A Unified Graphical Criterion for Characterizing the Causal Interpretation of Partial Regression Coefficients in Linear Structural Equation Models

Abstract

This paper provides a graph-based characterization of partial regression coefficients in linear structural equation models. First, we derive a generalized graphical criterion that unifies the d-separation, single-door, and back-door criteria. This criterion provides a generically necessary and sufficient condition under which a partial regression coefficient coincides with a linear causal effect that is not mediated by other explanatory variables. Second, we clarify the mechanism underlying post-treatment bias and provide a quantitative characterization of this bias. This characterization offers a unified framework for analyzing graph structures that induce post-treatment bias, which have previously been studied on a case-by-case basis. These results are derived from the algebraic properties of acyclic directed mixed graphs and do not rely on any specific probability distribution. Consequently, they apply to a broad class of linear structural equation models.

v6: Added Theorem 3.7. v7: Corrected a typo in Definition 2.5. v8: Changed the title. Added a Discussion section. Removed Lemma 5.9. The correction does not affect the main results. v9: Focused the discussion on the main theme. v10: Changed the title