The Central Idea
A causal diagram represents a proposed causal system using variables and directed arrows. If changing \(X\) could change \(Y\), possibly through intermediate steps, the diagram may include
\[ X\longrightarrow Y. \]The arrow is a causal claim, not merely a statement that \(X\) and \(Y\) are correlated. It says that \(X\) is a direct cause of \(Y\) relative to the variables represented in the diagram.
\[ \boxed{ \text{diagram} = \text{variables} + \text{causal arrows} + \text{explicit assumptions} } \]The value of the diagram is not decorative. It helps determine which variables should be measured, randomized, controlled, adjusted for, or deliberately left unconditioned.
Nodes and Arrows
A node represents a variable, state, event, or characteristic. A directed edge, usually called an arrow, points from a proposed cause to a proposed direct effect.
| Symbol | Interpretation |
|---|---|
| \(X\longrightarrow Y\) | \(X\) is proposed to directly influence \(Y\) |
| \(X\longleftarrow C\longrightarrow Y\) | \(C\) is a common cause of \(X\) and \(Y\) |
| \(X\longrightarrow M\longrightarrow Y\) | \(M\) carries part or all of the effect of \(X\) to \(Y\) |
| \(X\longrightarrow S\longleftarrow Y\) | \(S\) is a common effect of \(X\) and \(Y\) |
The absence of an arrow is also an assumption: it states that no direct causal effect is needed between those nodes after the represented variables are taken into account.
Direct and Indirect Effects
A direct effect follows one arrow:
\[ X\longrightarrow Y. \]An indirect effect follows a directed path through one or more intermediate variables:
\[ X\longrightarrow M\longrightarrow Y. \]In a system containing both paths,
\[ \begin{aligned} X&\longrightarrow Y,\\ X&\longrightarrow M\longrightarrow Y, \end{aligned} \]the total effect of \(X\) on \(Y\) may contain both a direct component and an indirect component operating through \(M\).
Directed Acyclic Graphs
Many introductory causal diagrams are directed acyclic graphs, abbreviated DAGs.
- Directed: every arrow has a direction.
- Acyclic: no directed path returns to its starting node.
- Graph: the structure consists of nodes and edges.
The following is acyclic:
\[ A\longrightarrow B\longrightarrow C. \]The following contains a directed cycle and is not a DAG:
\[ A\longrightarrow B\longrightarrow C\longrightarrow A. \]Real systems can contain feedback. A DAG can sometimes represent feedback by indexing variables across time:
\[ A_1\longrightarrow B_2\longrightarrow A_3. \]The time-indexed diagram remains acyclic because causes precede their later effects.
Time Gives Arrows a Necessary Direction
A cause must occur before its effect. Baseline characteristics can influence treatment and later outcomes, but a later outcome cannot cause an earlier baseline value.
\[ C_{\text{baseline}} \longrightarrow X_{\text{treatment}} \longrightarrow Y_{\text{follow-up}}. \]Temporal order rules out impossible arrows, but it does not determine every remaining arrow. Scientific knowledge is still needed to decide whether a plausible earlier variable actually causes a later one.
Three Fundamental Path Structures
Nearly every introductory adjustment problem can be understood by recognizing chains, forks, and colliders.
| Structure | Diagram | Middle Variable | Usual Effect of Conditioning |
|---|---|---|---|
| Chain | \(X\to M\to Y\) | Mediator | Blocks the indirect path through \(M\) |
| Fork | \(X\leftarrow C\to Y\) | Common cause or confounder | Blocks the confounding path through \(C\) |
| Collider | \(X\to S\leftarrow Y\) | Common effect | Opens a path that was closed |
Conditioning means restricting, stratifying, matching, controlling, selecting, or statistically adjusting based on a variable.
Chains and Mediation
In the chain
\[ X\longrightarrow M\longrightarrow Y, \]\(M\) is a mediator. Part or all of the effect of \(X\) on \(Y\) passes through \(M\).
Suppose a training program \(X\) increases skill \(M\), which increases earnings \(Y\). If the goal is the total effect of the program on earnings, adjusting for post-program skill blocks part of the effect being estimated.
If the goal is instead to separate direct and indirect effects, mediation analysis requires additional assumptions. The diagram helps state the new question but does not make those assumptions automatically true.
Forks and Common Causes
In the fork
\[ X\longleftarrow C\longrightarrow Y, \]\(C\) is a common cause. The path creates an association between \(X\) and \(Y\) even without a causal arrow from \(X\) to \(Y\).
Suppose warm weather \(C\) increases both ice cream sales \(X\) and swimming exposure, which affects drowning incidents \(Y\). Comparing days with similar temperature can close the common-cause path and clarify the relationship of interest.
