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Bidirectional causal inference for binary outcomes in the presence of unmeasured confounding

Yafang Deng, Kang Shuai, Shanshan Luo

TL;DR

This work addresses bidirectional causal inference for binary outcomes in the presence of unmeasured confounding by mapping binary indicators to latent continuous variables through a linear structural equation framework with two instrumental variables. It derives identification formulas under a Probit-like system, proposes a practical two-step estimation procedure with Delta-method asymptotics (bootstrapped variance recommended), and conducts comprehensive sensitivity analyses to assess robustness to violations of exclusion restrictions and unmeasured confounding. The method is validated via simulation, showing favorable finite-sample performance relative to naive regression, and is applied to the BRFSS heart disease–diabetes data, revealing robust bidirectional causal effects in both directions. The framework offers a principled approach for binary bidirectional causality in biomedical and social science settings and provides practical tools for sensitivity analysis and real-data interpretation.

Abstract

Bidirectional causal relationships arising from mutual interactions between variables are commonly observed within biomedical, econometrical, and social science contexts. When such relationships are further complicated by unobserved factors, identifying causal effects in both directions becomes especially challenging. For continuous variables, methods that utilize two instrumental variables from both directions have been proposed to explore bidirectional causal effects in linear models. However, the existing techniques are not applicable when the key variables of interest are binary. To address these issues, we propose a structural equation modeling approach that links observed binary variables to continuous latent variables through a constrained mapping. We further establish identification results for bidirectional causal effects using a pair of instrumental variables. Additionally, we develop an estimation method for the corresponding causal parameters. We also conduct sensitivity analysis under scenarios where certain identification conditions are violated. Finally, we apply our approach to investigate the bidirectional causal relationship between heart disease and diabetes, demonstrating its practical utility in biomedical research.

Bidirectional causal inference for binary outcomes in the presence of unmeasured confounding

TL;DR

This work addresses bidirectional causal inference for binary outcomes in the presence of unmeasured confounding by mapping binary indicators to latent continuous variables through a linear structural equation framework with two instrumental variables. It derives identification formulas under a Probit-like system, proposes a practical two-step estimation procedure with Delta-method asymptotics (bootstrapped variance recommended), and conducts comprehensive sensitivity analyses to assess robustness to violations of exclusion restrictions and unmeasured confounding. The method is validated via simulation, showing favorable finite-sample performance relative to naive regression, and is applied to the BRFSS heart disease–diabetes data, revealing robust bidirectional causal effects in both directions. The framework offers a principled approach for binary bidirectional causality in biomedical and social science settings and provides practical tools for sensitivity analysis and real-data interpretation.

Abstract

Bidirectional causal relationships arising from mutual interactions between variables are commonly observed within biomedical, econometrical, and social science contexts. When such relationships are further complicated by unobserved factors, identifying causal effects in both directions becomes especially challenging. For continuous variables, methods that utilize two instrumental variables from both directions have been proposed to explore bidirectional causal effects in linear models. However, the existing techniques are not applicable when the key variables of interest are binary. To address these issues, we propose a structural equation modeling approach that links observed binary variables to continuous latent variables through a constrained mapping. We further establish identification results for bidirectional causal effects using a pair of instrumental variables. Additionally, we develop an estimation method for the corresponding causal parameters. We also conduct sensitivity analysis under scenarios where certain identification conditions are violated. Finally, we apply our approach to investigate the bidirectional causal relationship between heart disease and diabetes, demonstrating its practical utility in biomedical research.
Paper Structure (19 sections, 7 theorems, 14 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 19 sections, 7 theorems, 14 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Framework
  3. Binary bidirectional model
  4. Assumptions
  5. Identification
  6. Estimation and inference
  7. Sensitivity Analysis
  8. Sensitivity analysis for Assumption \ref{['assu:U']}
  9. Further sensitivity analysis for ER assumption
  10. Numerical Experiments
  11. Simulation studies
  12. Sensitivity analysis for Assumption \ref{['assu:U']}
  13. Sensitivity analysis simulation for ER assumption
  14. Application
  15. Description
  16. ...and 4 more sections

Key Result

Proposition 1

Under Model eq:bidirectional_model and Assumption assu:U, the bidirectional causal effects are identifiable: where the terms $\xi_{xz}$, $\xi_{yz}$, $\xi_{xw}$, and $\xi_{yw}$ correspond to the coefficients of $Z$ and $W$ in Model eq:probit_for_assu1.

Figures (8)

  • Figure 1: Graphic illustration of the bidirectional causal model \ref{['eq:bidirectional_model']} with instrumental variables $Z$ and $W$.
  • Figure 2: Three graphical illustrations of Model \ref{['eq:sensitivity-eq']}.
  • Figure 3: Corresponding regional division for the application of causal effect identification expressions.
  • Figure 4: Boxplot of simulation results under bidirectional framework.
  • Figure 5: Bias and standard deviation estimation results.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Corollary 2
  • Corollary 3