covariate $x$ is available, a natural alternative to adjusting the intercept of the imputation model based on this information is to weight the complete records in the imputation model (which we term ‘weighted multiple imputation’), in order to match the post-imputation distribution of $x$ to the population. In the supporting information section we explore two such weighting approaches, marginal and conditional weighted MI; we show analytically that while these methods can provide more accurate results compared to standard MI under certain MNAR mechanisms, they do not provide a general solution as does calibrated-$\delta$ adjustment MI.

Simulation studies {#sec3}
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This section presents univariate simulation studies to evaluate performance measures of the calibrated-$\delta$ adjustment MI method for an incomplete binary covariate $x$, when the fully observed outcome variable $y$ is also binary. The term ‘univariate’ is used here to refer to the setting where missingness occurs in a single covariate. The aims of these simulation studies are (i) to examine finite-sample properties of calibrated-$\delta$ adjustment MI including bias in parameter estimates, efficiency in terms of the empirical and average model standard errors (SE), and coverage of $95\%$ confidence intervals (CI); and (ii) to compare the method with standard MI and complete 