Colliders and Selection
In the collider
\[ X\longrightarrow S\longleftarrow Y, \]\(S\) is a common effect. Before conditioning on \(S\), the path through \(S\) is closed. Conditioning on \(S\), or on one of its descendants, can open the path and create a noncausal association between \(X\) and \(Y\).
Suppose academic ability \(X\) and athletic ability \(Y\) both increase admission \(S\) to a selective school. Among admitted students, lower academic ability may need to be offset by higher athletic ability, creating a negative association even if the abilities are independent among all applicants.
This is why adjusting for every available variable is unsafe.
Paths and Directed Paths
A path is a sequence of connected nodes, ignoring arrow direction while tracing the connection. A directed path follows arrows consistently from cause toward effect.
In
\[ C\longrightarrow X\longrightarrow M\longrightarrow Y, \]\(X\to M\to Y\) is a directed causal path from \(X\) to \(Y\). The connection \(X\leftarrow C\to Y\), if the \(C\to Y\) arrow also exists, is a nondirected path between \(X\) and \(Y\) because the first arrow points into \(X\).
Backdoor Paths
A backdoor path from treatment or exposure \(X\) to outcome \(Y\) begins with an arrow pointing into \(X\). It represents a noncausal route by which \(X\) and \(Y\) can become associated.
\[ X\longleftarrow C\longrightarrow Y. \]This path enters \(X\) through the back door. Adjusting for the common cause \(C\) blocks the path, provided the diagram is otherwise adequate and \(C\) is measured appropriately.
The main design question is:
\[ \boxed{ \text{Which pre-treatment variables block all open backdoor paths from }X\text{ to }Y? } \]Open and Blocked Paths
A path is open when it can transmit association under the current conditioning choices. A path is blocked when its structure prevents association from traveling through it.
In an introductory DAG:
- conditioning on a noncollider in a chain blocks that path;
- conditioning on a noncollider in a fork blocks that path;
- a collider blocks a path when neither it nor its descendants are conditioned on; and
- conditioning on a collider or its descendant can open that path.
These rules form the basic logic of d-separation, the graphical criterion used to determine which variables are separated by a conditioning set.
Adjustment Sets
An adjustment set is a collection of variables conditioned on to identify a causal effect under a stated diagram.
Consider
\[ \begin{aligned} C&\longrightarrow X,\\ C&\longrightarrow Y,\\ X&\longrightarrow M\longrightarrow Y,\\ X&\longrightarrow Y. \end{aligned} \]To estimate the total effect of \(X\) on \(Y\), adjusting for \(C\) blocks the backdoor path. Adjusting for \(M\) would block an intended causal pathway and change the estimand.
A valid set is not necessarily unique. One should generally prefer a sufficient, scientifically defensible set that avoids unnecessary variables and preserves adequate overlap.
Minimal Adjustment
A minimal sufficient adjustment set blocks the relevant noncausal paths, and no proper subset of it does so.
Minimal does not mean smallest at any cost. Measurement quality, missingness, overlap, precision, and the credibility of the diagram all matter. Still, avoiding unnecessary adjustment can reduce model dependence and prevent accidental control of mediators or colliders.
How Random Assignment Appears in a Diagram
Let \(R\) be a randomized treatment assignment. Baseline variables may affect the outcome, but they do not cause \(R\):
\[ R\longrightarrow Y, \qquad C\longrightarrow Y, \]with no baseline arrow \(C\to R\). Randomization removes systematic backdoor paths into treatment assignment.
Post-assignment processes can still create new problems:
\[ R\longrightarrow A\longrightarrow Y, \]where \(A\) may represent adherence, and
\[ R\longrightarrow S\longleftarrow Y, \]where \(S\) may represent remaining under observation. Attrition, noncompliance, treatment switching, and measurement can complicate the effect of assignment.
Unmeasured Variables
An unmeasured variable is often represented by \(U\). For example,
\[ X\longleftarrow U\longrightarrow Y \]states that an unmeasured common cause confounds the relationship between \(X\) and \(Y\).
Drawing \(U\) does not solve the problem. It documents that identification of the causal effect requires another strategy, stronger assumptions, better measurement, a valid proxy, a natural experiment, an instrumental variable, or sensitivity analysis.
Omitting \(U\) from the page does not make the unmeasured cause disappear.
Latent Variables and Proxies
A latent variable is not observed directly. A proxy variable is an observed measurement related to that latent construct.
If socioeconomic resources \(U\) are imperfectly represented by neighborhood income \(P\), a diagram might include
\[ U\longrightarrow P, \qquad U\longrightarrow X, \qquad U\longrightarrow Y. \]Adjusting for \(P\) may reduce confounding without eliminating it because the proxy does not fully measure \(U\). The diagram makes the residual pathway visible.
