically best. We discuss the tradeoffs among the benefits of randomization, the benefits of attaining the lower bound, and the benefits of targeting an optimal allocation. We conclude that none of these procedures possesses all of these benefits.

Resumen
Doubly adaptive biased coin design (DBCD) is an important family of response-adaptive randomization procedures for clinical trials. It uses sequentially updated estimation to skew the allocation probability to favor the treatment that has performed better thus far. An important assumption for the DBCD is the homogeneity assumption for the patient responses. However, this assumption may be violated in many sequential experiments. Here we prove the robustness of the DBCD against certain time trends in patient responses. Strong consistency and asymptotic normality of the design are obtained under some widely satisfied conditions. Also, we propose a general weighted likelihood method to reduce the bias caused by the heterogeneity in the inference after a trial. Some numerical studies are also presented to illustrate the finite sample properties of DBCD.