The Annals of Statistics

Asymptotic properties of covariate-adaptive randomization

Yanqing Hu and Feifang Hu

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Balancing treatment allocation for influential covariates is critical in clinical trials. This has become increasingly important as more and more biomarkers are found to be associated with different diseases in translational research (genomics, proteomics and metabolomics). Stratified permuted block randomization and minimization methods [Pocock and Simon Biometrics 31 (1975) 103–115, etc.] are the two most popular approaches in practice. However, stratified permuted block randomization fails to achieve good overall balance when the number of strata is large, whereas traditional minimization methods also suffer from the potential drawback of large within-stratum imbalances. Moreover, the theoretical bases of minimization methods remain largely elusive. In this paper, we propose a new covariate-adaptive design that is able to control various types of imbalances. We show that the joint process of within-stratum imbalances is a positive recurrent Markov chain under certain conditions. Therefore, this new procedure yields more balanced allocation. The advantages of the proposed procedure are also demonstrated by extensive simulation studies. Our work provides a theoretical tool for future research in this area.

Article information

Ann. Statist., Volume 40, Number 3 (2012), 1794-1815.

First available in Project Euclid: 16 October 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems 62G10: Hypothesis testing
Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Balancing covariates clinical trial marginal balance Markov chain Pocock and Simon’s design stratified permuted block


Hu, Yanqing; Hu, Feifang. Asymptotic properties of covariate-adaptive randomization. Ann. Statist. 40 (2012), no. 3, 1794--1815. doi:10.1214/12-AOS983.

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Supplemental materials

  • Supplementary material: Additional proofs. We provide additional proofs that are omitted in Section 6. They include: (1) derivation of $\Delta V(\mathbf{D})$; (2) the appropriate choice of $d_i$ when $M(\mathbf{D},\mathbf{\delta})=i$ ($i=4,3,2,1$); (3) proof of Corollary 3.1.