  We often seek to estimate the causal effect of an exposure on a particular
outcome in both randomized and observational settings. One such estimation
method is the covariate-adjusted residuals estimator, which was designed for
individually or cluster randomized trials. In this manuscript, we study the
properties of this estimator and develop a new estimator that utilizes both
covariate adjustment and inverse probability weighting We support our
theoretical results with a simulation study and an application in an infectious
disease setting. The covariate-adjusted residuals estimator is an efficient and
unbiased estimator of the average treatment effect in randomized trials;
however, it is not guaranteed to be unbiased in observational studies. Our
novel estimator, the covariate-adjusted residuals estimator with inverse
probability weighting, is unbiased in randomized and observational settings,
under a reasonable set of assumptions. Furthermore, when these assumptions
hold, it provides efficiency gains over inverse probability weighting in
observational studies. The covariate-adjusted residuals estimator is valid for
use in randomized trials, but should not be used in observational studies. The
covariate-adjusted residuals estimator with inverse probability weighting
provides an efficient alternative for use in randomized and observational
settings.
