Directed Acyclic Graphs

Summary

A Directed Acyclic Graph (DAG) is a causal diagram representing proposed cause-and-effect relationships between variables. DAGs make confounding explicit and provide algorithmic tools (backdoor adjustment, d-separation) to identify valid adjustment sets — the minimum set of variables to condition on to isolate the causal effect of a treatment on an outcome.

Overview

Causal inference requires more than data — it requires a model of the causal structure. A DAG supplements data with encoded assumptions about which variables cause which, enabling us to:

1. Identify confounding sources (backdoor paths)
2. Determine the optimal adjustment set (which variables to control for)
3. Derive the do-calculus formula for the intervention distribution $P(Y\mid do(X))$

Causal Inference

Causal inference is the process of reasoning and drawing conclusions about cause-and-effect relationships between variables while accounting for potential confounding factors and biases. It asks: “What is the effect of $X$ on $Y$, independent of other variables that affect both?”

DAG Structure

Directed Acyclic Graph (DAG)

A DAG consists of:

• Nodes: variables (treatments, outcomes, confounders, mediators)
• Directed edges (arrows): causal relationships $A\to B$ means ”$A$ causes $B$”
• Acyclic: no variable can cause itself (no cycles)

Key roles:

• Treatment (exposure): the intervention or variable whose causal effect we want to measure
• Outcome: the variable we measure the effect on
• Confounder: a variable that causes both treatment and outcome, creating spurious association

Confounders and the Identification Problem

Confounder

A confounder is a variable $C$ that causes both the treatment $T$ and the outcome $Y$ (i.e., $T\leftarrow C\to Y$). Its presence creates a spurious association between $T$ and $Y$ that does not reflect a direct causal effect.

Example: Gender (G) affects both whether someone takes a drug (D) and their natural recovery (R). Simply observing D and R gives a biased estimate of the drug’s effect.

Approaches to de-confounding:

Method	Applicable	Mechanism
Randomized Controlled Trial (RCT)	Future studies	Breaks $C\to T$ by random assignment
Stratification	Observed data	Compute effects separately per stratum of $C$; weight-average
Conditioning/Backdoor Adjustment	Observed data (DAG-based)	Mathematical formula using observational data
Inverse Probability Weighting	Observed data	See Bayesian Inverse Probability Weighting
Instrumental Variables	Observed data	See Instrumental Variables
Difference-in-Differences	Panel data	See Differences-in-Differences

Paths and Junctions

Path

A path is a sequence of edges connecting treatment $X$ and outcome $Y$ in a DAG, regardless of the direction of the arrows. Example in a DAG $X\leftarrow Z_1\to Z_3\to Y$: one path is $X\leftarrow Z_1\to Z_3\to Y$.

Three Junction Patterns

Fork

A fork at node $B$: $A\leftarrow B\to C$

$B$ is a common cause of $A$ and $C$. Without conditioning on $B$, there is a spurious correlation between $A$ and $C$ (e.g., age → shoe size AND age → reading ability → shoe size correlates with reading ability).

Rule: Conditioning on $B$ blocks the path. Not conditioning leaves it open.

Chain

A chain at node $B$: $A\to B\to C$

$B$ mediates the effect of $A$ on $C$. If we condition on the intermediate variable $B$, we “freeze” it, blocking the flow of information from $A$ to $C$.

Rule: Conditioning on $B$ blocks the path. Not conditioning leaves it open. (Same rule as forks — counterintuitive but true)

Collider

A collider at node $B$: $A\to B\leftarrow C$

$B$ is jointly caused by $A$ and $C$. A collider is naturally blocked (no spurious association between $A$ and $C$). BUT conditioning on $B$ opens the path and creates a spurious association.

Rule: Conditioning on $B$ unblocks the path. Not conditioning leaves it blocked. (Opposite of forks and chains!)

Collider in Action: Sports College

• Sporting ability (S) → Bursary (B) ← Academic ability (A)
• $B$ is a collider: without conditioning, S and A are uncorrelated (athletes are not necessarily more/less academic)
• After conditioning on $B$ (studying only bursary recipients): S and A become negatively correlated! Students with low sports ability must have high academic ability to receive the bursary, and vice versa.
• This is Berkson’s bias (selection bias from conditioning on a collider).

Three Rules for Paths

Path Blocking Rules

1. Fork ($A\leftarrow B\to C$): conditioning on $B$ blocks the path; leaving unconditioned leaves it open
2. Chain ($A\to B\to C$): conditioning on $B$ blocks the path; leaving unconditioned leaves it open
3. Collider ($A\to B\leftarrow C$): conditioning on $B$ opens the path; leaving unconditioned leaves it blocked

Additionally: if a descendant of a collider $B$ is conditioned on, the same effect occurs (the path is opened).

Simpson’s Paradox

When a confounder creates a spurious correlation, we see an aggregate correlation that reverses or changes within subgroups. Fork structures generate Simpson’s paradox: an association that exists “overall” disappears when you condition on the fork node.

Backdoor Criterion and Adjustment Sets

Backdoor Path

A backdoor path from treatment $X$ to outcome $Y$ is any path that starts with an arrow pointing into $X$ (i.e., a path that goes “backwards” from $X$). Equivalently, any path containing a fork with the fork pointing into $X$.

More precisely (Pearl): A back-door path is any path from X to Y that starts with an arrow pointing into X. (The Book of Why, p158)

Front-door paths start with an arrow pointing out of $X$.

