To determine if $$X$$ is ignorable relative to outcome $$Y$$, conditional on a set $$Z$$ of matching variables, we need only test to see if $$Z$$ blocks all the back-door paths between $$X$$ and $$Y$$ and no member of $$Z$$ is a descendant of $$X$$.

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Ignorability

When deconfounding variables $$X$$ and $$Y$$, a back-door path is any path from $$X$$ to $$Y$$ that starts with an arrow pointing into $$X$$. (Notice that if there is a collider on the back-door path, the path is already blocked by the collider.) The path $$A \leftarrow B \leftarrow C \rightarrow D \rightarrow E$$ is a back-door path from $$A$$ to $$E$$. Forks are also back-door paths.

Back-Door Path

Pearl, J., & Mackenzie, D. (2018). $$\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$$ Basic Books.

The Book of Why

Determining Conditional Ignorability Using Causal Diagrams

In the following causal diagram, which of the following are confounding back-door paths for A to F?

For Causal Nodes X,Y:
X and Y will be deconfounded if we block every backdoor, because such paths allow for spurious correlations. 

When eliminating backdoors, we additionally need to ensure that anything closed off from X is not a descendent of X.

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