Estimation accuracy under covariate-adaptive randomization procedures

In this paper we provide some general asymptotic properties of covariate-adaptive (CA) randomized designs aimed at balancing the allocations of two treatments across a set of chosen covariates. In particular, we establish the central limit theorem for a vast class of covariate-adaptive procedures characterized by i) a different allocation function for each covariate profile and ii) sequences of allocation rules instead of a pre-fixed one. This result allows one to derive theoretically the asymptotic expressions of the loss of information induced by imbalance and the selection bias due to the lack of randomness, that are the fundamental properties for estimation of every CA rule, widely used in order to compare different CA procedures. Besides providing the proofs of unsolved conjectures about some CA designs suggested in the literature, explored up to now almost exclusively through simulations, our results provide substantial insight for future suggestions and represent an accurate tool for the large sample comparisons between CA designs. A numerical study is also performed to assess the validity of the suggested approach.