Doubly-adaptive biased coin design, as one of the response-adaptive randomization schemes, is proposed to appeal to ethical concerns by skewing the probability of assigning subjects based on the responses obtained thus far. Over the past decades, substantial progress has been made on the theoretical properties of doubly-adaptive biased coin design. However, most properties are established based on the standard model assumptions of subject responses. Concerns have been raised among clinical trial designers regarding whether model-misspecification can impact on both the design of, and inference in, the general doubly-adaptive biased coin design. In this paper, we exploited covariate-response relationships, and confirmed that the consistency and asymptotic normality under doubly-adaptive biased coin designs can be inherited albeit with model misspecification. Then, we extensively investigate three intuitively and commonly used linear regression models, i.e., the difference-in-means, the analysis of covariance I (ANCOVA I), and the analysis of covariance II (ANCOVA II) model, for estimating and inferring the treatment effect. Furthermore, we derived the consistency and asymptotic normality for the three estimators with respect to treatment effect estimation. The asymptotic properties show that the ANCOVA II model, which takes interaction terms into account, gives the most efficient estimator. These results provide theoretical supports for doubly-adaptive biased coin design with model misspecification and thus put this randomization procedure into wide application.
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