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 adjusted and unadjusted estimators. 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 both the adjusted and unadjusted estimators.

翻译：本文在协变量自适应随机化（CARs）下的因果推断中，发现了使用回归调整（RAs）的新权衡。一方面，RAs通过引入随机化中未使用的协变量信息，可以提高因果估计量的效率。另一方面，RAs的估计误差可能降低估计效率——当回归变量数量与样本量同阶时，该误差在渐近意义上不可忽略。忽略RAs的估计误差会导致原假设下因果推断的严重过度拒绝。为解决该问题，我们通过最优线性组合调整估计量与未调整估计量，构建了一个新的平均处理效应估计量。随后，我们建立了该估计量在CARs下的统一推断理论，具有以下两个特征：（1）无论协变量数量固定还是发散至不超过样本量，基于该估计量的Wald检验在原假设下均能达到精确渐近大小；（2）它保证在弱效率上优于调整估计量与未调整估计量。