Propensity score matching after multiple imputation when a confounder has missing data

One of the main challenges when using observational data for causal inference is the presence of confounding. A classic approach to account for confounding is the use of propensity score techniques that provide consistent estimators of the causal treatment effect under four common identifiability assumptions for causal effects, including that of no unmeasured confounding. Propensity score matching is a very popular approach which, in its simplest form, involves matching each treated patient to an untreated patient with a similar estimated propensity score, that is, probability of receiving the treatment. The treatment effect can then be estimated by comparing treated and untreated patients within the matched dataset. When missing data arises, a popular approach is to apply multiple imputation to handle the missingness. The combination of propensity score matching and multiple imputation is increasingly applied in practice. However, in this article we demonstrate that combining multiple imputation and propensity score matching can lead to over-coverage of the confidence interval for the treatment effect estimate. We explore the cause of this over-coverage and we evaluate, in this context, the performance of a correction to Rubin's rules for multiple imputation proposed by finding that this correction removes the over-coverage.