Figure \[fig:sim\_base\]. Full data and CRA both give the results that the theory predicts. Analysis of full data is always unbiased with coverage close to the $95\%$ level and the smallest standard errors of all methods. CRA is unbiased under M1 and M3 as expected,[@White2010b] but bias is observed under the other two missingness mechanisms. Coverage is correspondingly low when bias is present, and efficiency is lower than that in full data.

Under M1, when $x$ is MCAR, all methods appear unbiased, with comparable empirical and average model standard errors and correct coverage. This is as expected.

Under M2, when $x$ is MAR conditional on $y$, CRA is severely biased in the estimate of $\beta_{0}$ and the corresponding coverage of 95% CIs falls to 0. However, the method provides an unbiased estimate of $\beta_{x}$ with correct coverage. This result is specific to this simulation set-up, where the probability of being a complete record depends on the outcome, and the analysis model is a logistic regression. This mimics case-control sampling, where the log odds of the logistic regression is biased in case-control studies but the log odds ratio is not.[@White2010b; @Bartlett2015] The outcome–covariate association can therefore be estimated consistently among the 