Multiple imputation has become one of the most popular approaches for handling missing data in statistical analyses. Part of this success is due to Rubins simple combination rules. These give frequentist valid inferences when the imputation and analysis procedures are so called congenial and the complete data analysis is valid, but otherwise may not. Roughly speaking, congeniality corresponds to whether the imputation and analysis models make different assumptions about the data. In practice imputation and analysis procedures are often not congenial, such that tests may not have the correct size and confidence interval coverage deviates from the advertised level. We examine a number of recent proposals which combine bootstrapping with multiple imputation, and determine which are valid under uncongeniality and model misspecification. Imputation followed by bootstrapping generally does not result in valid variance estimates under uncongeniality or misspecification, whereas bootstrapping followed by imputation does. We recommend a particular computationally efficient variant of bootstrapping followed by imputation.
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