  In epidemiology and social sciences, propensity score methods are popular for
estimating treatment effects using observational data, and multiple imputation
is popular for handling covariate missingness. However, how to appropriately
use multiple imputation for propensity score analysis is not completely clear.
This paper aims to bring clarity on the consistency (or lack thereof) of
methods that have been proposed, focusing on the within approach (where the
effect is estimated separately in each imputed dataset and then the multiple
estimates are combined) and the across approach (where typically propensity
scores are averaged across imputed datasets before being used for effect
estimation). We show that the within method is valid and can be used with any
causal effect estimator that is consistent in the full-data setting. Existing
across methods are inconsistent, but a different across method that averages
the inverse probability weights across imputed datasets is consistent for
propensity score weighting. We also comment on methods that rely on imputing a
function of the missing covariate rather than the covariate itself, including
imputation of the propensity score and of the probability weight. Based on
consistency results and practical flexibility, we recommend generally using the
standard within method. Throughout, we provide intuition to make the results
meaningful to the broad audience of applied researchers.
