Doubly Robust Variance Estimation for Average Causal Effects with Parametric Working Models: A Comparison of the Empirical Sandwich Variance Estimator, Nonparametric Bootstrap, and Influence Function Based Variance Estimator

Grunnleggende konsepter

The empirical sandwich variance estimator and the nonparametric bootstrap are doubly robust variance estimators for the average causal effect with observational data, while the commonly used influence function based variance estimator is not.

Sammendrag

Bibliographic Information:

Shook-Sa, B. E., Zivich, P. N., Lee, C., Xue, K., Ross, R. K., Edwards, J. K., Stringer, J. S. A., & Cole, S. R. (2024). Double robust variance estimation with parametric working models. arXiv preprint arXiv:2404.16166v2.

Research Objective:

This paper aims to compare three variance estimation methods for doubly robust estimators of the average causal effect (ACE) with observational data: the influence function based variance estimator, the empirical sandwich variance estimator, and the nonparametric bootstrap. The authors seek to demonstrate the superior performance of the empirical sandwich and bootstrap methods, which are doubly robust, over the influence function method, which is not.

Methodology:

The authors first describe three common doubly robust estimators of the ACE: the classic augmented inverse probability weighted (AIPW) estimator, the weighted regression AIPW estimator, and targeted maximum likelihood estimation (TMLE). They then detail the three variance estimation approaches and apply them to data from the Improving Pregnancy Outcomes with Progesterone (IPOP) study to estimate the effect of maternal anemia on birth weight among women with HIV. Finally, they conduct a simulation study to compare the empirical properties of the three variance estimators under various model misspecification scenarios.

Key Findings:

The simulation study demonstrates that both the empirical sandwich variance estimator and the nonparametric bootstrap provide valid estimates of the variance and achieve nominal confidence interval coverage when at least one of the working models (propensity score or outcome model) is correctly specified. In contrast, the influence function based variance estimator is biased and inconsistent when either model is misspecified, leading to either conservative or anti-conservative confidence intervals.

Main Conclusions:

The authors conclude that the empirical sandwich variance estimator and the nonparametric bootstrap are preferable to the influence function based variance estimator for doubly robust estimation of the ACE with observational data. They advocate for wider adoption of these doubly robust variance estimators in practice and highlight the limitations of the commonly used influence function approach.

Significance:

This research has important implications for causal inference in epidemiology and other fields where doubly robust estimators are employed. By demonstrating the limitations of the influence function based variance estimator and advocating for alternative doubly robust approaches, the authors contribute to more accurate and reliable estimation of causal effects in observational studies.

Limitations and Future Research:

The study focuses on the average causal effect with a binary exposure and parametric working models. Future research could extend these findings to other estimands, exposures, and modeling approaches, including machine learning methods. Additionally, the authors acknowledge limitations in the IPOP data analysis, suggesting alternative estimands and methods for handling competing events in future work.

Statistikk

A systematic review published in October 2023 found that 67% of papers published using TMLE did not indicate how variances were estimated.
Of the 33% of papers that did report the variance estimation method, the majority (72%) relied on the influence function based variance estimator.
Among published studies that reported the variance estimation approach, 21% applied the bootstrap.
Simulations were conducted with a sample size of 800, similar to the sample size in the IPOP study.
The nonparametric bootstrap was based on 1000 resamples in each simulation iteration.

Sitater

"Doubly robust estimators have gained popularity in the field of causal inference due to their ability to provide consistent point estimates when either an outcome or exposure model is correctly specified."
"However, for nonrandomized exposures the influence function based variance estimator frequently used with doubly robust estimators of the average causal effect is only consistent when both working models (i.e., outcome and exposure models) are correctly specified."
"That is, they are expected to provide valid estimates of the variance leading to nominal confidence interval coverage when only one working model is correctly specified."

Viktige innsikter hentet fra

Double Robust Variance Estimation with Parametric Working Models

by Bonnie E. Sh... klokken arxiv.org 11-06-2024

https://arxiv.org/pdf/2404.16166.pdf

Dypere Spørsmål

How might the performance of these variance estimators differ when using machine learning methods for propensity score and outcome modeling, and what alternative approaches exist for doubly robust variance estimation in that context?

When using machine learning methods like SuperLearner for propensity score and outcome modeling, the performance of the variance estimators discussed in the context differs significantly. Here's a breakdown:

Influence Function Based Variance Estimator: This method, while computationally straightforward, relies on the correct specification of both the outcome and propensity score models. With the increased complexity and potential "black box" nature of machine learning models, ensuring correct specification becomes even more challenging. Consequently, this variance estimator might lead to unreliable confidence intervals and potentially misleading inferences.

Empirical Sandwich Variance Estimator: The empirical sandwich variance estimator, as described in the context, relies on the estimating equations approach. This approach is generally not compatible with many machine learning methods, making it unsuitable for doubly robust variance estimation in such scenarios.

