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Moments of Causal Effects

Yuta Kawakami, Jin Tian

TL;DR

The paper advances causal inference by defining and identifying the moments and product moments of causal effects within an SCM framework, leveraging PoC results to express moments via conditional distributions. It provides Fréchet-based bounds that relax monotonicity and demonstrates consistent estimation from finite samples, including applications to real medical data. The work reveals how higher-order moments and covariances capture heterogeneity and inter-treatment relationships beyond the ACE, with practical implications for personalized policies. Overall, it offers a comprehensive toolkit for distributional analysis of causal effects and motivates future work on sensitivity analysis and tighter bounding techniques.

Abstract

The moments of random variables are fundamental statistical measures for characterizing the shape of a probability distribution, encompassing metrics such as mean, variance, skewness, and kurtosis. Additionally, the product moments, including covariance and correlation, reveal the relationships between multiple random variables. On the other hand, the primary focus of causal inference is the evaluation of causal effects, which are defined as the difference between two potential outcomes. While traditional causal effect assessment focuses on the average causal effect, this work provides definitions, identification theorems, and bounds for moments and product moments of causal effects to analyze their distribution and relationships. We conduct experiments to illustrate the estimation of the moments of causal effects from finite samples and demonstrate their practical application using a real-world medical dataset.

Moments of Causal Effects

TL;DR

The paper advances causal inference by defining and identifying the moments and product moments of causal effects within an SCM framework, leveraging PoC results to express moments via conditional distributions. It provides Fréchet-based bounds that relax monotonicity and demonstrates consistent estimation from finite samples, including applications to real medical data. The work reveals how higher-order moments and covariances capture heterogeneity and inter-treatment relationships beyond the ACE, with practical implications for personalized policies. Overall, it offers a comprehensive toolkit for distributional analysis of causal effects and motivates future work on sensitivity analysis and tighter bounding techniques.

Abstract

The moments of random variables are fundamental statistical measures for characterizing the shape of a probability distribution, encompassing metrics such as mean, variance, skewness, and kurtosis. Additionally, the product moments, including covariance and correlation, reveal the relationships between multiple random variables. On the other hand, the primary focus of causal inference is the evaluation of causal effects, which are defined as the difference between two potential outcomes. While traditional causal effect assessment focuses on the average causal effect, this work provides definitions, identification theorems, and bounds for moments and product moments of causal effects to analyze their distribution and relationships. We conduct experiments to illustrate the estimation of the moments of causal effects from finite samples and demonstrate their practical application using a real-world medical dataset.
Paper Structure (19 sections, 14 theorems, 99 equations, 2 tables)

This paper contains 19 sections, 14 theorems, 99 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and Backgrounds
  3. Moments of Causal Effects
  4. Definition of the moments of causal effects
  5. Identification of the moments of causal effects
  6. Bounding the moments of causal effects
  7. Product Moments of Causal Effects
  8. Definition of the product moment of causal effects
  9. Identification of the product moment of causal effects
  10. Bounding the product moment of causal effects
  11. Numerical Experiments
  12. Application to Real-World
  13. Conclusion
  14. Proofs
  15. Identification and Bounds of the Central Moments of Causal Effects
  16. ...and 4 more sections

Key Result

Lemma 1

Under SCM ${\cal M}$, we have

Theorems & Definitions (37)

  • Definition 1: The moments of causal effects
  • Definition 2: The central moment of causal effects
  • Lemma 1
  • Theorem 1: Identification of the moments of causal effect
  • Lemma 2
  • Theorem 2: Bounds of the moments of causal effect
  • Definition 3: The product moment of causal effects
  • Definition 4: Covariance of causal effects
  • Definition 5: Correlation of causal effects
  • Lemma 3
  • ...and 27 more