  Multiple imputation (MI) has become popular for analyses with missing data in
medical research. The standard implementation of MI is based on the assumption
of data being missing at random (MAR). However, for missing data generated by
missing not at random (MNAR) mechanisms, MI performed assuming MAR might not be
satisfactory. For an incomplete variable in a given dataset, its corresponding
population marginal distribution might also be available in an external data
source. We show how this information can be readily utilised in the imputation
model to calibrate inference to the population, by incorporating an
appropriately calculated offset termed the `calibrated-$\delta$ adjustment'. We
describe the derivation of this offset from the population distribution of the
incomplete variable and show how in applications it can be used to closely (and
often exactly) match the post-imputation distribution to the population level.
Through analytic and simulation studies, we show that our proposed
calibrated-$\delta$ adjustment MI method can give the same inference as
standard MI when data are MAR, and can produce more accurate inference under
two general MNAR missingness mechanisms. The method is used to impute missing
ethnicity data in a type 2 diabetes prevalence case study using UK primary care
electronic health records, where it results in scientifically relevant changes
in inference for non-White ethnic groups compared to standard MI.
Calibrated-$\delta$ adjustment MI represents a pragmatic approach for utilising
available population-level information in a sensitivity analysis to explore
potential departure from the MAR assumption.
