Asymptotic Properties of Multi-Treatment Covariate Adaptive Randomization Procedures for Balancing Observed and Unobserved Covariates

TL;DR

This paper develops a comprehensive framework for multi-treatment covariate adaptive randomization (CAR) in clinical trials, demonstrating superior balancing of covariates and unobserved factors, which enhances treatment effect testing accuracy.

Contribution

It introduces a general, unifying framework for multi-treatment CAR procedures, establishing their asymptotic properties and broadening the class of effective balancing methods.

Findings

Achieves $O_P(1)$ convergence rate for covariate imbalance with discrete and continuous covariates.

Attains $O_P( ext{sqrt } n)$ convergence rate for unobserved covariate imbalance.

Provides consistent treatment effect tests with asymptotic correct type I error even with unobserved covariates.

Abstract

Applications of CAR for balancing continuous covariates remain comparatively rare, especially in multi-treatment clinical trials, and the theoretical properties of multi-treatment CAR have remained largely elusive for decades. In this paper, we consider a general framework of CAR procedures for multi-treatment clinal trials which can balance general covariate features, such as quadratic and interaction terms which can be discrete, continuous, and mixing. We show that under widely satisfied conditions the proposed procedures have superior balancing properties; in particular, the convergence rate of imbalance vectors can attain the best rate $O_P(1)$ for discrete covariates, continuous covariates, or combinations of both discrete and continuous covariates, and at the same time, the convergence rate of the imbalance of unobserved covariates is $O_P(\sqrt n)$, where $n$ is the sample size. The general framework unifies many existing methods and related theories, introduces a much broader class of new and useful CAR procedures, and provides new insights and a complete picture of the properties of CAR procedures. The favorable balancing properties lead to the precision of the treatment effect test in the presence of a heteroscedastic linear model with dependent covariate features. As an application, the properties of the test of treatment effect with unobserved covariates are studied under the CAR procedures, and consistent tests are proposed so that the test has an asymptotic precise type I error even if the working model is wrong and covariates are unobserved in the analysis.