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In epidemiology and social sciences, propensity score methods are popular forestimating treatment effects using observational data, and multiple imputationis popular for handling covariate missingness. However, how to appropriatelyuse multiple imputation for propensity score analysis is not completely clear.This paper aims to bring clarity on the consistency (or lack thereof) ofmethods that have been proposed, focusing on the within approach (where theeffect is estimated separately in each imputed dataset and then the multipleestimates are combined) and the across approach (where typically propensityscores are averaged across imputed datasets before being used for effectestimation). We show that the within method is valid and can be used with anycausal effect estimator that is consistent in the full-data setting. Existingacross methods are inconsistent, but a different across method that averagesthe inverse probability weights across imputed datasets is consistent forpropensity score weighting. We also comment on methods that rely on imputing afunction of the missing covariate rather than the covariate itself, includingimputation of the propensity score and of the probability weight. Based onconsistency results and practical flexibility, we recommend generally using thestandard within method. Throughout, we provide intuition to make the resultsmeaningful to the broad audience of applied researchers.

Multiple imputation for propensity score analysis with covariates missing at random: some clarity on within and across methods