Inverse Confounding Analysis: An Exact Method for Quantifying the Significance of Confounding
Abstract
The presence of unmeasured confounding factors during the collection of observational data may lead to biased estimates of the effect of an exposure on an outcome. Consequently, a central problem in causal inference based on observational data is sensitivity analysis with respect to unmeasured confounding. Existing sensitivity analyses generally focus on worst-case bounds. We propose an exact method for quantifying the significance of confounding, defined here in terms of the complete range of analytical estimates of the stratification-based Risk Ratio over the set of all joint distributions compatible with the observed characteristics. We refer to the proposed method as Inverse Confounding Analysis (ICA). The proposed ICA method extends the widely used E-value approach but, in contrast to it, does not restrict the analysis to a worst-case lower bound. Instead, it provides exact estimates over the entire set of admissible configurations. This requires several additional input parameters, namely the frequencies of the exposure, the confounder, and the outcome. The ICA method is based on an inverse problem: reconstructing the set of admissible joint distributions from specified frequencies and pairwise associations. We formulate this reconstruction problem as a system of nonlinear equations and obtain an analytical solution. Surprisingly, the complete solution set can be parameterized linearly by a single free parameter. The corresponding stratification-based Risk Ratio is then represented as a fractional-linear function of this parameter. This representation makes it possible to derive exact analytical measures of the significance of confounding over the entire set of admissible statistical configurations.
Related
- Spatial causal inference in the presence of unmeasured confounding and interference
- Robust Design and Evaluation of Predictive Algorithms under Unobserved Confounding
- The Effect of Residual Confounding in the Analysis of Large Observational Data Sets
- Bounding the Risk Difference Bias Given Incomplete Parameters for Externally Adjusting Uncontrolled Confounding
- Reinforcement Learning Under Runtime Confounding: Policy Learning and Off-Policy Evaluation
BibTeX
@misc{porotsky2026inverse,
title = {{Inverse Confounding Analysis: An Exact Method for Quantifying the Significance of Confounding}},
author = {Sergey Porotsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.11991},
eprint = {2608.11991},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11991},
}
RIS
TY - UNPB
TI - Inverse Confounding Analysis: An Exact Method for Quantifying the Significance of Confounding
AU - Sergey Porotsky
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.11991
UR - https://arxiv.org/abs/2608.11991
AB - The presence of unmeasured confounding factors during the collection of observational data may lead to biased estimates of the effect of an exposure on an outcome. Consequently, a central problem in causal inference based on observational data is sensitivity analysis with respect to unmeasured confounding. Existing sensitivity analyses generally focus on worst-case bounds. We propose an exact method for quantifying the significance of confounding, defined here in terms of the complete range of analytical estimates of the stratification-based Risk Ratio over the set of all joint distributions compatible with the observed characteristics. We refer to the proposed method as Inverse Confounding Analysis (ICA). The proposed ICA method extends the widely used E-value approach but, in contrast to it, does not restrict the analysis to a worst-case lower bound. Instead, it provides exact estimates over the entire set of admissible configurations. This requires several additional input parameters, namely the frequencies of the exposure, the confounder, and the outcome. The ICA method is based on an inverse problem: reconstructing the set of admissible joint distributions from specified frequencies and pairwise associations. We formulate this reconstruction problem as a system of nonlinear equations and obtain an analytical solution. Surprisingly, the complete solution set can be parameterized linearly by a single free parameter. The corresponding stratification-based Risk Ratio is then represented as a fractional-linear function of this parameter. This representation makes it possible to derive exact analytical measures of the significance of confounding over the entire set of admissible statistical configurations.
ER -