Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Role of Measured Covariates in Assessing Sensitivity to Unmeasured Confounding

Abhinandan Dalal, Iris Horng, Yang Feng, Dylan S. Small

TL;DR

The paper tackles how the structure of measured covariates shapes sensitivity to unmeasured confounding in observational causal inference. It derives a bias amplification result in linear regression, showing that the impact of unmeasured confounding on the estimated effect is magnified when a proxy $X$ for a latent factor $U$ is strongly associated with the exposure $A$ and measured with error, formalized through the observable ratio $\beta_{A\sim X}/(\mathrm{Var}(A)(1-R^2_{A\sim X}))$. The authors apply the framework to smoking and lung cancer using NHANES data from 1971–74 and 2015–16, revealing that evolving socioeconomic patterns have increased the exposure–covariate coupling and thus the sensitivity to residual confounding. They propose a practical diagnostic that combines the observed exposure–covariate association with the residual variance to gauge robustness of causal conclusions in proxy-adjusted analyses. Overall, the work provides a principled way to quantify and monitor how multicollinearity and proxy quality affect the credibility of causal claims in observational studies.

Abstract

Sensitivity analysis is widely used to assess the robustness of causal conclusions in observational studies, yet its interaction with the structure of measured covariates is often overlooked. When latent confounders cannot be directly adjusted for and are instead controlled using proxy variables, strong associations between exposure and measured proxies can amplify sensitivity to residual confounding. We formalize this phenomenon in linear regression settings by showing that a simple ratio involving the exposure model coefficient and residual exposure variance provides an observable measure of this increased sensitivity. Applying our framework to smoking and lung cancer, we document how growing socioeconomic stratification in smoking behavior over time leads to heightened sensitivity to unmeasured confounding in more recent data. These results highlight the importance of multicollinearity when interpreting sensitivity analyses based on proxy adjustment.

The Role of Measured Covariates in Assessing Sensitivity to Unmeasured Confounding

TL;DR

The paper tackles how the structure of measured covariates shapes sensitivity to unmeasured confounding in observational causal inference. It derives a bias amplification result in linear regression, showing that the impact of unmeasured confounding on the estimated effect is magnified when a proxy for a latent factor is strongly associated with the exposure and measured with error, formalized through the observable ratio . The authors apply the framework to smoking and lung cancer using NHANES data from 1971–74 and 2015–16, revealing that evolving socioeconomic patterns have increased the exposure–covariate coupling and thus the sensitivity to residual confounding. They propose a practical diagnostic that combines the observed exposure–covariate association with the residual variance to gauge robustness of causal conclusions in proxy-adjusted analyses. Overall, the work provides a principled way to quantify and monitor how multicollinearity and proxy quality affect the credibility of causal claims in observational studies.

Abstract

Sensitivity analysis is widely used to assess the robustness of causal conclusions in observational studies, yet its interaction with the structure of measured covariates is often overlooked. When latent confounders cannot be directly adjusted for and are instead controlled using proxy variables, strong associations between exposure and measured proxies can amplify sensitivity to residual confounding. We formalize this phenomenon in linear regression settings by showing that a simple ratio involving the exposure model coefficient and residual exposure variance provides an observable measure of this increased sensitivity. Applying our framework to smoking and lung cancer, we document how growing socioeconomic stratification in smoking behavior over time leads to heightened sensitivity to unmeasured confounding in more recent data. These results highlight the importance of multicollinearity when interpreting sensitivity analyses based on proxy adjustment.
Paper Structure (9 sections, 1 theorem, 12 equations, 1 figure, 3 tables)

This paper contains 9 sections, 1 theorem, 12 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. The measurement error problem in explanatory variables
  4. Multicollinearity and the Role of Measured Covariates
  5. The Smoking Example: Data from NHANES
  6. The Effect of Residual Confounding on Sensitivity
  7. Implications on Omitted Variable Bias
  8. Are smoking effect estimates today more sensitive to unmeasured confounding?
  9. Discussion

Key Result

Proposition 1

Suppose Equations ysem and xsem hold, and all random variables have finite second moments. Let $\beta_{Y\sim A,X}$ denote the coefficient of $A$ in the linear regression of $Y\sim A,X$. Then, under Assumption nocom where $\beta_{A\sim X}$ and $R^2_{A\sim X}$ are the coefficient of $X$ and the coefficient of determination respectively, in a linear regression of $A$ on $X$.

Figures (1)

  • Figure 1: Comparison of two hypothetical studies. (Top) Observed Exposure and Covariate Relationship (Bottom) Effect estimates and Sensitivity Analysis for Observed Outcome.

Theorems & Definitions (1)

  • Proposition 1