Selection Nodes
A selection indicator \(S\) can represent entering, remaining in, responding to, or being analyzed in a study. If both exposure and outcome affect selection,
\[ X\longrightarrow S\longleftarrow Y, \]then restricting analysis to \(S=1\) conditions on a collider.
Selection may also depend on causes of \(X\) and \(Y\). A full diagram should represent the actual pathway rather than labeling all missingness as one generic bias.
Measurement Nodes
A diagram can distinguish a true variable \(Y\) from its recorded measurement \(Y^*\):
\[ Y\longrightarrow Y^*. \]If treatment knowledge \(K\) also influences the recorded outcome,
\[ K\longrightarrow Y^*, \]then \(Y^*\) mixes the true outcome with treatment-aware measurement. Blinding an assessor aims to remove or weaken the arrow \(K\to Y^*\).
Measurement diagrams help separate bias in the causal system from bias in the observation system.
What an Ordinary DAG Does Not Show
A basic DAG shows the existence and direction of proposed causal relationships. It does not ordinarily show:
- the numerical size of an effect;
- whether an effect is positive or negative;
- the functional form of the relationship;
- the amount of random variation;
- the reliability of a measurement;
- the certainty that an arrow is correct; or
- the population and time period without additional labels.
Two diagrams with the same arrows may describe very different effect sizes and statistical models.
The Diagram Is Not Discovered by Correlation Alone
A correlation matrix does not determine arrow direction. The same observed association between \(X\) and \(Y\) may be compatible with
\[ X\longrightarrow Y, \qquad X\longleftarrow Y, \qquad X\longleftarrow U\longrightarrow Y. \]Time order, experimental intervention, domain knowledge, institutional rules, natural variation, and prior evidence help distinguish these structures. Data can challenge a diagram, but a diagram is never a picture of correlations alone.
How to Build an Introductory Causal Diagram
- State the causal question. Which treatment or exposure and which outcome?
- Define the time order. Which variables occur before treatment, during treatment, and afterward?
- List common causes. What prior variables could influence both exposure and outcome?
- List mediators. Through which mechanisms might the treatment affect the outcome?
- List selection processes. What determines entry, retention, response, or analysis?
- List measurement processes. What determines the recorded versions of the variables?
- Draw causal arrows. Add an arrow only when a direct causal claim is intended.
- Check for cycles. Add time indices when feedback unfolds over time.
- Identify open backdoor paths. Determine what must be blocked.
- Choose an adjustment set. Avoid mediators and colliders for a total-effect question.
Example: Exercise and Blood Pressure
Let \(X\) be exercise participation and \(Y\) later blood pressure. Age \(A\) affects both exercise participation and blood pressure. Exercise may reduce body mass \(M\), which may reduce blood pressure.
\[ \begin{aligned} A&\longrightarrow X,\\ A&\longrightarrow Y,\\ X&\longrightarrow M\longrightarrow Y,\\ X&\longrightarrow Y. \end{aligned} \]What Is the Backdoor Path?
\[ X\longleftarrow A\longrightarrow Y. \]What Should Be Adjusted for?
For the total effect of exercise on blood pressure, adjust for baseline age \(A\) to block the backdoor path.
What Should Not Be Adjusted for Automatically?
Body mass \(M\), measured after exercise begins and affected by it, is a mediator. Adjusting for \(M\) would remove part of the total effect.
Example: Tutoring, Motivation, and Test Scores
In an observational study, motivation \(C\) affects participation in tutoring \(X\) and later test scores \(Y\). Tutoring also affects study time \(M\), which affects the score.
\[ \begin{aligned} C&\longrightarrow X,\\ C&\longrightarrow Y,\\ X&\longrightarrow M\longrightarrow Y. \end{aligned} \]Motivation is a confounder; study time is a mediator. Adjusting for motivation helps compare similarly motivated students. Adjusting for post-tutoring study time changes the question from total tutoring effect to an effect not transmitted through study time.
If motivation is poorly measured, residual confounding may remain.
Example: Hospital-Based Selection
Suppose exposure \(X\) and disease \(Y\) both increase the probability of hospitalization \(S\):
\[ X\longrightarrow S\longleftarrow Y. \]A study including only hospitalized patients conditions on \(S=1\). Even if \(X\) does not cause \(Y\), exposure and disease can become associated within the hospital sample.
The diagram shows that adding hospitalization as a regression control does not solve the problem; the sample was already selected through the collider.
