  Covariate adjustment can improve precision in analyzing randomized
experiments. With fully observed data, regression adjustment and propensity
score weighting are asymptotically equivalent in improving efficiency over
unadjusted analysis. When some outcomes are missing, we consider combining
these two adjustment methods with inverse probability of observation weighting
for handling missing outcomes, and show that the equivalence between the two
methods breaks down. Regression adjustment no longer ensures efficiency gain
over unadjusted analysis unless the true outcome model is linear in covariates
or the outcomes are missing completely at random. 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 by the missingness indicator method, which imputes
all missing covariates by zero and uses the union of the completed covariates
and corresponding missingness indicators as the new, fully observed covariates.
Based on these findings, we recommend using regression adjustment in
combination with the missingness indicator method if the linear outcome model
or missing complete at random assumption is plausible and using propensity
score weighting with the missingness indicator method otherwise.
