Covariate-adaptive designs are widely used to balance covariates and maintain randomization in clinical trials. Adaptive designs for discrete covariates and their asymptotic properties have been well studied in the literature. However, important continuous covariates are often involved in clinical studies. Simply discretizing or categorizing continuous covariates can result in loss of information. The current understanding of adaptive designs with continuous covariates lacks a theoretical foundation as the existing works are entirely based on simulations. Consequently, conventional hypothesis testing in clinical trials using continuous covariates is still not well understood. In this paper, we establish a theoretical framework for hypothesis testing on adaptive designs with continuous covariates based on linear models. For testing treatment effects and significance of covariates, we obtain the asymptotic distributions of the test statistic under null and alternative hypotheses. Simulation studies are conducted under a class of covariate-adaptive designs, including the p-value-based method, the Su's percentile method, the empirical cumulative-distribution method, the Kullback-Leibler divergence method, and the kernel-density method. Key findings about adaptive designs with independent covariates based on linear models are (1) hypothesis testing that compares treatment effects are conservative in terms of smaller type I error, (2) hypothesis testing using adaptive designs outperforms complete randomization method in terms of power, and (3) testing on significance of covariates is still valid.
Keywords: Covariate-adaptive designs; conservative test; continuous covariates; linear models; power; type I error.