7.2 Directed Acyclic Graphs (DAGs)

Directed acyclic graphs (DAGs) are another approach that can be used to examine causal inferences from observational data.

• They were developed primarily by the computer scientist Judea Pearl
• DAGs provide a visual representation of causal assumptions.
• Some overlap with structural equation models (SEMs).

Importantly, DAGs can provide insights on

• What variables should be controlled for?
• What variables should not be controlled for?
• In what situations might control worsen causal inference?

7.2.1 Introduction to DAGs

Below is a simple DAG depicting a model in which the relationship between maltreatment externalizing is confounded by a common cause, income.

DAGs consist of nodes (variables) and arrows (edges) between these nodes, which reflect causal relationships.

It is assumed that manipulation of a variable at which an arrow begins (e.g., a manipulation of child maltreatment with income held constant) would change the variable at the end of the arrow (e.g., externalizing).

7.2.2 Introduction to DAGs: Paths

From these two simple building blocks—nodes and arrows—one can visualize more complex situations and trace paths from variable to variable:

Paths

• A simple path leads just from one node to another (income → stress).
• Paths can also contain multiple nodes:
  • income → stress → child maltreatment
• Paths can also travel against the direction indicated by arrows
  • child maltreatment ← stress ← income → externalizing

7.2.3 Introduction to DAGs: Chains

Chains

• Chains have the structure A → B → C.
• Chains can transmit an association between the beginning and end nodes.
  • If income causally affects child maltreatment, and child maltreatment causally affects externalizing, then income and externalizing can be correlated.

7.2.4 Introduction to DAGs: Descendants and Ancestors

Chains: Descendants and Ancestors

• Along a chain, variables that are directly or indirectly causally affected by a certain variable are called its descendants
  • externalizing is a descendant of child maltreatment
• Variables that directly or indirectly affect a certain variable are considered its ancestors.
  • income is an ancestor of support, child maltreatment and internalizing

7.2.5 Introduction to DAGs: Forks

Forks

• Forks have the structure A ← B → C.
• A fork can transmit an association, but it is not causal.
  • In isolation, this fork indicates that child maltreatment and externalizing may be correlated because they share a common cause, income.
• Forks are the causal structure most relevant for the phenomenon of confounding.

7.2.6 Introduction to DAGs: Inverted Forks

Inverted Forks

• Inverted forks have the structure A → B ← C.
• An inverted fork does not transmit an association.
  • In isolation, If child maltreatment and income both affect externalizing, this does not imply that they are in any way correlated.
• Inverted forks are relevant to the problem of collider bias.

7.2.7 Introduction to DAGs: Acyclicity

Acyclicity

• DAGs are acyclic because they do not allow for cyclic paths in which variables become their own ancestors.
  • a variable cannot causally affect itself
• Developmental systems often contain feedback loops and reciprocal relationships.
  • Often feedback loops can be modeled in a DAG by taking the temporal order into account and adding nodes for repeated measures.