Directed Acyclic Graphs (DAGs) and Structural Causal Models (SCMs) are complementary frameworks for thinking about causality.
DAGs provide a visual, graphical representation of causal assumptions, while SCMs formalize these relationships mathematically. Together, they help us identify whether certain statistical strategies can support valid estimation of causal effects from observational data.
This tutorial introduces the key concepts, moving from basic graph terminology through to identifying causal effects using the backdoor criterion. Throughout this tutorial, we’ll use a running example from education research studying the causal impact of participating in Free and Reduced-Price Meals (FRPM) on high school grades (adapted from Feng, 2024).
Motivating Example: Does participating in FRPM improve high school academic performance? This is a classic observational study where students self-select into the program based on family income and household size, creating confounding that need to be addressed for valid causal inference.
Feng, Y. (2024). Introduction to causal graphs for education researchers. Asia Pacific Education Review, 25(3), 595-609.
A Directed Acyclic Graph (DAG) is a graphical representation of causal relationships where:
Key Purpose: DAGs help us communicate assumptions about causal structure and identify which design/statistical strategies might support valid causal estimation.
library(ggdag)
library(ggplot2)
# Simple DAG example: FRPM → GPA
dag1 <- dagify(
GPA ~ FRPM,
exposure = "FRPM",
outcome = "GPA",
labels = c(
FRPM = "FRPM\nParticipation",
GPA = "High School\nGPA"
)
)
ggdag(dag1) + theme_dag() +
labs(title = "Simple DAG: Does FRPM Participation Improve Grades?",
subtitle = "The focal treatment effect we want to identify")
Understanding DAG terminology is essential for reading and constructing causal graphs:
frpm_dag <- dagify(
GPA ~ FRPM + FI + PE,
FRPM ~ FI + HS,
HS ~ PE + FI,
FI ~ PE,
exposure = "FRPM",
outcome = "GPA",
labels = c(
FRPM = "FRPM",
GPA = "GPA",
FI = "Family\nIncome",
HS = "Household\nSize",
PE = "Parent\nEducation"
)
)
ggdag(frpm_dag) +
theme_dag() +
labs(title = "FRPM and GPA with Observed Confounders")
Example: In the path Family Income → FRPM → GPA:
A Structural Causal Model formalizes the causal relationships depicted in a DAG. An SCM consists of:
Example SCM for FRPM Study:
\[ \begin{align} PE_i &= f_{PE}(U_i, E_{PE_i}) \\ FI_i &= f_{FI}(PE_i, U_i, E_{FI_i}) \\ HS_i &= f_{HS}(PE_i, FI_i, U_i, E_{HS_i}) \\ FRPM_i &= f_{FRPM}(HS_i, FI_i, E_{FRPM_i}) \\ GPA_i &= f_{GPA}(FRPM_i, PE_i, FI_i, U_i, E_{GPA_i}) \end{align} \]
where: - \(PE\) = Parent education - \(FI\) = Family income
- \(HS\) = Household size - \(FRPM\) = Participation in free/reduced-price meals - \(GPA\) = High school GPA - \(U_i\) = Unobserved confounders (e.g., structural inequity, historical adversity) - \(E\) terms = Idiosyncratic errors (mutually independent)
The SCM specifies how each variable is generated from its parents and exogenous noise. The DAG visualizes this causal structure. Note that the functions \(f(\cdot)\) are nonparametric—they don’t assume any specific functional form (linear, exponential, etc.).
To understand when variables are associated, we need to understand three fundamental path structures:
chain_dag <- dagify(C ~ B, B ~ A)
ggdag(chain_dag) +
theme_dag() +
labs(title = "Chain Structure")
fork_dag <- dagify(A ~ B, C ~ B)
ggdag(fork_dag) +
theme_dag() +
labs(title = "Fork Structure (Common Cause)")
collider_dag <- dagify(B ~ A + C)
ggdag(collider_dag) +
theme_dag() +
labs(title = "Collider Structure")
Conditioning on a collider opens a path and can introduce bias.
d-separation (directional separation) formalizes when we can claim conditional independence between variables based on the graph structure.
Definition: Nodes \(A\) and \(C\) are d-separated by a set \(Z\) if and only if:
Implication: If \(A\) and \(C\) are d-separated by \(Z\), then \(A \perp\!\!\!\perp C \mid Z\) in any dataset generated by that graph.
Summary:
The Cardinal Sin of Causal Inference: Assuming that “how the distribution would look if we made \(D=d\)” is the same as “the distribution we observe where \(D=d\) already.”
Key Distinction:
The do-operator represents a hypothetical intervention: \(do(D=d)\) means setting treatment \(D\) to value \(d\) for all units.
Average Treatment Effect: \[\text{ATE} = E[Y|do(D=1)] - E[Y|do(D=0)]\]
When we perform \(do(A=a)\), we create a modified SCM/graph where:
# Original graph
orig <- dagify(Y ~ A + X, A ~ X)
ggdag(orig) +
theme_dag() +
labs(title = "Original Graph: X → A → Y with X → Y")
# After do(A=a) - arrows into A are removed
intervened <- dagify(Y ~ A + X)
ggdag(intervened) +
theme_dag() +
labs(title = "After do(A=a): Remove X → A arrow")
Example: Ice cream sales (\(A\)), crime (\(Y\)), and temperature (\(X\)):
The question becomes: Can we use observational data to recover the interventional distribution \(P(Y|do(X=x))\)?
This is the identification problem.
We want to: - Keep frontdoor paths open (they carry the causal effect) - Block backdoor paths (they create confounding)
Definition: A set of variables \(Z\) satisfies the backdoor criterion relative to \((D,Y)\) if:
Practical Application: If \(Z\) satisfies the backdoor criterion, then:
\[P(Y|do(d)) = \sum_z P(Y|d,z)P(z)\]
This is the adjustment formula: we can identify the causal effect by conditioning on \(Z\).
