Graphical models, potential outcomes and causal inference: Comment on Linquist and Sobel

Dear Editor, I read with interest the comment by Lindquist and Sobel (L&S) entitled: “Graphical models, potential outcomes and causal inference” (NeuroImage, 2010) in which they advocate the use of counterfactual language to explicate causal assumptions, and raise doubts on whether graphical models are generally useful for estimating causal effects. Their comment creates the impression, perhaps unintentionally, that counterfactual language is somehow superior, more rigorous or more principled than the graphical language used by structural equation modelers (SEM) in fMRI research. The purpose of this communication is to supplement L&S comment with certain mathematical results regarding the relations between the two notational systems. It has been proven (Balke and Pearl, 1994; Galles and Pearl, 1998; Halpern, 1998; Pearl, 2000, Ch.7) that the two notational systems are logically equivalent in the sense that a theorem in one is a theorem in the other, and an assumption in one has a parallel interpretation in the other. The translation between the two is given by two simple rules (Pearl, 2000, p. 101) that rewrite assumptions conveyed in graphical form into symbolic counterfactual notation. In particular, assumptions A1–A4(b) that L&S present in their paper are faithfully represented by the causal chain Z → X → Y which they aim to replace or supplement.