We can describe causal models with directed graphs. The graph perspective allows us to specify precise procedures for determining when variables (vertices) are independent (d-separated) and dependent (d-connected).
We can represent causal models as directed graphs. The vertices in the graph represent different random variables—causes and effects—and the edges represent causal relationships. If two vertices do not have an edge between them, there is no direct causal relationship between them. For example:
Some technical details:
- These graphs must be acyclic. In a strict sense, something can’t be both a cause and an effect of something else. Thing A at time 1 can effect thing B at time 2 which affects thing A at time 3. Causation only flows forward in time and time is acyclic.
- A path on a directed graph is a sequence of edges joining a sequence of vertices. We can ignore direction of the edges when forming a path.
Now that we’ve presented the basic idea of modeling causal systems with graphs, we can start to use graphs as a tool to analyze causal models. We’ll start by looking at the smallest interesting part of a graph—a triplet consisting of three vertices and two edges. Such a triplet can be configured in one of three ways1. We give a name to each triplet and to the center vertex in each triplet.
- Chains are the most straightforward. If A causes B and B causes C (A → B → C), then A causes C2. We call the central vertex B a mediator or a traverse. For example, if smoking causes (increased risk of) cancer and cancer causes (increased risk of) death, then smoking causes (increased risk of) death.
- The next possible triplet configuration is what we call a fork. If B causes both A and C (A ← B → C), then A and C will not be independent in light of their common cause. For example, if smoking causes both yellowed fingers and lung cancer, we’d expect lung cancer and yellowed fingers to be correlated.
- Inverted forks
- The final possible triplet configuration is what we call an inverted fork. If A causes B and C causes B (A → B ← C), then A and C will be independent. We call the central vertex B a collider. For example, if smoking causes lung cancer and exposure to high doses of radiation also causes lung cancer, we wouldn’t expect smoking and exposure to high doses of radiation to be correlated.
|Name of triplet||Name of central vertex||Diagram||Ends (A and C) dependent?|
|Chain||Mediator/Traverse||A → B → C||Causally (probably)|
|Fork||Confounder/Common cause||A ← B → C||Noncausally|
|Inverted fork||Collider/Common effect||A → B ← C||No|
So we can determine the causal and non-causal dependence between three factors by turning them into a causal graph and looking at the configuration of the edges.Full post