complete records. Standard MI and calibrated-$\delta$ adjustment MI are unbiased for both parameter estimates. Standard MI yields comparable empirical and average model standard errors and coverage attains the nominal level. In calibrated-$\delta$ adjustment MI, empirical standard errors are slightly smaller than the average model counterparts, leading to a minimal increase in coverage.

[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/bias_base.pdf "fig:"){width="\textwidth"}

 

[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/se_base.pdf "fig:"){width="\textwidth"}

 

[0.69]{} ![Simulation study: bias in point estimates, empirical and average model SE, and coverage of 95% CIs under different missingness mechanism for $x$.[]{data-label="fig:sim_base"}](figures/cv_base.pdf "fig:"){width="\textwidth"}

Under M3, when $x$ is MNAR dependent on $x$, CRA yields unbiased estimates of both parameters. Standard MI is biased in the estimate of $\beta_{0}$ but provides an unbiased estimate of $\beta_{x}$ due to the symmetry property of the odds ratios. Generally, in logistic regression with an incomplete covariate $x$, when the missingness mechanism is such that both standard MI and CRA are unbiased, standard MI tends not to be more efficient than CRA in estimating 