Bootstrap Inference when Using Multiple Imputation (1602.07933v6)

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.

• The paper introduces four innovative methods that integrate multiple imputation with bootstrap resampling to construct valid confidence intervals.
• Simulation studies reveal that MI Boot and Boot MI deliver reliable inference, though computational efficiency varies with the imputation approach.
• Applied to HIV treatment data, the methods demonstrate practical advantages in handling missing values in complex causal inference models.

Bootstrap Inference When Using Multiple Imputation

The paper "Bootstrap Inference When Using Multiple Imputation" by Michael Schomaker and Christian Heumann tackles a notable challenge in statistical analysis concerning the inference of data with missing values. When dealing with missing entries in a dataset, multiple imputation (MI) is widely used to fill in these gaps. However, the combination of MI with bootstrap resampling for the purpose of constructing confidence intervals has not been thoroughly addressed in literature, especially when no analytic standard errors are available for the analysis model.

Context and Motivation

The main motivation arises from complex data analyses where estimators are reliant on bootstrapping due to a lack of analytic solutions. This is notably the case in causal inference frameworks, where the distribution of estimators is often intractable. Specifically, the paper applies its findings to HIV treatment research, assessing optimal treatment initiation using the g-formula. This estimation method lacks straightforward standard errors, pressing the need for alternative inference strategies.

Methodological Framework

The paper introduces four distinct methodologies combining MI and bootstrap:

1. MI Boot (PS): Bootstrapping within each imputed data set and pooling the estimates.
2. MI Boot: Bootstrapping each dataset to estimate standard errors and applying standard MI combinatory rules.
3. Boot MI (PS): Creating bootstrap samples of the original dataset, performing multiple imputations on each, and pooling these results.
4. Boot MI: As above, but point estimates are averaged within each bootstrap sample prior to pooling.

These methods are carefully analyzed concerning their validity to construct confidence intervals post-multiple imputation. They explore each method's capacity to achieve reliable inference using both theoretical grounds and simulation studies.

Simulation Studies and Results

The authors conducted a suite of simulation studies extending through diverse settings including linear regression and survival analyses. The empirical investigations reveal that MI Boot and Boot MI generally provide valid inference, though the performance may vary depending on the extent of missingness and the number of imputations. Specifically, MI Boot requires an appropriately large number of imputations to achieve consistent results, while MI Boot (PS) showed potential inefficiencies due to pooling.

Moreover, the computational expense is a crucial factor in choosing between methodologies, with MI Boot being substantially more computationally efficient than Boot MI due to the nesting of bootstrap operations.

Data Analysis and Practical Implications

In an applied setting, specifically the HIV treatment timing scenario, the paper demonstrates the pragmatic implications of each methodology. The aim was to understand the impact of ART initiation rules on child mortality, facing data scarcity common in clinical research. Applying these methods yielded varying confidence intervals, thus impacting inferential validity. This not only provides insight into methodological robustness but also signifies practical advantages in real-world data analysis.

Conclusions and Future Recommendations

The paper concludes that while multiple approaches can generate valid confidence intervals when coupling MI with bootstrap, careful selection dependent on data characteristics and computational capability is imperative. It notably discourages the use of MI Boot pooled due to unreliable efficiency.

For future developments, enhanced focus on computationally efficient algorithms and deeper exploration into nonparametric settings are indicated. As these methodologies evolve, they stand to improve the reliability of statistical inference in fields routinely encountering missing data, further cemented by the insights shared in this paper.

In summary, this work provides a significant contribution to the integration of bootstrap resampling with multiple imputation, enriching the toolkit available for statistical inference in the face of missing data. Its applications extend beyond epidemiology, offering potential avenues for refined data analysis efforts in numerous scientific domains.