43.1 Basic Notations

Basic building blocks of DAG

• Mediators (chains): \(X \to Z \to Y\)

  • controlling for Z blocks (closes) the causal impact of \(X \to Y\)

• Common causes (forks): \(X \leftarrow Z \to Y\)

  • Z (i.e., confounder) is a common cause in which it induces a non-causal association between \(X\) and \(Y\).

  • Controlling for \(Z\) should close this association.

  • \(Z\) d-separates \(X\) from \(Y\) when it blocks (closes) all paths from \(X\) to \(Y\) (i.e., \(X \perp Y |Z\)). This applies to both common causes and mediators.

• Common effects (colliders): \(X \to Z \leftarrow Y\)

  • Not controlling for \(Z\) does not induce an association between \(X\) and \(Y\)

  • Controlling for \(Z\) induces a non-causal association between \(X\) and \(Y\)

Notes:

• A descendant of a variable behavior similarly to that variable (e.g., a descendant of \(Z\) can behave like \(Z\) and partially control for \(Z\))

• Rule of thumb for multiple Controls: o have Causal inference \(X \to Y\), we must

  • Close all backdoor path between \(X\) and \(Y\) (to eliminate spurious correlation)

  • Do not close any causal path between \(X\) and \(Y\) (any mediators).