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Decomposition of Probabilities of Causation with Two Mediators

Yuta Kawakami, Jin Tian

TL;DR

This work extends causal mediation analysis for probabilities of causation by introducing path-specific PNS components with two mediators and proving a decomposition of the total PNS into four pathway-specific terms. It develops a nonparametric identification framework under a monotonicity assumption, providing theta-based integral representations to identify the path-specific PNSs and explicit formulas for practical estimation. Through simulations on linear and nonlinear SCMs, the authors demonstrate finite-sample performance and robustness, and they corroborate the approach with a real-world educational dataset, revealing how different causal pathways contribute to outcomes. The paper also discusses extensions to more mediators, path-deactivation interpretations, and potential bounds when monotonicity fails, highlighting the method's relevance for nuanced causal explanations in AI explainability contexts.

Abstract

Mediation analysis for probabilities of causation (PoC) provides a fundamental framework for evaluating the necessity and sufficiency of treatment in provoking an event through different causal pathways. One of the primary objectives of causal mediation analysis is to decompose the total effect into path-specific components. In this study, we investigate the path-specific probability of necessity and sufficiency (PNS) to decompose the total PNS into path-specific components along distinct causal pathways between treatment and outcome, incorporating two mediators. We define the path-specific PNS for decomposition and provide an identification theorem. Furthermore, we conduct numerical experiments to assess the properties of the proposed estimators from finite samples and demonstrate their practical application using a real-world educational dataset.

Decomposition of Probabilities of Causation with Two Mediators

TL;DR

This work extends causal mediation analysis for probabilities of causation by introducing path-specific PNS components with two mediators and proving a decomposition of the total PNS into four pathway-specific terms. It develops a nonparametric identification framework under a monotonicity assumption, providing theta-based integral representations to identify the path-specific PNSs and explicit formulas for practical estimation. Through simulations on linear and nonlinear SCMs, the authors demonstrate finite-sample performance and robustness, and they corroborate the approach with a real-world educational dataset, revealing how different causal pathways contribute to outcomes. The paper also discusses extensions to more mediators, path-deactivation interpretations, and potential bounds when monotonicity fails, highlighting the method's relevance for nuanced causal explanations in AI explainability contexts.

Abstract

Mediation analysis for probabilities of causation (PoC) provides a fundamental framework for evaluating the necessity and sufficiency of treatment in provoking an event through different causal pathways. One of the primary objectives of causal mediation analysis is to decompose the total effect into path-specific components. In this study, we investigate the path-specific probability of necessity and sufficiency (PNS) to decompose the total PNS into path-specific components along distinct causal pathways between treatment and outcome, incorporating two mediators. We define the path-specific PNS for decomposition and provide an identification theorem. Furthermore, we conduct numerical experiments to assess the properties of the proposed estimators from finite samples and demonstrate their practical application using a real-world educational dataset.
Paper Structure (17 sections, 6 theorems, 68 equations, 7 figures, 1 table)

This paper contains 17 sections, 6 theorems, 68 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Backgrounds and Notations
  3. Path-specific PNS with Two Mediators
  4. Identification of Path-Specific PNS with Two Mediators
  5. Numerical Experiments
  6. Application to Real-World
  7. Conclusion
  8. Total orders
  9. Marginal Distribution of Counterfactual under Special Cases
  10. Pathway Representations
  11. Proofs
  12. Path-Specific PNS with Three Mediators
  13. Additional Numerical Experiments
  14. Special Cases
  15. Sensitivity Analysis for Violation of Monotonicity
  16. ...and 2 more sections

Key Result

Lemma 2.1

Daniel2015 Under SCM ${\cal M}_2$ and Assumption SCAS, the conditional CDF of potential outcome $\mathbb{P}(Y_{x,{M}_{x'},{N}_{x",{M}_{x"'}}}\prec y|C=c)$ is given by for any $x, x', x", x"' \in \Omega_X$, $y \in \Omega_Y$, and $c \in \Omega_C$.

Figures (7)

  • Figure 1: A causal graph representing SCM ${\cal M}_1$.
  • Figure 2: A causal graph representing SCM ${\cal M}_2$.
  • Figure 4: Pathway representation of $\text{\normalfont PNS}^{X \rightarrow Y}(y;x',x,{\cal E},c)$.
  • Figure 5: Pathway representation of $\text{\normalfont PNS}^{X \rightarrow {N} \rightarrow Y}(y;x',x,{\cal E},c)$.
  • Figure 6: Pathway representation of $\text{\normalfont PNS}^{X \rightarrow {M} \rightarrow {N} \rightarrow Y}(y;x',x,{\cal E},c)$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Definition 2.1: PNS with Evidence
  • Definition 2.2: CD-PNS, ND-PNS, and NI-PNS with Evidence
  • Lemma 2.1
  • Definition 3.1: Path-specific PNS for decomposition
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Theorem 4.1
  • Theorem 4.2: Identification of path-specific PNS
  • Definition A.1: Total order
  • ...and 7 more