DAG (Directed Acyclic Graph)

Definition

A DAG (Directed Acyclic Graph) is a graph that visually represents the causal relationships among variables. It is a core tool in causal inference for grasping confounding structure and deciding an identification strategy.

Components:

• Node: represents a variable
• Directed Edge (arrow): represents a direct causal effect ($A \rightarrow B$ means “A affects B”)
• Acyclic: no variable can cause itself (no cycles)

Key features:

• Non-parametric: an arrow can represent any functional form (linear, nonlinear, etc.)
• Qualitative: represents only the existence of an effect, not its magnitude
• Requires domain knowledge: a DAG cannot be constructed from data alone

Three Elementary Structures

The three basic structures that determine how association is transmitted in a DAG:

1. Chain

A → B → C

• Meaning: A indirectly affects C through B
• Association: transmits causal association between A and C
• Conditioning: conditioning on B blocks the A-C association

2. Fork

A ← B → C

• Meaning: B is the common cause of both A and C (a confounder)
• Association: transmits a non-causal (spurious) association between A and C
• Conditioning: conditioning on B blocks the A-C spurious association

3. Inverted Fork / Collider

A → B ← C

• Meaning: B is the effect of both A and C (a Collider)
• Association: no association between A and C (blocked by default)
• Conditioning: conditioning on B creates a spurious association between A and C (collider bias)

Path and Association

Types of Paths

1. Causal Path: a path that follows the direction of the arrows (A → B → C)
2. Non-causal Path: a path that contains a segment going against the direction of the arrows

Rules for Transmitting Association

• A path transmits association unless it is blocked
• Blocking conditions:
  • There is a collider on the path and that collider is not conditioned on
  • There is a non-collider on the path and that variable is conditioned on

Back-door Criterion

Back-door Criterion (Pearl, 1993): a condition for identifying a causal effect

Definition: To identify the causal effect of $X \rightarrow Y$:

1. Block every path beginning with an arrow pointing into $X$ (back-door path)
2. Do not condition on any descendant of $X$

Example:

    Z
   ↙ ↘
  X   Y

• $Z$ is a confounder: $X \leftarrow Z \rightarrow Y$ (back-door path)
• Conditioning on $Z$ makes the causal effect identifiable

Guide to Drawing a DAG

Variables to Include

1. Treatment (independent variable)
2. Outcome (dependent variable)
3. Confounders (common causes)
4. Mediators (treatment → mediator → outcome)
5. Colliders (treatment → collider ← outcome)

Cautions

• Include all relevant variables
• Arrow direction follows causal direction (consider temporal order)
• Mark unmeasured variables too (dashed line or U)

Example: Education and Income

Intelligence
    ↓
Education → Income
    ↑
Intelligence

More precisely:

      Intelligence
       ↙        ↘
  Education  →  Income

• Back-door path: Education ← Intelligence → Income
• Solution: condition on Intelligence to block the back-door path
• Causal effect: Education → Income becomes identifiable

DAG vs SEM

Aspect	DAG	SEM
Parametric	No (qualitative)	Yes (functional form specified)
Focus	Identification	Estimation
Arrows meaning	Any causal effect	Specific functional relationship
Use	Conceptual reasoning	Statistical modeling

Limitations

1. Untestable assumptions: whether the DAG is correct cannot be verified from data
2. Complexity: a real-world DAG quickly becomes complicated
3. Temporal dynamics: a static DAG struggles to represent feedback loops
4. Unmeasured variables: measuring all relevant variables is difficult

Related Concepts

• Confounder - Common cause, creates back-door paths
• Collider - Common effect, creates bias when conditioned on
• Mediator - A variable on the causal pathway
• Back-door Criterion - Conditions for causal effect identification
• d-separation - Rules for determining independence in a DAG
• SCM - Structural Causal Model
• Propensity Score - A method for adjusting for confounding

References

• Pearl, J. (2009). Causality: Models, Reasoning, and Inference
• rohrerThinkingClearlyCorrelations - Psychological applications of DAGs