Single world intervention graphs are very helpful

Researchers across quantitative social science currently use two different formalisms to think about causal inference: potential outcomes and directed acyclic graphs (DAGs). Textbooks say that the two formalisms are mathematically equivalent (e.g. Wasserman, 2004). But at least on the surface DAGs and potential outcomes appear quite different. Single world intervention graphs (SWIGs) offer a clear way to think about DAGs and potential outcomes using a unified framework. I find SWIG graphs very intuitive and very helpful.

The basic idea of SWIGs is to split the treatment node on a DAG in two.[^1 This is a slight oversimplification because there can be more than one treatment node, but it’s fine for now.] The left side of the split node remains a random variable, as in a traditional DAG. The right side of the split node represents a random variable fixed to a given value.

It’s worth pausing for a minute to think about what that really means. In a traditional DAG, nodes represent random variables, which are formally functions mapping members of a sample space to real numbers. For example, say we have a sample space of people represented as vectors $w=[l, ses, smokes]$ where $l$ is their lifespan, $ses$ is their socioeconomic status and $smokes$ is true if the person is a smoker. Say we have a r.v. $X$ representing if a person smokes. When we see the node $X$ in a DAG it actually represents a function $X(w)$ which returns the value 1 if $w_3$ (i.e. component 3) is 1 and 0 otherwise. I’ve drawn this two ways in the picture below.

. The r.v. $X(w)$ could represent

The right side is a literal value, like “True.” The left side is a random variable which formally a function from members of the sample space