  Many modern estimators require bootstrapping to calculate confidence
intervals because either no analytic standard error is available or the
distribution of the parameter of interest is non-symmetric. It remains however
unclear how to obtain valid bootstrap inference when dealing with multiple
imputation to address missing data. We present four methods which are
intuitively appealing, easy to implement, and combine bootstrap estimation with
multiple imputation. We show that three of the four approaches yield valid
inference, but that the performance of the methods varies with respect to the
number of imputed data sets and the extent of missingness. Simulation studies
reveal the behavior of our approaches in finite samples. A topical analysis
from HIV treatment research, which determines the optimal timing of
antiretroviral treatment initiation in young children, demonstrates the
practical implications of the four methods in a sophisticated and realistic
setting. This analysis suffers from missing data and uses the $g$-formula for
inference, a method for which no standard errors are available.
