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Assessing Interactive Causes of an Occurred Outcome Due to Two Binary Exposures

Shanshan Luo, Wei Li, Xueli Wang, Shaojie Wei, Zhi Geng

TL;DR

This work develops a backward-looking causal attribution framework for two binary exposures and a binary outcome by introducing posterior interaction probabilities $\pi_{rstu}(\mathcal{O})$ tied to the latent four-way potential outcomes $G=(Y_{0,0},Y_{0,1},Y_{1,0},Y_{1,1})$. Identifiability is achieved by leveraging a post-treatment secondary outcome $W$, with a Gaussian-mixture model underpinning the identification under monotonicity, and alternative latent-class distinguishability conditions used when monotonicity does not hold. The framework is applied to the smoking–asbestos lung cancer example, showing substantial attribution to synergistic effects under dual exposure, and extended to a JOBS II study to demonstrate practical applicability and model-selection considerations through AIC. The contributions include a principled causal-attribution methodology for two binary exposures, formal identifiability results with and without monotonicity, and likelihood-based estimation via EM, offering a tool for legal, policy, and epidemiological decision-making where attribution of effects to interacting causes is essential.

Abstract

In contrast to evaluating treatment effects, causal attribution analysis focuses on identifying the key factors responsible for an observed outcome. For two binary exposure variables and a binary outcome variable, researchers need to assess not only the likelihood that an observed outcome was caused by a particular exposure, but also the likelihood that it resulted from the interaction between the two exposures. For example, in the case of a male worker who smoked, was exposed to asbestos, and developed lung cancer, researchers aim to explore whether the cancer resulted from smoking, asbestos exposure, or their interaction. Even in randomized controlled trials, widely regarded as the gold standard for causal inference, identifying and evaluating retrospective causal interactions between two exposures remains challenging. In this paper, we define posterior probabilities to characterize the interactive causes of an observed outcome. We establish the identifiability of posterior probabilities by using a secondary outcome variable that may appear after the primary outcome. We apply the proposed method to the classic case of smoking and asbestos exposure. Our results indicate that for lung cancer patients who smoked and were exposed to asbestos, the disease is primarily attributable to the synergistic effect between smoking and asbestos exposure.

Assessing Interactive Causes of an Occurred Outcome Due to Two Binary Exposures

TL;DR

This work develops a backward-looking causal attribution framework for two binary exposures and a binary outcome by introducing posterior interaction probabilities tied to the latent four-way potential outcomes . Identifiability is achieved by leveraging a post-treatment secondary outcome , with a Gaussian-mixture model underpinning the identification under monotonicity, and alternative latent-class distinguishability conditions used when monotonicity does not hold. The framework is applied to the smoking–asbestos lung cancer example, showing substantial attribution to synergistic effects under dual exposure, and extended to a JOBS II study to demonstrate practical applicability and model-selection considerations through AIC. The contributions include a principled causal-attribution methodology for two binary exposures, formal identifiability results with and without monotonicity, and likelihood-based estimation via EM, offering a tool for legal, policy, and epidemiological decision-making where attribution of effects to interacting causes is essential.

Abstract

In contrast to evaluating treatment effects, causal attribution analysis focuses on identifying the key factors responsible for an observed outcome. For two binary exposure variables and a binary outcome variable, researchers need to assess not only the likelihood that an observed outcome was caused by a particular exposure, but also the likelihood that it resulted from the interaction between the two exposures. For example, in the case of a male worker who smoked, was exposed to asbestos, and developed lung cancer, researchers aim to explore whether the cancer resulted from smoking, asbestos exposure, or their interaction. Even in randomized controlled trials, widely regarded as the gold standard for causal inference, identifying and evaluating retrospective causal interactions between two exposures remains challenging. In this paper, we define posterior probabilities to characterize the interactive causes of an observed outcome. We establish the identifiability of posterior probabilities by using a secondary outcome variable that may appear after the primary outcome. We apply the proposed method to the classic case of smoking and asbestos exposure. Our results indicate that for lung cancer patients who smoked and were exposed to asbestos, the disease is primarily attributable to the synergistic effect between smoking and asbestos exposure.
Paper Structure (31 sections, 3 theorems, 47 equations, 3 figures, 20 tables)

This paper contains 31 sections, 3 theorems, 47 equations, 3 figures, 20 tables.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. Identification of Posterior Probabilities
  4. Identification with the monotonicity assumption
  5. Identification without monotonicity assumption
  6. Analysis of Interactive Effects and Causal Attribution in Lung Cancer
  7. Descriptive and review analysis
  8. Analysis with a secondary outcome
  9. Discussion
  10. Additional graphic illustration
  11. Estimation with monotonicity assumption \ref{['ass:monotonicity']}
  12. EM algorithms
  13. Estimating posterior probabilities
  14. Estimation without monotonicity assumption \ref{['ass:monotonicity']}
  15. Numerical experiments
  16. ...and 16 more sections

Key Result

Theorem 1

Under Assumptions ass:monotonicity, assump:no-confounding, and assumption:normal, the posterior probabilities $\pi_{rstu}$ and $\pi_{rstu}(\mathcal{O})$ are identifiable for $r, s, t, u \in \{0,1\}$, where $\mathcal{O} = (Z=z, M=m, Y=y)$ with $z, m, y \in \{0,1\}$.

Figures (3)

  • Figure 1: The posterior probabilities $\textnormal{pr}(G = rstu \mid Z = z, M = m, Y = 1, W)$ are plotted along the horizontal axis $W$ for four observed scenarios. The vertical dashed lines mark the points where the two curves intersect. The thick black line represents the zero baseline on the vertical axis; for example, in the upper left panel, it covers all cases except the posterior probability $\textnormal{pr}(G = 1111 \mid Z =0, M = 0, Y = 1, W)$.
  • Figure 2: Study subjects (21319) grouped according to questionnaire information about asbestos exposure and smoking habits (shown as boxes), with the number of lung cancers expected to occur in each group during the next decade (shown as columns). Following vanderweele2014tutorial, we classify ex-smokers and current smokers as $Z = 1$ and never smokers as $Z = 0$. We categorize those who have been exposed to asbestos as $M = 1$ and those who have not been exposed as $M = 0$. Here, $Y = 1$ denotes individuals with lung cancer, and $Y = 0$ denotes individuals without lung cancer.
  • Figure 3: The posterior probabilities $\textnormal{pr}(G = rstu \mid Z = z, M = m, Y = 1, W)$ are plotted along the horizontal axis $W$ for four observed scenarios using QOLS as the secondary outcome. The vertical dashed lines mark the points where the two curves intersect. The thick black line represents the zero baseline on the vertical axis; for example, in the upper left panel, it covers all cases except the posterior probability $\textnormal{pr}(G = 1111 \mid Z = 0, M = 0, Y = 1, W)$.

Theorems & Definitions (7)

  • Example 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • proof
  • proof