## 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.

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