Causal Graphs and DAGs

A directed acyclic graph (DAG) makes a causal story inspectable. Nodes are variables; an arrow states that the source is a direct cause of the destination relative to the variables represented. “Acyclic” means no variable can eventually cause itself along the directed arrows.

Prerequisites

Complete Counterfactual Thinking. A DAG complements potential outcomes: the graph organizes structural assumptions, while potential outcomes define intervention contrasts and estimands.

Learning goals

• Translate a domain account into a directed acyclic graph.
• Recognize backdoor paths, descendants, and selection mechanisms.
• Explain why observational associations do not discover a unique causal graph.

Build a graph from domain knowledge

For observational Study Sprint enrollment, let $L$ be prior preparedness, $A$ enrollment, and $Y$ final score. Domain knowledge suggests:

$L \rightarrow A, \qquad L \rightarrow Y, \qquad A \rightarrow Y.$

The path $A \leftarrow L \rightarrow Y$ is a backdoor path. It creates association between enrollment and score even apart from the program effect. If $L$ is measured before enrollment with sufficient quality, conditioning on it may block that path. The graph does not show that $L$ is sufficient merely because a column named prior_score exists.

Add $M$ for study hours after enrollment: $A \rightarrow M \rightarrow Y$. If the question is the total effect of assignment, adjusting for $M$ blocks part of the effect. Add $C$ for completing a voluntary post-course survey, influenced by both engagement and outcome. Restricting analysis to $C=1$ can open a noncausal path through this common effect.

Reading, not discovering, the graph

A DAG should be drafted before outcome-driven model selection. Use subject-matter knowledge, timing, measurement processes, and competing plausible explanations. Observational data can contradict some graphs through conditional independences, but many different causal graphs imply the same observational distribution. Therefore a fitted network or correlation pattern does not, by itself, discover causal direction.

Absence of an arrow is a strong claim: no direct causal effect remains at the graph's level of detail. Hidden common causes can be represented with an unobserved node such as $U$. Selection and measurement processes deserve nodes when they determine which values enter the dataset.

Assumptions and limits

For a specified graph, graphical criteria can identify adjustment sets. “Control for everything” is not a criterion: descendants of treatment, mediators, colliders, or their descendants can change the question or introduce bias. A valid set must block relevant noncausal paths without blocking the causal paths that define the target effect.

Graphical identification is conditional on the graph. It does not verify arrow directions, rule out omitted causes, establish positivity, repair measurement error, or guarantee transport to another population.

Common mistakes

• Drawing arrows from whichever variable was measured first rather than from a causal account.
• Using a DAG as a correlation diagram or an estimated path model.
• Omitting the sampling, missingness, or survey-completion mechanism.
• Choosing adjustment variables from p-values alone.
• Assuming one plausible DAG is the only plausible DAG.

Practice

Draw two plausible DAGs for Study Sprint: one where motivation causes both enrollment and score, and one where a tutor recommendation causes enrollment but has no other route to score. For each, identify what additional measurement or design information would distinguish the stories.

Next steps

Confounders, Mediators, and Colliders turns graph positions into adjustment decisions.

Primary sources

Checked 2026-10-15: Greenland, Pearl, and Robins, “Causal Diagrams for Epidemiologic Research” and Hernán and Robins, Causal Inference: What If.