Multiple Regression Analysis of Unmeasured Confounding

2025-08-11

AI Summary

This paper addresses bias in causal effect identification arising from unmeasured confounding in observational data. We propose a quantitative sensitivity analysis method grounded in a multiple regression framework. Our key innovation extends the confounding interval approach—previously limited to single-regression settings—to multivariate regression, leveraging observed covariates and domain knowledge (particularly the coefficient of determination, $R^2$) to derive theoretical bounds on omitted-variable bias and thereby achieve partial identification of causal effects. The method supports $R^2$-based bound sensitivity analysis, enabling quantification of estimation uncertainty induced by unmeasured confounders, and is accompanied by an open-source implementation. Simulation studies and empirical applications demonstrate its robustness even under natural stochasticity, offering an interpretable and actionable tool for uncertainty assessment in causal inference.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Action, Change, and Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Whereas confidence intervals are used to assess uncertainty due to unmeasured individuals, confounding intervals can be used to assess uncertainty due to unmeasured attributes. Previously, we have introduced a methodology for computing confounding intervals in a simple regression setting in a paper titled ``Regression Analysis of Unmeasured Confounding." Here we extend that methodology for more general application in the context of multiple regression. Our multiple regression analysis of unmeasured confounding utilizes subject matter knowledge about coefficients of determination to bound omitted variables bias, while taking into account measured covariate data. Our generalized methodology can be used to partially identify causal effects. The methodology is demonstrated with example applications, to show how coefficients of determination, being complementary to randomness, can support sensitivity analysis for causal inference from observational data. The methodology is best applied when natural sources of randomness are present and identifiable within the data generating process. Our main contribution is an algorithm that supports our methodology. The purpose of this article is to describe our algorithm in detail. In the paper we provide a link to our GitHub page for readers who would like to access and utilize our algorithm.

Problem

• Extends methodology to assess unmeasured confounding in multiple regression
• Bounds omitted variables bias using coefficients of determination knowledge
• Supports sensitivity analysis for causal inference from observational data

Innovation

• Extends confounding intervals to multiple regression
• Bounds bias using coefficients of determination
• Algorithm supports causal effect identification