Causal Diagrams: Five Elementary Structures
PSYC 434 — Week 2
Motivating example: the Salk vaccine
In 1954, investigators first evaluated the polio vaccine with an observational comparison: children whose parents consented versus children whose parents did not.
That comparison suggested higher polio rates among vaccinated children. The design was confounded.
Parental consent was not random. Wealthier, more health-conscious families were more likely to consent. These same families lived in more hygienic environments, which paradoxically increased their children’s susceptibility to polio (less early natural exposure). The consent mechanism, not the vaccine, drove the observed association.
The reversal
A large randomised, double-blind trial assigned vaccine or placebo by chance. The conclusion reversed: the vaccine reduced paralytic polio.
Same question. Different assignment mechanism. Different answer.
The randomised trial enrolled over 400,000 children across 217 areas. By assigning treatment by coin flip, the trial broke the link between family background and vaccination. The difference between the two study designs is not a statistical subtlety; it is structural. This week’s tools let us see why.
Why this week matters
In Week 1 we defined causal questions informally. This week we learn how to represent structural assumptions as directed acyclic graphs (DAGs).
A DAG does not create causal knowledge. It makes assumptions explicit, checkable, and discussable.
The value of drawing a DAG is not that it proves your assumptions are correct. The value is that it forces you to state them. Once assumptions are on paper, collaborators can challenge them, reviewers can evaluate them, and you can check which statistical adjustments they justify.
Randomisation and exchangeability
Under random assignment, potential outcomes are independent of treatment:
Y(a) \coprod A.
This is unconditional exchangeability. Under this condition, a simple difference in means identifies the ATE:
\widehat{\text{ATE}}=\hat{\mathbb{E}}[Y\mid A=1]-\hat{\mathbb{E}}[Y\mid A=0].
In the Salk trial, randomisation ensured that the children who received the vaccine and those who received the placebo were, on average, comparable in every respect except treatment. No unmeasured confounder could systematically differ between groups. In observational studies, this condition usually fails without adjustment.
Working definitions
Internal validity The study contrast estimates the target causal contrast in the study population
External validity That causal contrast transports to the target population
Confounding bias At least one backdoor path from A to Y remains open
These are design properties, not instrument properties.
Confounding bias is defined structurally: it refers to open backdoor paths, not to omitted variables in a regression. A variable can be omitted from a model without causing confounding bias, and a variable can be included in a model while introducing bias (collider conditioning). The DAG tells us which case we are in.
DAG notation
We use a consistent notation throughout the course. A is the treatment or exposure, Y is the outcome, L is a set of measured covariates, U denotes unmeasured causes, and M is a mediator. Solid arrows represent known or assumed causal relationships. Dashed arrows represent relationships we are uncertain about.
DAG elements
A : treatment or exposureY : outcomeY(a) : potential outcome under intervention level a L : measured confounder setU : unmeasured causeM : mediator
The potential outcome Y(a) is the outcome a person would have experienced under treatment level a , regardless of their actual treatment. We never observe both Y(1) and Y(0) for the same person at the same time. This is the fundamental problem of causal inference, formalised in Week 5.
Independence language
A \coprod Y(a) Independence
A \cancel\coprod Y(a) Dependence
A \coprod Y(a) \mid L Conditional independence given L
Independence means that knowing the value of A tells us nothing about the distribution of Y(a) . Conditional independence means that within strata of L , knowing A tells us nothing about Y(a) . The backdoor criterion gives us conditions under which conditioning on L achieves conditional independence.
Three identification assumptions
Causal consistency: If person i receives A_i = a , then Y_i = Y_i(a) .
Conditional exchangeability:
Y(a) \coprod A \mid L.
Positivity:
P(A = a \mid L = l) > 0.
Consistency links the observed outcome to the potential outcome framework. It fails when treatment versions are mixed under one label (e.g., “social media use” that bundles scrolling, messaging, and content creation). Exchangeability is the no-unmeasured-confounding assumption. Positivity ensures that every covariate stratum contains both treated and untreated individuals. All three are needed for identification.
