Multiple Imputation Guided by Full Law and Target Law Identifiability

The central challenges in missing data models concern the identifiability of two distributions: the target law and the full law. The target law refers to the joint distribution of the data variables, whereas the full law refers to the joint distribution of the data variables and their corresponding response indicators. However, the relationship between the identifiability of these two distributions and the feasibility of multiple imputation has not been clearly established. We present a procedure where the choice of the imputation method is guided by identifiability considerations. We show that imputations can be drawn from the correct conditional distributions for all possible missing data patterns if and only if the full law is identifiable. This result implies that standard multiple imputation methods -- which keep observed values unchanged and replace missing values with imputed values -- are invalid when the target law is identifiable but the full law is not. We demonstrate that alternative imputation strategies can sometimes enable the estimation of the target law in such cases. Specifically, we introduce factorizable imputation where certain observed values are also imputed and the imputed data are weighted in the analysis.