A unified analysis of regression adjustment in randomized experiments

Regression adjustment is broadly applied in randomized trials under the premise that it usually improves the precision of a treatment effect estimator. However, previous work has shown that this is not always true. To further understand this phenomenon, we develop a unified comparison of the asymptotic variance of a class of linear regression-adjusted estimators. Our analysis is based on the classical theory for linear regression with heteroscedastic errors and thus does not assume that the postulated linear model is correct. For a randomized Bernoulli trial, we provide sufficient conditions under which some regression-adjusted estimators are guaranteed to be more asymptotically efficient than others. We comment on the extension of our theory to other settings such as general treatment mechanisms and generalized linear models, and find that the variance dominance phenomenon no longer occurs