providing a way to incorporate population distribution information into MI.

This paper focuses on missing data in an incomplete binary/categorical covariate in an analysis model, where the outcome variable and other covariates are all binary/categorical and fully observed. The remainder of this paper is structured as follows. works through a simple example analytically to describe the derivation of the calibrated-$\delta$ adjustment. In \[sec3\], we formally introduce the procedure of calibrated-$\delta$ adjustment MI and evaluate the performance of the method in simulation studies. illustrates the use of this MI method in a case study which uses electronic health records to examine the association between ethnicity and the prevalence of type 2 diabetes diagnoses in UK primary care. We conclude the paper with a discussion in \[sec5\].

Analytic study – bias in a $2\times 2$ contingency table {#sec2}
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In this section, we present the development of calibrated-$\delta$ adjustment MI in a simple setting of a $2 \times 2$ contingency table and describe the derivation of the calibrated-$\delta$ adjustment.

Suppose it is of interest to study the association between a binary variable $x$ taking values $j=0,1$ and a binary outcome $y$ taking values $k=0,1$, whose full-data distribution is given in Table \[tab:analytic\_study\_full\_data\]. 