A General (Non-Markovian) Framework for Covariate Adaptive Randomization: Achieving Balance While Eliminating the Shift

TL;DR

This paper introduces a flexible, non-Markovian covariate adaptive randomization framework that ensures covariate balance and eliminates the shift problem, addressing limitations of existing methods in unequal allocation scenarios.

Contribution

It develops a new allocation function and a non-Markovian procedure that adaptively balances covariates while removing the shift issue in covariate adaptive randomization.

Findings

The proposed procedure achieves bounded covariate imbalance.

It guarantees asymptotic covariate balance for additional covariates.

The framework is theoretically validated with key properties established.

Abstract

Emerging applications increasingly demand flexible covariate adaptive randomization (CAR) methods that support unequal targeted allocation ratios. While existing procedures can achieve covariate balance, they often suffer from the shift problem. This occurs when the allocation ratios of some additional covariates deviate from the target. We show that this problem is equivalent to a mismatch between the conditional average allocation ratio and the target among units sharing specific covariate values, revealing a failure of existing procedures in the long run. To address it, we derive a new form of allocation function by requiring that balancing covariates ensures the ratio matches the target. Based on this form, we design a class of parameterized allocation functions. When the parameter roughly matches certain characteristics of the covariate distribution, the resulting procedure can balance covariates. Thus, we propose a feasible randomization procedure that updates the parameter based on collected covariate information, rendering the procedure non-Markovian. To accommodate this, we introduce a CAR framework that allows non-Markovian procedure. We then establish its key theoretical properties, including the boundedness of covariate imbalance in probability and the asymptotic distribution of the imbalance for additional covariates. Ultimately, we conclude that the feasible randomization procedure can achieve covariate balance and eliminate the shift.