SEM and Neyman-Rubin potential-outcome model are considered as two major statistical tools for explaining the causal relationships of two variables. However, as Pearl（1997） says,SEM is used by many people and understood by few, while potential-outcome model is understood by few and used by fewer. People apply SEMs to solve the problems in their academic field without guaranteeing their power of causal interpretation. Therefore, as Pearl points out, “ the current dominating philosophy treats SEM as just a convenient way to encode density functions (in economics) or covariance information (in social science). ” So John Fox introduces a cynical view of SEMs in the appendix on structural equation model of his book（http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-sems.pdf） that the popularity of SEMs may be attributed to their pretentious ability of causal interpretation of observational data although they’re not less problematic than other regression models, or they just translate the informal thinking of causal relationships into a formal data analysis.
However, Pearl（2000） believes there is causal interpretations for observational data from SEMs. He argues that almost all the academic educations and publications overlook the presuming condition raised by the fathers of SEM, that is in a equation like y=βx + ε, if we make sure that it is structural, β should be the unique causal connection between x and y, the statistical relationships of x and εcan not be changed because of the different interpretation of β. That is the condition called self-containment. In the language of graph favored by Pearl, it is also named d-separation.
d-separation is clarified by Pearl in his blog article “d-Separation Without Tears” （http://www.mii.ucla.edu/causality/?m=200001） as a criteria to determine whether a set X of variables is independent of a set Y of variables given a third set Z. If X and Y satisfy the conditions of d-separation, we can say X and Y is d-separated or disconnected. If we use the language of graph, x and y is d-connected if there is an unblocked path between them. In the graph
\inline $x\rightarrow m\rightarrow o\rightarrow n\rightarrow y$ , x and y is d-connected conditioned on the set of Z, if X and Y have no collider-tree path that traverses the member of Z, we can say X and Y is d-separated by Z. if r and v are the members of Z like graph \inline $x\rightarrow m'\rightarrow o\rightarrow n'\rightarrow y$ , X and Y is d-separated by Z. however, if a collider is the member of conditioning set Z, like m’ and v’ in the paragraph below, s and y is d-connected because t has p’ as the member of z.
（quoted from Pearl’s blog article “d-Separation Without Tears” http://www.mii.ucla.edu/causality/?m=200001)
In another words, x and y is d-separated by z means there is the null partial correlation between x and y by z.
D-separation is the a key assumption for the causal faithfulness, which promises every conditional independence of variable in the causal relationship so that the causal structure can be testable. Therefore, d-separation is used to make the causal interpretation of SEMs more reliable. If we test the causality of structural equation model, we should carefully test the self-containment of equation by using d-separation to compute the conditional independence relations. Now I just primarily understand the conception of d-separation. A lot of efforts are still needed for me to study the statistical reading of causality by applying d-separation.
Reference:
d-separation, http://www.andrew.cmu.edu/user/scheines/tutor/d-sep.html#d-sepapplet2
Pearl’s blog article “d-Separation Without Tears” January 1, 2000
Judea Pearl (2000), Causality: Models of Reasoning and Inference, Cambridge University Press
John Fox, Web appendix to “An R and S-PLUS Companion to Applied Regression”, Structural-Equation Models http://socserv.mcmaster.ca/jfox/Books/Companion/appendix-sems.pdf
Pearl(1997), The New Challenge: From a Century of Statistics to the Age of Causation，Computing Science and Statistics，29，415--423
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