One Variable, Different Roles
The label of a variable does not determine its causal role. Consider stress \(Z\):
| Diagram | Role of \(Z\) | Implication for a Total Effect |
|---|---|---|
| \(X\leftarrow Z\to Y\) | Confounder | Often adjust for \(Z\) |
| \(X\to Z\to Y\) | Mediator | Do not adjust if estimating the total effect |
| \(X\to Z\leftarrow Y\) | Collider | Do not condition on \(Z\) or its descendants |
The research question, timing, and arrow structure determine the role.
Every Diagram Is an Assumption Set
A causal diagram should be treated as a transparent proposal, not as established fact. Good practice includes:
- explaining why each important arrow is included;
- explaining why important alternative arrows are excluded;
- including plausible unmeasured common causes;
- comparing multiple reasonable diagrams;
- consulting subject-matter experts;
- using time and institutional rules to constrain directions; and
- testing how adjustment choices change across plausible diagrams.
Disagreement about a diagram is useful when it exposes disagreement about the causal system.
Common Errors
Drawing Arrows for Correlations
An arrow asserts direct causal influence relative to the graph. Correlation alone does not justify its direction.
Adjusting for Every Node
Confounders, mediators, and colliders require different treatment. More adjustment is not automatically better adjustment.
Leaving Time Unspecified
A variable measured before treatment and the same named variable measured after treatment may occupy different causal roles. Time indices prevent silent reversal.
Ignoring Selection and Measurement
A diagram containing only treatment and true outcome can miss how units enter the analysis and how variables become recorded data.
Believing the Diagram Proves Causation
A DAG organizes causal assumptions and their implications. The credibility of the conclusion still depends on design, measurement, data, and the assumptions.
Guided Practice
Problem 1
In \(X\leftarrow C\to Y\), identify the role of \(C\) and state the usual adjustment decision for estimating the effect of \(X\) on \(Y\).
Show the solution
\(C\) is a common cause and potential confounder. Adjusting for \(C\) blocks the backdoor path \(X\leftarrow C\to Y\), assuming the diagram is adequate and \(C\) is measured appropriately.
Problem 2
In \(X\to M\to Y\), what happens to the total-effect question if the analysis adjusts for \(M\)?
Show the solution
\(M\) is a mediator. Adjusting for it blocks the indirect path through \(M\) and removes that component from the estimated effect. The analysis no longer estimates the total effect without additional structure and assumptions.
Problem 3
In \(X\to S\leftarrow Y\), is the path through \(S\) open or blocked before conditioning? What happens after selecting only \(S=1\)?
Show the solution
The collider blocks the path before conditioning. Selecting only \(S=1\) conditions on the collider and can open the path, creating a noncausal association between \(X\) and \(Y\).
Problem 4
A randomized treatment \(R\) and a baseline variable \(C\) both affect outcome \(Y\). Should the graph contain \(C\to R\)?
Show the solution
Under valid random assignment, baseline \(C\) does not cause treatment assignment \(R\), so the graph should not include \(C\to R\). It may include \(C\to Y\) because baseline characteristics can affect the outcome.
Problem 5
Consider \(C\to X\), \(C\to Y\), \(X\to M\to Y\), and \(X\to Y\). Which variable should be adjusted for to estimate the total effect of \(X\) on \(Y\): \(C\), \(M\), both, or neither?
Show the solution
Adjust for \(C\) to block the backdoor path. Do not adjust for \(M\) when estimating the total effect because \(M\) lies on a causal pathway from \(X\) to \(Y\).
A Final Diagram Check
Before selecting variables for adjustment, ask:
\[ \boxed{ \begin{aligned} &\text{Which paths carry the causal effect I want?}\\ &\text{Which open paths create noncausal association?}\\ &\text{Which conditioning choices could open a collider path?} \end{aligned} } \]The adjustment plan should preserve the first set, block the second, and avoid opening the third.
Summary
- A causal diagram represents variables as nodes and proposed direct causal relationships as directed arrows.
- A DAG is directed and contains no directed cycles.
- Chains represent mediation, forks represent common causes, and colliders represent common effects.
- Backdoor paths enter the treatment or exposure through an incoming arrow and can create noncausal association.
- A sufficient adjustment set blocks open noncausal paths without blocking intended causal paths or opening collider paths.
- Random assignment removes systematic baseline causes of treatment assignment, but post-assignment selection, adherence, and measurement still matter.
- Diagrams encode assumptions; they do not establish arrow direction or effect size from correlation alone.
Next Step
Causal diagrams organize pathways among observed variables. The next lesson develops counterfactual reasoning—the comparison between the outcome observed under one condition and the outcome that would have occurred for the same unit under an alternative condition.