Propensity score estimation with missing values using a multiple imputation missingness pattern (MIMP) approach

Abstract

Propensity scores have been used widely as a bias reduction method to estimate the treatment effect in nonrandomized studies. Since many covariates are generally included in the model for estimating the propensity scores, the proportion of subjects with at least one missing covariate could be large. While many methods have been proposed for propensity score-based estimation in the presence of missing covariates, little has been published comparing the performance of these methods. In this article we propose a novel method called multiple imputation missingness pattern (MIMP) and compare it with the naive estimator (ignoring propensity score) and three commonly used methods of handling missing covariates in propensity score-based estimation (separate estimation of propensity scores within each pattern of missing data, multiple imputation and discarding missing data) under different mechanisms of missing data and degree of correlation among covariates. Simulation shows that all adjusted estimators are much less biased than the naive estimator. Under certain conditions MIMP provides benefits (smaller bias and mean-squared error) compared with existing alternatives.

Inverse Probability Weighting with Missing Predictors of Treatment Assignment or Missingness

“…This latter estimator is complicated, but has the advantage of being valid even when the parameter estimator is not a MLE, as is the case here. Qu and Lipkovich (2009) use proper MI. We describe the analogous improper MI procedure and its closed-form variance estimator.…”

“…A second drawback is the difficulty of interpreting the parameter constraints needed to make the joint model for (X, T , R) estimable. Qu and Lipkovich (2009) proposed multiply imputing missing values of X using the observed values of X, T, and the outcome, thus creating M multiple data sets in which X is complete. For each completed data set, the PS model is fitted, PS's are estimated and the inverse PS's are used as weights in the estimator of treatment effect.…”

“…The M treatment effect estimates are then averaged. In a refinement of this approach, Qu and Lipkovich (2009) propose including R as an additional covariate in the PS model. They explain that this may reduce bias when X is missing not at random.…”

“…Their descriptions of the methods they used are somewhat limited, but these methods would appear to be the same as, or very similar to, that of Qu and Lipkovich (2009). Hayes and Groner (2008) multiply impute missing predictors and calculate PSs for each imputed data set.…”

“…Second, as these estimators are not maximum likelihood estimators (MLEs), it is not obvious that Rubin's Rules will apply in this case (Robins and Wang, 2000;Nielsen, 2003). Qu and Lipkovich (2009) proposed that variance estimates be obtained by bootstrapping, a computationally intensive procedure. We investigate how the simple Rubin's Rules variance estimator performs in this setting.…”