38.1 Basic Notation and Graph Structures

Directed Acyclic Graphs are composed of basic building blocks that define relationships between variables.

$X \to Z \to Y$

Variable $Z$ mediates the effect of $X$ on $Y$ .
Controlling for $Z$ blocks the indirect effect of $X$ on $Y$ .
Use case in marketing: Email promotion ( $X$ ) → customer interest ( $Z$ ) → purchase ( $Y$ ). Controlling for interest removes the indirect path, isolating the direct impact.

$X \leftarrow Z \to Y$

$Z$ is a confounder, creating a spurious association between $X$ and $Y$ .
To estimate the causal effect of $X$ on $Y$ , $Z$ must be controlled.
Use case in finance: An economic indicator ( $Z$ ) affects both stock investment decisions ( $X$ ) and market returns ( $Y$ ).

Key concept: If $Z$ is not controlled, $X$ and $Y$ may appear correlated due to a shared cause rather than a causal link.

$X \to Z \leftarrow Y$

$Z$ is a collider, and controlling for it induces a spurious association between $X$ and $Y$ .
Do not control for $Z$ or its descendants.
Use case in HR analytics: Two independent hiring factors ( $X$ = education, $Y$ = experience) both influence a decision variable $Z$ (hiring outcome). Conditioning on being hired can create an artificial correlation between education and experience.

Other Concepts

Descendants: Any variable downstream from a node; controlling for a descendant can have similar effects to controlling for the ancestor.
d-Separation: A graphical criterion to determine conditional independence. If all paths between $X$ and $Y$ are blocked by controlling for a set of variables $Z$ , then $X$ is d-separated from $Y$ given $Z$ .