Valid Adjustment Set

A valid adjustment set $Z$ is any set of nodes such that, when conditioned on:

1. All backdoor paths from $X$ to $Y$ are blocked
2. At least one front-door path from $X$ to $Y$ remains open
3. No new spurious paths are created

Pearl’s official rules:

1. Block all spurious paths between $X$ and $Y$
2. Leave all directed paths from $X$ to $Y$ unperturbed (or open them if needed)
3. Create no new spurious paths

There can be zero, one, or multiple valid adjustment sets. The optimal adjustment set minimizes the number of variables conditioned on.

Finding Adjustment Sets in a Complex DAG

Given paths:

1. $X\leftarrow Z_1\to Z_3\to Y$ (backdoor, via fork at $Z_1$)
2. $X\leftarrow Z_1\to Z_3\leftarrow Z_2\to Y$ (complex — $Z_3$ is collider here!)
3. $X\leftarrow Z_3\to Y$ (backdoor, via fork at $Z_3$)
4. $X\leftarrow Z_3\leftarrow Z_2\to Y$ (backdoor)
5. $X\to W\leftarrow Y$ (front-door, but $W$ is a collider — naturally blocked)

Analysis:

• Condition on $Z_3$: blocks paths 1, 3, 4. BUT opens path 2 (since $Z_3$ is a collider in path 2).
• Add conditioning on $Z_1$ or $Z_2$: closes path 2.
• Path 5 ($X\to W\leftarrow Y$): $W$ is a collider. Must condition on $W$ to open this front-door path!

Valid adjustment sets: $\{Z_1,Z_3,W\}$ and $\{Z_2,Z_3,W\}$ and $\{Z_1,Z_2,Z_3,W\}$ Optimal: $\{Z_1,Z_3,W\}$ or $\{Z_2,Z_3,W\}$ (minimum size)

Backdoor Adjustment Formula

Backdoor Adjustment (do-Calculus)

If $Z$ is a valid adjustment set, the causal effect of intervention $do(X=x)$ on $Y$ is:

$$P(Y\mid do(X=x))=\sum _{z}P(Y\mid X=x,Z=z)P(Z=z)$$

The left side is an interventional distribution (what would happen if we set $X=x$). The right side is expressed entirely in terms of observational distributions (what we can measure from data).

This is what makes conditioning on a valid adjustment set equivalent to (simulating) a randomized controlled trial on historical observational data.

d-Separation and d-Connection

d-Separation

A path $p$ is d-separated by a set of conditioning nodes $Z$ if and only if:

1. $p$ contains a fork $A\leftarrow B\to C$ or chain $A\to B\to C$ such that middle node $B\in Z$, OR
2. $p$ contains a collider $A\to B\leftarrow C$ such that $B\notin Z$ and no descendant of $B$ is in $Z$

d-separation = the path is blocked by the conditioning set $Z$

d-Connection

A path is d-connected by $Z$ (i.e., unblocked) when:

1. It contains a chain/fork and the middle node is NOT in $Z$, OR
2. It contains a collider and the collider OR a descendant is in $Z$

Note: the empty conditioning set $Z=\varnothing$ is valid — a path with only chains/forks and no colliders is naturally d-connected without conditioning.

Glossary of Additional Terms

Term	Definition
Exogenous variable	A node with no incoming arrows; its causes are outside the model. Denoted $U$ in structural causal models
Endogenous variable	A node with incoming arrows; its causes are inside the model. Denoted $V$
Unobserved confounder	A confounder not measured or included; shown as a dashed U-node with bidirected arrows to both treatment and outcome
Unconditional dependence	A path is naturally open with no conditioning
Unconditional independence	A path is naturally blocked (collider) with no conditioning
Conditional dependence	A path that has been opened by conditioning (on a collider)
Conditional independence	A path that has been closed by conditioning (on a fork/chain middle node)

Key Insights from DAG Analysis

1. More conditioning is not always better: conditioning on a collider opens a spurious path; conditioning on a mediator blocks the causal pathway you want to measure
2. The optimal adjustment set is minimal: condition on just enough to block all backdoors, no more
3. DAGs make assumptions explicit: writing down a DAG requires committing to a causal story, making the assumptions falsifiable
4. Data alone cannot reveal causality: DAGs must supplement the data with domain knowledge

Connections

• The Selection Problem — DAGs formalize the problem of selection bias
• The Experimental Ideal — RCTs break all backdoor paths; DAGs show why
• Bayesian Inverse Probability Weighting — IPW uses the adjustment set from a DAG
• Nonparametric Causal Inference — BART + propensity scores also uses DAGs to identify confounders
• Instrumental Variables — IVs are variables that affect treatment but have no direct arrow to the outcome
• Differences-in-Differences — DiD adjusts for time-invariant confounders not in the DAG

See Also

• The Selection Problem — Potential outcomes framework for the same problem
• Bayesian Inverse Probability Weighting — Practical application of DAG adjustment sets
• Instrumental Variables — When backdoor adjustment is insufficient (unobserved confounders)
• Regression and the CEF — regression is the estimator applied once a valid adjustment set is identified from the DAG
• Conditional Independence Assumption — CIA is the statistical assumption that a valid DAG adjustment set justifies
• PC Algorithm and Constraint-Based Discovery — learning the DAG from data (vs reasoning about a given DAG)