Our paper discovers a new trade-off of using regression adjustments (RAs) in causal inference under covariate-adaptive randomizations (CARs). On one hand, RAs can improve the efficiency of causal estimators by incorporating information from covariates that are not used in the randomization. On the other hand, RAs can degrade estimation efficiency due to their estimation errors, which are not asymptotically negligible when the number of regressors is of the same order as the sample size. Ignoring the estimation errors of RAs may result in serious over-rejection of causal inference under the null hypothesis. To address the issue, we construct a new ATE estimator by optimally linearly combining the estimators with and without RAs. We then develop a unified inference theory for this estimator under CARs. It has two features: (1) the Wald test based on it achieves the exact asymptotic size under the null hypothesis, regardless of whether the number of covariates is fixed or diverges no faster than the sample size; and (2) it guarantees weak efficiency improvement over estimators both with and without RAs.

翻译：本文揭示了在协变量自适应随机化（CARs）框架下，使用回归调整（RAs）进行因果推断时存在的新权衡。一方面，RAs 能够通过纳入随机化过程中未使用的协变量信息来提高因果估计量的效率；另一方面，由于估计误差的存在，RAs 也可能降低估计效率——当回归元数量与样本量同阶时，这些估计误差在渐近意义上不可忽略。忽略 RAs 的估计误差可能导致在零假设下因果推断出现严重的过度拒绝问题。为解决该问题，我们通过最优线性组合使用与不使用 RAs 的估计量，构建了一种新的平均处理效应（ATE）估计量。随后，我们在 CARs 下为该估计量建立了统一的推断理论。该理论具有两个特征：（1）基于该估计量的 Wald 检验在零假设下能达到精确的渐近水平，无论协变量数量是固定的还是以不超过样本量的速度发散；（2）该估计量能保证相对于使用与不使用 RAs 的估计量均实现弱效率提升。