# Example: Drug, Blood Pressure, SES, Survival
backdoor_ex <- dagify(
Live ~ Drug + BP,
Drug ~ SES,
BP ~ SES,
exposure = "Drug",
outcome = "Live"
)
ggdag(backdoor_ex) + theme_dag() +
labs(title = "Backdoor Example: Which sets block backdoor paths?",
subtitle = "SES is unobserved, BP can block the backdoor path")
In this example:
Let’s apply the backdoor criterion to identify the causal effect of FRPM on GPA. Consider the complete DAG with unobserved confounders:
# Complete FRPM DAG with unobserved confounder
frpm_complete <- dagify(
GPA ~ FRPM + FI + PE + U,
FRPM ~ FI + HS,
HS ~ PE + FI + U,
FI ~ PE + U,
PE ~ U,
exposure = "FRPM",
outcome = "GPA"
)
ggdag(frpm_complete) + theme_dag() +
labs(title = "Complete FRPM DAG with Unobserved Confounder U")
Step 1: Identify all paths between FRPM and GPA
Causal path:
Open backdoor (confounding) paths include:
Step 2: Find variables that block all backdoor paths
To satisfy the backdoor criterion, we need a set \(Z\) that: 1. Blocks all backdoor paths 2. Contains no descendants of FRPM 3. Doesn’t open new paths by conditioning on colliders
Solution: The set \(\{FI, HS\}\) (Family Income and Household Size) satisfies the backdoor criterion.
ggdag(frpm_complete) +
theme_dag() +
labs(title = "Adjustment Set: {Family Income, Household Size}",
subtitle = "Conditioning on FI and HS blocks all backdoor paths")
Step 3: Apply the adjustment formula
\[ \begin{align} &E[GPA|do(FRPM=1)] - E[GPA|do(FRPM=0)] \\ &= \sum_{fi, hs} E[GPA|FRPM=1, FI=fi, HS=hs] \cdot P(FI=fi, HS=hs) \\ &\quad - \sum_{fi, hs} E[GPA|FRPM=0, FI=fi, HS=hs] \cdot P(FI=fi, HS=hs) \end{align} \]
Alternative adjustment sets: - \(\{FI, HS, PE\}\) also works (though PE is not necessary once we condition on FI and HS) - \(\{FI\}\) alone is NOT sufficient (doesn’t block paths through HS) - \(\{HS\}\) alone is NOT sufficient (doesn’t block paths through FI)
confounder_dag <- dagify(
D ~ X,
Y ~ X + D,
exposure = "D",
outcome = "Y"
)
ggdag(confounder_dag) + theme_dag() +
labs(title = "Confounder: X is a common cause")
Structure: \(D \leftarrow X \rightarrow Y\)
collider_dag2 <- dagify(
Z ~ D + Y,
exposure = "D",
outcome = "Y"
)
ggdag(collider_dag2) + theme_dag() +
labs(title = "Collider: Z is a common effect")
Structure: \(D \rightarrow Z \leftarrow Y\)
combined_dag <- dagify(
D ~ X + A,
Y ~ X + B + D,
Z ~ D + Y
)
ggdag(combined_dag) + theme_dag() +
labs(title = "Mixed Structure: Confounder X and Collider Z",
subtitle = "X: confounder, Z: collider, A: cause of D only, B: cause of Y only")
To identify the effect of \(D\) on \(Y\):
Consider a common scenario in education research: you can only collect data from college students (e.g., a retrospective study). This creates selection bias because college enrollment is a collider:
# FRPM DAG with college enrollment as collider
frpm_college <- dagify(
College ~ FRPM + GPA + PE + FI + U,
GPA ~ FRPM + FI + PE + U,
FRPM ~ FI + HS,
HS ~ PE + FI + U,
FI ~ PE + U,
PE ~ U,
exposure = "FRPM",
outcome = "GPA"
)
ggdag(frpm_college) +
theme_dag() +
labs(title = "College Enrollment as a Collider",
subtitle = "Restricting sample to college students conditions on the collider")
The Problem: When you restrict your sample to college students, you are conditioning on College Enrollment, which:
Why? Students with lower FRPM participation rates might need higher GPAs to get into college, creating a negative association within the selected sample (Berkson’s paradox).
Solution: Collect data from all high school students, not just those who enrolled in college. If that’s impossible, the causal effect may not be identifiable with conditioning methods alone.
Key Lesson: Always consider your sampling mechanism. If sample selection depends on both treatment and outcome (or their causes), you’ve conditioned on a collider.
Another important consideration is mediators - variables that lie on the causal path between treatment and outcome. Consider school attendance as a mediator:
# FRPM with attendance as mediator
frpm_mediator <- dagify(
Attendance ~ FRPM + U,
GPA ~ FRPM + Attendance + FI + PE + U,
FRPM ~ FI + HS,
HS ~ PE + FI,
FI ~ PE,
exposure = "FRPM",
outcome = "GPA"
)
ggdag(frpm_mediator) + theme_dag() +
labs(title = "School Attendance as a Mediator",
subtitle = "Attendance lies on a causal path from FRPM to GPA")
The Causal Paths: 1. Direct effect: FRPM → GPA 2. Indirect effect: FRPM → Attendance → GPA
The total causal effect includes both paths.
What Happens If We Condition on Attendance?
The adjustment set {FI, HS} identifies the total effect. Adding Attendance to this set would:
Rule of thumb: To estimate the total causal effect, do NOT condition on variables that:
This tutorial is based on lecture materials from Stat 256: Causality (UCLA) and EDUC 255C: Introduction to Causal Inference in Education Research (UCLA), Spring 2025.