Causal reasoning in difference graphs

TL;DR

This paper introduces difference graphs as a framework to compare causal mechanisms across populations and derives identifiability conditions for causal changes using these graphs. It provides nonparametric identifiability results for total effects (and total causal changes) via a common back-door under a no-hidden-confounding assumption, and linear-setting identifiability results for direct effects (and direct causal changes) via a common single-door, with extensions to cycles. The work includes theoretical lemmas and theorems, plus small simulation studies that illustrate practical adjustment-set choices. It positions difference graphs within the broader causal-graph literature and suggests avenues for future work on nonparametric direct/path-specific effects and handling hidden confounding.

Abstract

Understanding causal mechanisms across different populations is essential for designing effective public health interventions. Recently, difference graphs have been introduced as a tool to visually represent causal variations between two distinct populations. While there has been progress in inferring these graphs from data through causal discovery methods, there remains a gap in systematically leveraging their potential to enhance causal reasoning. This paper addresses that gap by establishing conditions for identifying causal changes and effects using difference graphs. It specifically focuses on identifying total causal changes and total effects in a nonparametric setting, as well as direct causal changes and direct effects in a linear setting. In doing so, it provides a novel approach to causal reasoning that holds potential for various public health applications.

Figures (3)

• Figure 1: Three difference graphs ((c), (h), and (m)), each associated with two pairs of causal DAGs (one pair on the left and one on the right). The two causal DAGs in each pair share the same topological ordering, ensuring that Assumption \ref{['assumption:order']} is satisfied. In each subfigure, red and blue vertices represent the cause and effect of interest. In (c), neither the total nor the direct effect are identifiable. In (h), only the total effect is identifiable, while in (m), only the direct effect is identifiable.
• Figure 2: Three difference graphs ((c), (f), and (k)), each associated with two pairs of causal DAGs (one pair on the left and one on the right). The two causal DAGs in each pair do not share the same topological ordering, therefore Assumption \ref{['assumption:order']} is not satisfied. In each subfigure, the red and blue vertices represent the cause and effect of interest, respectively. In the first difference graph (c), neither the total nor the direct effect is identifiable. In the second graph (f), only the total effect is identifiable, while in the third graph (k), only the direct effect is identifiable.
• Figure 3: Mean absolute error and standard deviation for estimating the total and direct causal changes of $X$ on $Y$. Subfigure (a) compares errors in total causal change estimation using (i) the set of parents of $X$, (ii) a minimal set satisfying the back-door criterion, and (iii) the set derived from Theorem \ref{['theorem:total_effect']} and \ref{['theorem:total_effect_cycle']} using a difference graph. Subfigure (b) evaluates direct causal change estimation using (i) the set of parents of $Y$, (ii) a minimal set satisfying the single-door criterion, and (iii) the set from Theorem \ref{['theorem:direct_effect']} and \ref{['theorem:direct_effect_cycle']}. Results are reported over $100$ pairs of dataset, consisting of $1000$ observations and $10$ variables, with each $10$ pairs sharing the same difference graph for which the causal effect of interest is identifiable by our theorems. Each dataset is simulated using a linear SCM, where coefficients are randomly selected between $0$ and $1$, with Gaussian noise.

Theorems & Definitions (24)

• Definition 1: Difference graphs
• Definition 2: Causal effect identifiability using difference graph
• Definition 3: Back-door criterion, Pearl_1995
• Definition 4: Single-door criterion, Spirtes_1998Pearl_1998Pearl_2000
• Theorem 1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof