A Unified Framework for Causal Inference with Multiple Imputation Using Martingale
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Multiple imputation is widely used to handle confounders missing at random in causal inference. Although Rubin's combining rule is simple, it is not clear weather or not the standard multiple imputation inference is consistent when coupled with the commonly-used average causal effect (ACE) estimators. This article establishes a unified martingale representation for the average causal effect (ACE) estimators after multiple imputation. This representation invokes the wild bootstrap inference to provide consistent variance estimation. Our framework applies to asymptotically normal ACE estimators, including the regression imputation, weighting, and matching estimators. We extend to the scenarios when both outcome and confounders are subject to missingness and when the data are missing not at random.
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