in MI such that the post-imputation distribution much more closely (and often exactly) matches the population distribution. The idea of the calibrated-$\delta$ adjustment is motivated by van Buuren et al.’s $\delta$ adjustment (offset) approach in MI.[@VanBuuren1999] However, while values of $\delta$ are often chosen arbitrarily (and independently of covariates in the imputation model) in van Buuren et al.’s approach, the incomplete variable’s population distribution is used to derive the value of $\delta$ in calibrated-$\delta$ adjustment MI. We show that our proposed method gives equivalent inference to standard MI when data are MAR, and can produce unbiased inference under two general MNAR mechanisms.

From a practical point of view, the development of calibrated-$\delta$ adjustment MI is motivated by the issue incomplete recording of ethnicity data in UK primary care electronic health records. Routine recording of ethnicity has been incorporated at the general practice level in the UK, and the variable is therefore available in many large primary care databases. However, research addressing ethnicity has been constrained by the low level of recording.[@Kumarapeli2006; @Aspinall2007; @Mathur2013b] Studies often handle missing data in ethnicity by either dropping ethnicity from the analysis [@Osborn2015], performing a complete record analysis (i.e. excluding individuals with missing data), or 