  Multiple imputation has become one of the most popular approaches for
handling missing data in statistical analyses. Part of this success is due to
Rubin's 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.
