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Causal Inference: A Tale of Three Frameworks

Linbo Wang, Thomas Richardson, James Robins

TL;DR

The paper addresses how three foundational causal frameworks—potential outcomes, NPSEM-IE, and DAGs—can be understood, translated, and applied in a unified way for interventional questions. It articulates precise translations between frameworks, showing how NPSEM-IE induces potential outcomes and how DAGs can be derived from or embedded within NPSEMs, with SWIGs bridging the perspectives. It discusses when the stronger, cross-world assumptions of NPSEM-IE yield identification advantages and when they may be overly restrictive, highlighting examples in mediation, monotonicity, and additive-noise models. The work offers a practical workflow for causal inference that combines estimands, graphical assumptions, and algebro-geometric identification tools to guide robust analysis across a range of substantive domains.

Abstract

Causal inference is a central goal across many scientific disciplines. Over the past several decades, three major frameworks have emerged to formalize causal questions and guide their analysis: the potential outcomes framework, structural equation models, and directed acyclic graphs. Although these frameworks differ in language, assumptions, and philosophical orientation, they often lead to compatible or complementary insights. This paper provides a comparative introduction to the three frameworks, clarifying their connections, highlighting their distinct strengths and limitations, and illustrating how they can be used together in practice. The discussion is aimed at researchers and graduate students with some background in statistics or causal inference who are seeking a conceptual foundation for applying causal methods across a range of substantive domains.

Causal Inference: A Tale of Three Frameworks

TL;DR

The paper addresses how three foundational causal frameworks—potential outcomes, NPSEM-IE, and DAGs—can be understood, translated, and applied in a unified way for interventional questions. It articulates precise translations between frameworks, showing how NPSEM-IE induces potential outcomes and how DAGs can be derived from or embedded within NPSEMs, with SWIGs bridging the perspectives. It discusses when the stronger, cross-world assumptions of NPSEM-IE yield identification advantages and when they may be overly restrictive, highlighting examples in mediation, monotonicity, and additive-noise models. The work offers a practical workflow for causal inference that combines estimands, graphical assumptions, and algebro-geometric identification tools to guide robust analysis across a range of substantive domains.

Abstract

Causal inference is a central goal across many scientific disciplines. Over the past several decades, three major frameworks have emerged to formalize causal questions and guide their analysis: the potential outcomes framework, structural equation models, and directed acyclic graphs. Although these frameworks differ in language, assumptions, and philosophical orientation, they often lead to compatible or complementary insights. This paper provides a comparative introduction to the three frameworks, clarifying their connections, highlighting their distinct strengths and limitations, and illustrating how they can be used together in practice. The discussion is aimed at researchers and graduate students with some background in statistics or causal inference who are seeking a conceptual foundation for applying causal methods across a range of substantive domains.

Paper Structure

This paper contains 22 sections, 3 theorems, 41 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Three Frameworks for Causal Inference
  3. The Potential Outcomes Framework
  4. Structural Equation Models
  5. Directed Acyclic Graphs
  6. Translating Between the Three Causal Frameworks
  7. The Potential Outcomes and NPSEM-IE Frameworks
  8. From NPSEM-IE to Potential Outcomes.
  9. Implications of missing input variables in NPSEM-IE
  10. From Potential Outcomes to NPSEM-IE via one-step-ahead counterfactuals
  11. NPSEM-IE and Causal DAG
  12. Potential Outcomes and DAGs: Single World Intervention Graphs (SWIGs)
  13. Comparison between the three frameworks
  14. Minimalist vs. mechanistic perspectives on causality
  15. Single-world versus cross-world causal inference
  16. ...and 7 more sections

Key Result

Proposition 1

Any NPSEM-IE implies a unique causal DAG model. In particular, it induces (1) an observational distribution that factorizes according to the DAG induced by the NPSEM; (2) interventional distributions that satisfy the truncated factorization formula eqn:truncated-distribution.

Figures (5)

  • Figure 1: Two example DAGs over variables $L$, $A$, and $Y$.
  • Figure 2: Left: the original model including the mediator $S$. Right: the simplified model where $S$ is removed and its influence is captured via direct arrows.
  • Figure 3: The single world intervention graph (template) corresponding to intervening on $A$ in the causal DAG in Figure \ref{['fig:dag_complete']}.
  • Figure 4: Does the causal DAG model imply $Y(a,b) \perp\!\!\!\perp B \mid Z, A=a$?
  • Figure 5: Mediation model used to derive the mediation formula.

Theorems & Definitions (22)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Definition 1: Markov Properties
  • Remark 9
  • ...and 12 more