Bootstrap Inference when Using Multiple Imputation

TL;DR

This paper introduces four methods for valid bootstrap inference with multiple imputation, addressing the challenge of estimating confidence intervals when data is missing and standard errors are unavailable.

Contribution

It proposes and compares four intuitive bootstrap approaches for multiple imputation, clarifying their validity and performance in finite samples and real-world applications.

Findings

Three methods yield valid inference under certain conditions.

Performance varies with the number of imputations and missingness extent.

Simulation studies illustrate finite-sample behavior of the methods.

Abstract

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.