On the achievability of efficiency bounds for covariate-adjusted response-adaptive randomization

In the context of precision medicine, covariate-adjusted response-adaptive (CARA) randomization has garnered much attention from both academia and industry due to its benefits in providing ethical and tailored treatment assignments based on patients' profiles while still preserving favorable statistical properties. Recent years have seen substantial progress in inference for various adaptive experimental designs. In particular, research has focused on two important perspectives: how to obtain robust inference in the presence of model misspecification, and what the smallest variance, i.e., the efficiency bound, an estimator can achieve. Notably, Armstrong (2022) derived the asymptotic efficiency bound for any randomization procedure that assigns treatments depending on covariates and accrued responses, thus including CARA, among others. However, to the best of our knowledge, no existing literature has addressed whether and how this bound can be achieved under CARA. In this paper, by connecting two strands of adaptive randomization literature, namely robust inference and efficiency bound, we provide a definitive answer in an important practical scenario where only discrete covariates are observed and used for stratification. We consider a special type of CARA, i.e., a stratified version of doubly-adaptive biased coin design and prove that the stratified difference-in-means estimator achieves Armstrong (2022)'s efficiency bound, with possible ethical constraints on treatment assignments. Our work provides new insights and demonstrates the potential for more research on CARA designs that maximize efficiency while adhering to ethical considerations. Future studies could explore achieving the asymptotic efficiency bound for CARA with continuous covariates, which remains an open question.