reported in the applied literature. Instead, in practice, researchers more often try to enhance the plausibility of the MAR assumption as much as possible by including many variables in the imputation model.[@White2011; @Collins2001]

The extra model specification requirement in MI for MNAR data raises several issues. Firstly, the underlying MAR and MNAR mechanisms are not verifiable from the observed data alone. Secondly, there can be an infinite number of possible MNAR models for any dataset, and it is very rare to know which of these models is appropriate for the missingness mechanism. However, for an incomplete variable in a given dataset, its corresponding population marginal distribution might be available from an external data source, such as a population census or survey. If our study sample in truth comes from such a population, it is sensible to feed this population information into the imputation model, in order to calibrate inference to the population.

In this paper, we propose a version of MI for an incomplete binary/categorical variable, termed *calibrated-$\delta$ adjustment MI*, which exploits such external information. In this approach, the population distribution of the incomplete variable can be used to calculate an adjustment in the imputation model’s intercept, which is used 