Structural Causal Model
Often Randomized Controlled Trials (RCT) are not feasible, typically for ethical reasons. A Structural Causal Model (SCM) allows one to answer questions about the observational data without intervention. Originally invented by Judea Pearl, which defines them as an ordered triplet $\langle U, V, E \rangle$ where
- $U$: set of exogenous variables (values are determined by factors outside the model)
- $V$: set of endogenous variables (values are determined by factors within the model)
- $E$: set of structural equations defining values in $U$ based on values in $U$ and $V$.
Judea Pearl talks about the ladder of causation:
- Association: $A$ is associated to $B$ if observing $A$ changes the probability of observing $B$. Can be computed via correlations. These have no causal implication. We write $P(A \mid B)$.
- Intervention: Assesses casuality by performing an experimental intervention. We write $P(A \mid \text{do} B)$.
- Counterfactuals: Considering an alternate version of a past event, what would have happened under different circumstances to the same experimental unit? Can be used to indicate causal relationships.
Typically in SCMs we use causal graphs which displays causal relationships between variables in the model. If there is an arrow from node $A$ to node $B$ it mean that $A$ causes $B$. There are several types of graphs, including:
- Causal Loop Diagrams
- Directed Acyclic Graphs
- Ishikawa Diagrams
Elements of a causal graph:
- Chain: $A\longrightarrow B \longrightarrow C$, here $B$ is an mediator.
- Fork: $A \longleftarrow B \longrightarrow C$: one cause has multiple effects. This creates a correlation between $A$ and $C$ which can be removed by conditioning on $B$.
- Collider: $A\longrightarrow B \longleftarrow C$ multiple causes affect one outcome.