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.

翻译：连续协变量平衡的协变量自适应随机化（CAR）在临床试验中的应用仍相对少见，尤其在多处理临床试验中，数十年来多处理CAR的理论性质在很大程度上一直悬而未决。本文考虑了一类用于多处理临床试验的CAR程序通用框架，该框架能够平衡广义协变量特征（如离散、连续及混合类型的二次项与交互项）。我们证明，在广泛满足的条件下，所提出的程序具有优越的平衡性质：特别地，对于离散协变量、连续协变量或两者混合的情形，不平衡向量的收敛速率可达最优阶$O_P(1)$，同时未观测协变量的不平衡收敛速率为$O_P(\sqrt n)$，其中$n$为样本量。该通用框架统一了现有多种方法及理论，引入了一类更广泛的新型实用CAR程序，并为CAR程序的性质提供了全新见解与完整图景。在异方差线性模型且协变量特征相依的情况下，这种优越的平衡性质可提升处理效应检验的精度。作为应用，我们在CAR程序下研究了含未观测协变量的处理效应检验性质，并提出了一致性检验方法，使即使工作模型错误且分析中未观测到协变量时，检验也能渐近保持精确的第一类错误率。