The theocratical properties of the power of the conventional testing hypotheses and the selection bias are usually unknown under covariate-adaptive randomized clinical trials. In the literature, most studies are based on simulations. In this article, we provide theoretical foundation of the power of the hypothesis testing and the selection bias under covariate-adaptive randomization based on linear models. We study the asymptotic relative loss of power of hypothesis testing to compare the treatment effects and the asymptotic selection bias. Under the covariate-adaptive randomization, (i) the hypothesis testing usually losses power, the more covariates in testing model are not incorporated in the randomization procedure, the more the power is lost; (ii) the hypothesis testing is usually more powerful than the one under complete randomization; and (iii) comparing to complete randomization, most of the popular covariate-adaptive randomization procedures in the literature, for example, Pocock and Simon's marginal procedure, stratified permuted block design, etc, produce nontrivial selection bias. A new family of covariate-adaptive randomization procedures are proposed for considering the power and selection bias simultaneously, under which, the covariate imbalances are small enough so that the power of testing the treatment effects would be asymptotically the largest and at the same time, the selection bias is asymptotically the optimal. The theocratical properties give a full picture how the power of the hypothesis testing, the selection bias of the randomization procedure, and the randomization method affect each other.