A Unified Graphical Criterion for Characterizing a Linear Causal Interpretation of Partial Regression Coefficients

This paper characterizes the values of partial regression coefficients, defined as projection coefficients onto the space spanned by explanatory variables, for random variables generated by linear structural equation models using graphical structures. 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 the linear causal effect not mediated by other explanatory variables. Second, we reveal the mechanism underlying post-treatment bias and characterize it quantitatively. This provides a unified framework for discussing the graph structures that generate post-treatment bias, which have previously been examined individually, and clarifies the existence of graph structures that cannot be prevented by the conventional concept of path-blocking. These results are based on the algebraic properties of acyclic directed mixed graphs and do not rely on any specific probability distribution, making them applicable to a broad class of linear models.