same missingness scenario, but a different set of datasets is generated for each missingness scenario.[@Burton2006] The parameters of interest are $\beta_{0}$ and $\beta_{x}$, although in practice $\beta_{x}$ is usually of more interest. Bias, efficiency of $\hat{\beta}_{0}$ and $\hat{\beta}_{x}$ in terms of the empirical standard errors, and coverage of 95% CIs are calculated over $2\,000$ repetitions for each combination of simulation settings,[@White2010a] with analyses of full data (i.e. before any values in $x$ are set to missing) and complete records also provided for comparison.

All simulations are performed in Stata 14 [@StataCorp2015b]; `mi impute logit` is used for standard MI, the community-contributed command `uvis logit` [@Royston2004] for calibrated-$\delta$ adjustment MI, and `mi estimate: logit` for fitting the analysis model to the completed datasets and combining the results using Rubin’s rules.[@Rubin1987; @Barnard1999] Simulated datasets are analysed using the community-contributed command `simsum`.[@White2010a]

Based on the analytic calculations presented in \[sec2\], we propose the following procedure for imputing missing values in the covariate $x$ using calibrated-$\delta$ adjustment MI.

1.  Fit a logistic regression imputation model for $x$ conditional on $y$ to the complete records to obtain the maximum likelihood estimates of the imputation models’ parameters $\hat{\theta}$ and their asymptotic sampling variance $\widehat{\boldsymbol{U}}$;

2.  Draw 