Covariate adjustment in randomized experiments with missing outcomes and covariates

Covariate adjustment can improve precision in estimating treatment effects from randomized experiments. With fully observed data, regression adjustment and propensity score weighting are two asymptotically equivalent methods for covariate adjustment in randomized experiments. We show that this equivalence breaks down in the presence of missing outcomes, with regression adjustment no longer ensuring efficiency gain when the true outcome model is not linear in covariates. Propensity score weighting, in contrast, still guarantees efficiency over unadjusted analysis, and including more covariates in adjustment never harms asymptotic efficiency. Moreover, we establish the value of using partially observed covariates to secure additional efficiency. Based on these findings, we recommend a simple double-weighted estimator for covariate adjustment with incomplete outcomes and covariates: (i) impute all missing covariates by zero, and use the union of the completed covariates and corresponding missingness indicators to estimate the probability of treatment and the probability of having observed outcome for all units; (ii) estimate the average treatment effect by the coefficient of the treatment from the least-squares regression of the observed outcome on the treatment, where we weight each unit by the inverse of the product of these two estimated probabilities.