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Complete randomization allows for consistent estimation of the averagetreatment effect based on the difference in means of the outcomes withoutstrong modeling assumptions on the outcome-generating process. Appropriate useof the pretreatment covariates can further improve the estimation efficiency.However, missingness in covariates is common in experiments and raises animportant question: should we adjust for covariates subject to missingness, andif so, how? The unadjusted difference in means is always unbiased. Thecomplete-covariate analysis adjusts for all completely observed covariates andimproves the efficiency of the difference in means if at least one completelyobserved covariate is predictive of the outcome. Then what is the additionalgain of adjusting for covariates subject to missingness? A key insight is thatthe missingness indicators act as fully observed pretreatment covariates aslong as missingness is not affected by the treatment, and can thus be used incovariate adjustment to bring additional estimation efficiency. This motivatesadding the missingness indicators to the regression adjustment, yielding themissingness-indicator method as a well-known but not so popular strategy in theliterature of missing data. We recommend it due to its many advantages. We alsopropose modifications to the missingness-indicator method based on asymptoticand finite-sample considerations. To reconcile the conflicting recommendationsin the missing data literature, we analyze and compare various strategies foranalyzing randomized experiments with missing covariates under the design-basedframework. This framework treats randomization as the basis for inference anddoes not impose any modeling assumptions on the outcome-generating process andmissing-data mechanism.

To adjust or not to adjust? Estimating the average treatment effect in randomized experiments with missing covariates