  In randomized controlled trials, adjusting for baseline covariates is often
applied to improve the precision of treatment effect estimation. However,
missingness in covariates is common. Recently, Zhao & Ding (2022) studied two
simple strategies, the single imputation method and missingness indicator
method (MIM), to deal with missing covariates, and showed that both methods can
provide efficiency gain. To better understand and compare these two strategies,
we propose and investigate a novel imputation framework termed cross-world
imputation (CWI), which includes single imputation and MIM as special cases.
Through the lens of CWI, we show that MIM implicitly searches for the optimal
CWI values and thus achieves optimal efficiency. We also derive conditions
under which the single imputation method, by searching for the optimal single
imputation values, can achieve the same efficiency as the MIM.
