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The theocratical properties of the power of the conventional testinghypotheses and the selection bias are usually unknown under covariate-adaptiverandomized clinical trials. In the literature, most studies are based onsimulations. In this article, we provide theoretical foundation of the power ofthe hypothesis testing and the selection bias under covariate-adaptiverandomization based on linear models. We study the asymptotic relative loss ofpower of hypothesis testing to compare the treatment effects and the asymptoticselection bias. Under the covariate-adaptive randomization, (i) the hypothesistesting usually losses power, the more covariates in testing model are notincorporated in the randomization procedure, the more the power is lost; (ii)the hypothesis testing is usually more powerful than the one under completerandomization; and (iii) comparing to complete randomization, most of thepopular covariate-adaptive randomization procedures in the literature, forexample, Pocock and Simon's marginal procedure, stratified permuted blockdesign, etc, produce nontrivial selection bias. A new family ofcovariate-adaptive randomization procedures are proposed for considering thepower and selection bias simultaneously, under which, the covariate imbalancesare small enough so that the power of testing the treatment effects would beasymptotically the largest and at the same time, the selection bias isasymptotically the optimal. The theocratical properties give a full picture howthe power of the hypothesis testing, the selection bias of the randomizationprocedure, and the randomization method affect each other.

Theory on Covariate-Adaptive Randomized Clinical Trials: Efficiency, Selection bias and Randomization Methods