Factorization of the Partial Covariance in Singly-Connected Path Diagrams
This work addresses a theoretical problem in statistics and causal inference for researchers, providing incremental insights into path analysis.
The paper tackles the problem of understanding contributions to partial covariance in singly-connected path diagrams by proving that it factorizes over nodes and edges, enabling determination of each element's contribution and demonstrating that Simpson's paradox cannot occur in such diagrams.
We extend path analysis by showing that, for a singly-connected path diagram, the partial covariance of two random variables factorizes over the nodes and edges in the path between the variables. This result allows us to determine the contribution of each node and edge to the partial covariance. It also allows us to show that Simpson's paradox cannot occur in singly-connected path diagrams.