Consistent covariances estimation for stratum imbalances under marginal design for covariate adaptive randomization

Marginal design, also called as the minimization method, is a popular approach used for covariate adaptive randomization when the number of strata is large in a clinical trial. It aims to achieve a balanced allocation of treatment groups at the marginal levels of some prespecified and discrete stratification factors. Valid statistical inference with data collected under covariate adaptive randomization requires the knowledge of the limiting covariance matrix of within-stratum imbalances. The existence of the limit under the marginal design is recently established, which can be estimated by Monte Carlo simulations when the distribution of the stratification factors is known. This assumption may not hold, however, in practice. In this work, we propose to replace the usually unknown distribution with an estimator, such as the empirical distribution, in the Monte Carlo approach and establish its consistency, in particular, by Le Cam's third lemma. As an application, we consider in simulation studies adjustments to existing robust tests for treatment effects with survival data by the proposed covariances estimator. It shows that the adjusted tests achieve a size close to the nominal level, and unlike other designs, the robust tests without adjustment may have an asymptotic size inflation issue under the marginal design.