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 develop a unified inference theory for the regression-adjusted average treatment effect (ATE) estimator under CARs. Our theory has two key features: (1) it ensures 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 the ATE estimator without adjustments.

翻译：本文发现了一种新的权衡：在协变量自适应随机化（CARs）下的因果推断中使用回归调整（RAs）。一方面，RAs通过纳入未用于随机化的协变量信息，能够提高因果估计量的效率。另一方面，由于RAs的估计误差（当回归元数量与样本量同阶时，这些误差并非渐近可忽略），它们可能降低估计效率。忽略RAs的估计误差可能导致原假设下因果推断的严重过度拒绝。为解决此问题，我们为CARs下经回归调整的平均处理效应（ATE）估计量发展了一套统一的推断理论。该理论具有两个关键特征：（1）无论协变量数量固定还是以不超过样本量的速度发散，它都能确保原假设下的精确渐近尺寸；（2）它保证了相较于未经调整的ATE估计量具有弱效率改进。