  Inverse probability of treatment weighting (IPTW) is a popular propensity
score (PS)-based approach to estimate causal effects in observational studies
at risk of confounding bias. A major issue when estimating the PS is the
presence of partially observed covariates. Multiple imputation (MI) is a
natural approach to handle missing data on covariates, but its use in the PS
context raises three important questions: (i) should we apply Rubin's rules to
the IPTW treatment effect estimates or to the PS estimates themselves? (ii)
does the outcome have to be included in the imputation model? (iii) how should
we estimate the variance of the IPTW estimator after MI? We performed a
simulation study focusing on the effect of a binary treatment on a binary
outcome with three confounders (two of them partially observed). We used MI
with chained equations to create complete datasets and compared three ways of
combining the results: combining treatment effect estimates (MIte); combining
the PS across the imputed datasets (MIps); or combining the PS parameters and
estimating the PS of the average covariates across the imputed datasets
(MIpar). We also compared the performance of these methods to complete case
(CC) analysis and the missingness pattern (MP) approach, a method which uses a
different PS model for each pattern of missingness. We also studied empirically
the consistency of these 3 MI estimators. Under a missing at random (MAR)
mechanism, CC and MP analyses were biased in most cases when estimating the
marginal treatment effect, whereas MI approaches had good performance in
reducing bias as long as the outcome was included in the imputation model.
However, only MIte was unbiased in all the studied scenarios and Rubin's rules
provided good variance estimates for MIte.
