39.1 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).