Nonparametric Bootstrap: While the nonparametric bootstrap remains a viable option for doubly robust variance estimation with machine learning methods, its performance can be computationally expensive and sensitive to specific settings. For instance, the presence of outliers or extreme values in the data can disproportionately influence the bootstrap resamples, leading to unstable variance estimates.
Alternative Approaches for Doubly Robust Variance Estimation with Machine Learning:
Given the limitations of traditional methods, alternative approaches have been developed for doubly robust variance estimation when using machine learning for nuisance parameter estimation:

Cross-Validation/Cross-Fitting: This involves partitioning the data into multiple subsets, fitting the nuisance models on one subset, and estimating the treatment effect and its variance on the remaining subsets. This process is repeated across different data partitions, and the variance estimates are aggregated. This approach helps mitigate overfitting and provides more robust variance estimates.

Influence Function Based Methods with Correction Terms: Researchers have proposed modifications to the influence function based variance estimator that incorporate correction terms to account for the potential bias introduced by machine learning methods. These corrections aim to provide more accurate variance estimates even when the nuisance models are not perfectly specified.

Targeted Minimum Loss-Based Estimation (TMLE) with SuperLearner: TMLE, as mentioned in the context, can be paired with machine learning methods like SuperLearner for nuisance parameter estimation. Specific algorithms within the TMLE framework are designed to provide valid statistical inference even when using data-adaptive methods like machine learning.

Could the choice of variance estimator impact the conclusions drawn from observational studies, particularly in settings with potential model misspecification, and how can researchers mitigate the risk of misleading inferences?

Yes, the choice of variance estimator can significantly impact the conclusions drawn from observational studies, especially when there's a risk of model misspecification.
Here's how:

Overly Conservative Inferences: Using a conservative variance estimator, like the influence function based estimator with a misspecified outcome model, can lead to wider confidence intervals. This might result in failing to reject the null hypothesis even when a true effect exists, leading to a Type II error. In simpler terms, researchers might conclude that there's no effect when there actually is, potentially overlooking important findings.

Overly Liberal Inferences: Conversely, using an anti-conservative variance estimator, like the influence function based estimator with a misspecified propensity score model, can lead to artificially narrow confidence intervals. This increases the likelihood of rejecting the null hypothesis when it's true, resulting in a Type I error. In this case, researchers might falsely conclude that there's a significant effect when there's none, potentially leading to spurious findings.
Mitigating the Risk of Misleading Inferences:

Sensitivity Analyses: Conduct thorough sensitivity analyses to assess the robustness of the findings to different model specifications. This involves exploring a range of plausible models for both the outcome and propensity score and examining how the estimated treatment effect and its variance change.

Doubly Robust Methods: As highlighted in the context, prioritize the use of doubly robust estimation methods like AIPW and TMLE. These methods provide consistent point estimates even when one of the working models is misspecified, offering a layer of protection against model misspecification.

Appropriate Variance Estimation: Critically, pair doubly robust estimators with doubly robust variance estimators like the empirical sandwich variance estimator (when applicable) or the nonparametric bootstrap. This ensures that the variance estimates are also robust to potential model misspecification, leading to more reliable confidence intervals and reducing the risk of misleading inferences.

Transparent Reporting: Clearly report the chosen estimation methods, including the specific variance estimator used, and justify these choices. Transparency allows other researchers to assess the validity of the findings and understand the potential limitations of the chosen approach.

What are the broader implications of relying on statistical methods for causal inference, and how can researchers ensure that their interpretations are grounded in both statistical rigor and substantive domain knowledge?

Relying solely on statistical methods for causal inference, without considering the broader context and domain knowledge, can have significant implications:

Spurious Correlations: Statistical methods can identify associations between variables, but they cannot inherently distinguish between correlation and causation. Without careful consideration of confounding variables and alternative explanations, researchers risk misinterpreting spurious correlations as causal relationships.

Misleading Policy Recommendations: Causal inferences often inform policy decisions. If these inferences are based on flawed statistical analyses or a lack of domain knowledge, they can lead to ineffective or even harmful policies. For example, a misidentified causal relationship between a particular social program and improved health outcomes might lead to increased funding for that program, even if the observed association is not truly causal.

Erosion of Trust: Publishing misleading or inaccurate findings due to over-reliance on statistical methods without proper grounding in domain knowledge can erode public trust in scientific research. This erosion of trust can have far-reaching consequences, making it harder to address important societal challenges.
Ensuring Statistically Rigorous and Substantively Grounded Interpretations:

Collaboration: Foster interdisciplinary collaborations between statisticians, epidemiologists, social scientists, and domain experts. This collaboration ensures that statistical analyses are appropriately tailored to the research question and that interpretations are grounded in a deep understanding of the subject matter.

Causal Framework: Develop a clear causal framework or directed acyclic graph (DAG) that outlines the hypothesized causal relationships between variables, potential confounders, and the outcome of interest. This framework guides the statistical analysis and helps identify potential sources of bias.

Sensitivity to Assumptions: Be transparent about the assumptions underlying the chosen statistical methods and assess the sensitivity of the findings to violations of these assumptions. For instance, acknowledge the potential impact of unmeasured confounding and explore how the results might change if such confounding were present.

Contextualization: Interpret statistical findings within the broader context of existing literature, theoretical frameworks, and domain knowledge. Avoid making causal claims that are not supported by the data or that contradict well-established knowledge in the field.

Replication and Validation: Encourage replication studies and seek to validate findings using different data sources or study designs. This helps ensure that the observed relationships are robust and not simply artifacts of a particular dataset or analytical approach.