In the context of precision medicine, covariate-adjusted response-adaptive randomization (CARA) 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 understanding the 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 the asymptotic efficiency bound can be achieved under CARA. In this paper, by connecting two strands of literature on adaptive randomization, namely robust inference and efficiency bound, we provide a definitive answer to this question for an important practical scenario where only discrete covariates are observed and used to form stratification. We consider a specific 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 regarding the design and analysis of CARA that maximizes efficiency while adhering to ethical considerations. Future studies could explore how to achieve the asymptotic efficiency bound for general CARA with continuous covariates, which remains an open question.

翻译：在精准医疗背景下，协变量调整响应自适应随机化（CARA）因其能够根据患者特征提供符合伦理且个性化的治疗分配，同时保持优良的统计特性，已受到学术界与工业界的广泛关注。近年来，针对各类自适应实验设计的统计推断研究取得了显著进展。特别值得关注的是两个重要研究方向：如何在模型设定错误情况下获得稳健推断，以及估计量所能达到的最小方差（即效率界）为何。值得注意的是，Armstrong（2022）推导了任何依赖协变量与累积响应进行治疗分配的随机化程序（包括CARA在内）的渐近效率界。然而，据我们所知，现有文献尚未探讨在CARA框架下该渐近效率界是否可达及如何实现的问题。本文通过整合自适应随机化领域中稳健推断与效率界这两条研究脉络，针对仅观测离散协变量并依此进行分层的重要实际场景，对该问题给出了明确解答。我们研究一类特定的CARA设计——分层式双自适应偏币设计，并证明在治疗分配可能存在伦理约束的条件下，分层均值差分估计量能够达到Armstrong（2022）所提出的效率界。本研究为CARA的设计与分析提供了新见解，展示了在遵循伦理考量的同时实现效率最大化的研究潜力。未来研究可探索具有连续协变量的广义CARA如何达到渐近效率界，这仍是一个待解决的重要问题。