Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence Functions

Higher-Order Influence Functions (HOIF), developed in a series of papers over the past twenty years, is a fundamental theoretical device for constructing rate-optimal causal-effect estimators from observational studies. However, the value of HOIF for analyzing well-conducted randomized controlled trials (RCTs) has not been explicitly explored. In the recent U.S. Food and Drug Administration (FDA) and European Medicines Agency (EMA) guidelines on the practice of covariate adjustment in analyzing RCTs, in addition to the simple, unadjusted difference-in-mean estimator, it was also recommended to report the estimator adjusting for baseline covariates via a simple parametric working model, such as a linear model. In this paper, we show that a HOIF-motivated estimator for the treatment-specific mean has significantly improved statistical properties compared to popular adjusted estimators in practice when the number of baseline covariates $p$ is relatively large compared to the sample size $n$. We also characterize the conditions under which the HOIF-motivated estimator improves upon the unadjusted one. Furthermore, we demonstrate that a novel debiased adjusted estimator proposed recently by Lu et al. is, in fact, another HOIF-motivated estimator in disguise. Numerical and empirical studies are conducted to corroborate our theoretical findings.