# fixedpoint.jp

## A sufficient condition of backdoor adjustment for singleton exposure/outcome (2019-05-31)

The theorem on backdoor adjustment is one of major achievement of graph theoretic study for causal inference. Briefly speaking, the statement says that we can find the causal effect of $$X$$ on $$Y$$ without intervention if there is a set $$Z$$ of covariates satisfying the backdoor criterion.

Notice that here both $$X$$ and $$Y$$ are sets of random variables in general. If both $$X$$ and $$Y$$ in problem are just singletons of variables, then backdoor adjustment is always possible as long as all of covariates are observable. To put it more precisely: given a causal diagram with an exposure variable $$X$$ and an outcome variable $$Y$$, if all of covariates in the diagram are observable, then there is a set $$Z$$ of covariates satisfying the backdoor criterion to estimate the causal effect of $$X$$ on $$Y$$.

Proof. We provide an algorithm to find such a $$Z$$ as follows; initially $$Z := \{\}$$. First, enumerate all of paths from $$X$$ from $$Y$$ and call the set $$\mathcal{P}$$.

(1) Stop and return $$Z$$ if $$\mathcal{P}$$ is empty; otherwise, pick a path $$P$$ in $$\mathcal{P}$$ and remove it from $$\mathcal{P}$$.

(2) If $$P$$ is a causal path i.e. a directed path from $$X$$ to $$Y$$, then go to (1).

(3) If $$P$$ starts with an edge departing from $$X$$, i.e., is of form $$X \rightarrow A \cdots$$ for some variable $$A$$, then go to (1).

(4) Otherwise $$P$$ starts with an edge pointing to $$X$$, i.e., is of form $$X \leftarrow B \cdots$$ for some $$B$$. $$Z := Z \cup \{B\}$$ and go to (1).

With the resulting $$Z$$ all causal paths are open because of (2) together with acyclicity of the diagram. To see that all biasing paths are closed with $$Z$$, note that any biasing path of form $$X \rightarrow A \cdots$$ must have a collider. In fact its leftmost collider $$C$$ and $$C$$'s descendants cannot belong to $$Z$$ because no cycle exists in the diagram. Thus we can employ $$Z$$ for backdoor adjustment since all of covariates are observable. QED.