Li-Xin Zhang

Asymptotic properties of covariate-adjusted response-adaptive designs

Response-adaptive designs have been extensively studied and used in clinical trials. However, there is a lack of a comprehensive study of response-adaptive designs that include covariates, despite their importance in clinical trials. Because the allocation scheme and the estimation of parameters are affected by both the responses and the covariates, covariate-adjusted response-adaptive (CARA) designs are very complex to formulate. In this paper, we overcome the technical hurdles and lay out a framework for general CARA designs for the allocation of subjects to K (≥ 2) treatments. The asymptotic properties are studied under certain widely satisfied conditions. The proposed CARA designs can be applied to generalized linear models. Two important special cases, the linear model and the logistic regression model, are considered in detail.

EFFICIENT RANDOMIZED-ADAPTIVE DESIGNS

Response-adaptive randomization has recently attracted a lot of attention in the literature. In this paper, we propose a new and simple family of response-adaptive randomization procedures that attain the Cramer-Rao lower bounds on the allocation variances for any allocation proportions, including optimal allocation proportions. The allocation probability functions of proposed procedures are discontinuous. The existing large sample theory for adaptive designs relies on Taylor expansions of the allocation probability functions, which do not apply to nondifferentiable cases. In the present paper, we study stopping times of stochastic processes to establish the asymptotic efficiency results. Furthermore, we demonstrate our proposal through examples, simulations and a discussion on the relationship with earlier works, including Efrons biased coin design.

A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Fields

AbstractLet Xk; k ∈ Nd be a random field which is asymptotically negative dependent in a certain sense. Define the partial sum process in the usual way so that

A weak convergence for negatively associated fields

W_n \left( t \right) = \sigma _n^{{\text{ - 1}}} \sum\nolimits_{m \leqq n \cdot t} {\left( {X_m - EX_m } \right)} \quad {\text{for}}\quad t \in \left[ {0,1} \right]^d

Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications

, where

Donsker’s Invariance Principle Under the Sub-linear Expectation with an Application to Chung’s Law of the Iterated Logarithm

\sigma _n^{\text{2}} = {\text{Var}}\left( {\sum\nolimits_{m \leqq n} {X_m } } \right)