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Mediation Analysis for Probabilities of Causation

Yuta Kawakami, Jin Tian

TL;DR

The paper extends probabilities of causation ($PoC$) by introducing direct and indirect PNS variants (CD-PNS, ND-PNS, NI-PNS) to isolate treatment necessity and sufficiency across causal pathways via mediators. It develops identification theorems under monotonicity and sequential ignorability, including versions with post-treatment evidence, and expresses the quantities through counterfactual CDFs that are estimable from observational data. The authors validate the framework with simulations and apply it to the JOBS II dataset, revealing that the overall effect is largely mediated through the mediator, with explicit quantification of direct vs indirect contributions and subpopulation-specific insights. This work broadens PoC analysis to mediation contexts, enabling nuanced, pathway-specific decision-making and interpretability in complex causal structures, with practical guidance for estimation and reporting.

Abstract

Probabilities of causation (PoC) offer valuable insights for informed decision-making. This paper introduces novel variants of PoC-controlled direct, natural direct, and natural indirect probability of necessity and sufficiency (PNS). These metrics quantify the necessity and sufficiency of a treatment for producing an outcome, accounting for different causal pathways. We develop identification theorems for these new PoC measures, allowing for their estimation from observational data. We demonstrate the practical application of our results through an analysis of a real-world psychology dataset.

Mediation Analysis for Probabilities of Causation

TL;DR

The paper extends probabilities of causation () by introducing direct and indirect PNS variants (CD-PNS, ND-PNS, NI-PNS) to isolate treatment necessity and sufficiency across causal pathways via mediators. It develops identification theorems under monotonicity and sequential ignorability, including versions with post-treatment evidence, and expresses the quantities through counterfactual CDFs that are estimable from observational data. The authors validate the framework with simulations and apply it to the JOBS II dataset, revealing that the overall effect is largely mediated through the mediator, with explicit quantification of direct vs indirect contributions and subpopulation-specific insights. This work broadens PoC analysis to mediation contexts, enabling nuanced, pathway-specific decision-making and interpretability in complex causal structures, with practical guidance for estimation and reporting.

Abstract

Probabilities of causation (PoC) offer valuable insights for informed decision-making. This paper introduces novel variants of PoC-controlled direct, natural direct, and natural indirect probability of necessity and sufficiency (PNS). These metrics quantify the necessity and sufficiency of a treatment for producing an outcome, accounting for different causal pathways. We develop identification theorems for these new PoC measures, allowing for their estimation from observational data. We demonstrate the practical application of our results through an analysis of a real-world psychology dataset.

Paper Structure

This paper contains 38 sections, 10 theorems, 107 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Background
  3. Structural causal models (SCM).
  4. Probabilities of causation (PoC).
  5. Causal mediation analysis.
  6. Direct and Indirect PNS
  7. Definition of CD-PNS, ND-PNS, and NI-PNS
  8. Remark 1.
  9. Remark 2.
  10. Identification of CD-PNS, ND-PNS, and NI-PNS
  11. Assumptions
  12. Lemmas.
  13. Identification theorems.
  14. Direct and Indirect PNS with Evidence
  15. Definitions of CD-PNS, ND-PNS, and NI-PNS with evidence
  16. ...and 23 more sections

Key Result

Proposition 1

Imai2010aVanderWeele2014 Under SCM ${\cal M}$ and Assumption SCAS2, the counterfactual $\mathbb{P}(Y_{x',M_{x}}\prec y|C=c)$ is identifiable by for any $x', x \in \Omega_X$, $y \in \Omega_Y$, and $c \in \Omega_C$.

Figures (7)

  • Figure 1: A causal graph representing SCM ${\cal M}$.
  • Figure 3: Plots of T-PNS, ND-PNS, and NI-PNS. The solid line is T-PNS, the dashed line is NI-PNS, and the dotted line is NI-PNS. The x-axis represents the value of $C$, and the y-axis represents the values of T-PNS, ND-PNS, and NI-PNS, respectively.
  • Figure 4: Plots of T-PN, ND-PN, and NI-PN. The solid line is T-PN, the dashed line is NI-PN, and the dotted line is NI-PN. The x-axis represents the value of $C$, and the y-axis represents the values of T-PN, ND-PN, and NI-PN, respectively.
  • Figure 5: Plots of T-PS, ND-PS, and NI-PS. The solid line is T-PS, the dashed line is NI-PS, and the dotted line is NI-PS. The x-axis represents the value of $C$, and the y-axis represents the values of T-PS, ND-PS, and NI-PS, respectively.
  • Figure 6: Plots of TE, NDE, and NIE. The solid line is TE, the dashed line is NIE, and the dotted line is NIE. The x-axis represents the value of $C$, and the y-axis represents the values of TE, NDE, and NIE, respectively.
  • ...and 2 more figures

Theorems & Definitions (27)

  • Definition 1: PoC
  • Definition 2: TE, CDE, NDE, and NIE
  • Proposition 1: Identification of $\mathbb{P}(Y_{x',M_{x}}\prec y|C=c)$
  • Definition 3: CD-PNS, ND-PNS, and NI-PNS
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Theorem 1: Identification of CD-PNS
  • Theorem 2: Identification of ND-PNS and NI-PNS
  • Definition 4: CD-PNS, T-PNS, ND-PNS, and NI-PNS with evidence
  • ...and 17 more