$\beta_{x}$.[@White2010b] This is because without auxiliary variables in the imputation model, standard MI does not carry any extra information on the odds ratio compared to CRA. This is seen in the simulation results for $\beta_{x}$ under models M1–M3. Under M3, calibrated-$\delta$ adjustment MI is also unbiased in both parameter estimates. Given that all three methods are unbiased for $\beta_{x}$ under M3, there is a small gain in efficiency in the estimate of $\beta_{x}$ in calibrated-$\delta$ adjustment MI, as the empirical standard error for this parameter is slightly smaller than that in CRA. Under this missingness mechanism, empirical and average model standard errors are comparable across methods; for methods that are unbiased, their corresponding coverage of $95\%$ CIs generally attains the nominal level.

Under M4, when $x$ is MNAR dependent on $x$ and $y$, standard MI and CRA are again biased in both parameter estimates, leading to coverage close or equal to 0. In contrast, calibrated-$\delta$ adjustment MI produces unbiased estimates of both parameters. In this method, empirical standard errors are again slightly smaller than the average model counterparts (as seen previously under M2), which leads to coverage slightly exceeding the $95\%$ level.

When the population distribution is estimated with uncertainty 