Invited commentary: mixing multiple imputation and bootstrapping for variance estimation.

Multiple imputation (MI) is commonly implemented to mitigate potential selection bias due to missing data. The accompanying article by Nguyen and Stuart (Am J Epidemiol. 2024;193(10):1470-1476) examines the statistical consistency of several ways of integrating MI with propensity scores. As Nguyen and Stuart noted, variance estimation for these different approaches remains to be developed. One common option is the nonparametric bootstrap, which can provide valid inference when closed-form variance estimators are not available. However, there is no consensus on how to implement MI and nonparametric bootstrapping in analyses. To complement Nguyen and Stuart's article on MI and propensity score analyses, we review some currently available approaches on variance estimation with MI and nonparametric bootstrapping.