Balancing Unobserved Covariates With Covariate-Adaptive Randomized Experiments

Abstract

Balancing important covariates is often critical in clinical trials and causal inference. Stratified permuted block (STR-PB) and covariate-adaptive randomization (CAR) procedures are widely used to balance observed covariates in practice. The balance properties of these procedures with respect to the observed covariates have been well studied. However, it has been questioned whether these methods will also yield a good balance for the unobserved covariates. In this article, we develop a general framework for the analysis of the unobserved covariates imbalance. These results are applicable to develop and compare the balance properties of complete randomization (CR), STR-PB, and CAR procedures with respect to the unobserved covariates. To quantify the improvement obtained by using STR-PB and CAR procedures rather than CR, we introduce the percentage reduction in variance of the unobserved covariates imbalance and compare these quantities. Our results demonstrate the benefits of using CAR or STR-PB (when the number of strata is small relative to the sample size) in terms of balancing unobserved covariates. These results also pave the way for future research into the effect of unobserved covariates in covariate-adaptive randomized experiments in clinical trials, as well as many other applications. Supplementary materials for this article are available online.

KEYWORDS:

• Balancing covariates
• Correlation
• Hu and Hu’s procedure
• Pocock and Simon’s procedure
• Stratified permuted block design
• Unobserved covariates imbalance

Supplementary Materials

The supplementary materials contain a summary of the balance properties of CR, STR-PB, and CAR procedures, extra numerical studies for $D_n(\mathbf{s},\mathbf{r})$, and $D_n(\mathbf{r})$, additional details of the study in Elkashef et al. (2006), and proofs of the main theorems.

Acknowledgments

The authors thank the associate editor and two reviewers for helpful comments.

Additional information

Funding

This research project was supported by National Science Foundation grant DMS-1612970.