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, are 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 (RCT) has not been explicitly explored. In the recent U.S. Food and Drug Administration and European Medicines Agency guidelines on the practice of covariate adjustment in analyzing RCT, 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. However, when the number of baseline covariates $p$ is large, the recommendation is somewhat murky. In this paper, we show that HOIF-motivated estimators for the treatment-specific mean have significantly improved statistical properties compared to popular adjusted estimators in practice when $p$ is relatively large relative to the sample size $n$. We also characterize the conditions under which the HOIF-motivated estimator improves upon the unadjusted one. More importantly, we demonstrate that several state-of-the-art adjusted estimators proposed recently can be interpreted as particular HOIF-motivated estimators, thereby placing these estimators in a more unified framework. Numerical and empirical studies are conducted to corroborate our theoretical findings. An accompanying R package can be found on CRAN.