Estimate Std. Error t value Pr(>|t|) CI Lower CI Upper DF
(Intercept) 49.47 3.94 12.554 4.93e-31 41.7 57.22 432
anygirls -2.83 4.59 -0.617 5.38e-01 -11.8 6.19 432Week 5: Selection-on-observables
PS 813 - Causal Inference
February 16, 2026
Confounding and the direction of the bias
Example: Smoking and Cancer
- Back when the link between smoking and cancer was being debated, some researchers suggested that cigarettes might be a “healthy” alternative to pipe smoking
- Cochran (1968) uses this to illustrate adjustment by stratification
- What’s the omitted confounder?
- What’s the direction of the bias due to the omitted confounder?
Example: Smoking and Cancer
Example: Smoking and Cancer
Directed Acyclic Graphs
- Directed: Edges have arrows
- Acyclic: A node can’t be its own descendant
- Graph: Comprised of nodes and edges
- DAGs encode causal assumptions
- Absence of an edge - assume no causal effect
- Direction of the arrow - direction of effect
Directed Acyclic Graphs
- Chain: \(X \rightarrow D \rightarrow Y\)
- Fork: \(D \leftarrow X \rightarrow Y\)
- Collider: \(X \rightarrow Y \leftarrow D\)
A parent node is a direct cause of a child node.
An ancestor node is a direct or indirect cause of a descendent node.
Path: A sequence of nodes connected by edges (either direction!)
- Causal path: All arrows are pointed in the same direction
- Non-causal path: Some arrows go in the opposite direction
Directed Acyclic Graphs
\[\begin{align*} Y &= f_Y(D, X, \epsilon_Y)\\ D &= f_D(X, \epsilon_D)\end{align*}\]
- DAGs are a way of representing a particular nonparametric structural equation model
- The DAG also represents a factorization of the joint distribution if it’s compatible with the DAG.
\[P(X_1, X_2, \dotsc, X_K) = \prod_{k=1}^K P(X_K | \text{parents}(X_K))\]
- This lets us read (conditional) independence relationships directly from the DAG.
D-separation
- Are \(X_2\) and \(D\) \(d\)-separated?
- Paths:
- \(D \to M \to Y \leftarrow X_2\) (Blocked – collider at \(Y\))
- \(D \to M \leftarrow X_2\) (Blocked – collider at \(M\))
- \(D \leftarrow X_1 \to X_2\) (Unblocked)
D-separation
- Are \(X_2\) and \(D\) \(d\)-separated conditional on \(X_1\)?
- Paths:
- \(D \to M \to Y \leftarrow X_2\) (Blocked – collider at \(Y\))
- \(D \to M \leftarrow X_2\) (Blocked – collider at \(M\))
- \(D \leftarrow X_1 \to X_2\) (Blocked – conditioning on \(X_1\))
D-separation
- Are \(X_2\) and \(D\) \(d\)-separated conditional on \(X_1\) and \(M\)?
- Paths:
- \(D \to M \to Y \leftarrow X_2\) (Blocked – collider at \(Y\))
- \(D \to M \leftarrow X_2\) (Unblocked – conditioned collider at \(M\))
- \(D \leftarrow X_1 \to X_2\) (Blocked – conditioning on \(X_1\))
Good controls
- One non-causal path from \(D\) to \(Y\): \(D \leftarrow X \rightarrow Y\).
- Conditioning on \(X\) blocks that path
Bad controls
- No non-causal paths between \(D\) and \(Y\) w/o conditioning
- But conditioning on \(M\) opens up a non-causal path between \(D\) and \(Y\)
- \(D \rightarrow Z \leftarrow U \rightarrow Y\)
Bad controls
- Even though \(X\) is not causally related to \(D\) or \(Y\) (“irrelevant control”), conditioning on it still induces bias if there are common causes of \(X\) and treatment and \(X\) and outcome
- “M-bias”
Neutral Controls
- \(X\) is not a confounder but is predictive of \(Y\)
- Might improve precision!
SWIGs - a Unification
DAG Practice
- What is the minimum sufficient conditioning set needed to identify the effect of \(D\) on \(Y\)?
DAG Practice
