Back-door path

A back-door path in a causal graph is an indirect path with an arrow into the treatment/causal variable.

Formally, given a graph with nodes including $X$ (treatment/causal variable) and $Y$ (outcome variable), a direct path from $X$ to $Y$ is just a directed edge $X\to Y$ . An indirect path from $X$ to $Y$ is some traversal of edges from $X$ to $Y$ that isn't a direct path. Importantly, the edges don't have to be pointed in the "right way", so $X\to Z\leftarrow W\to Y$ is an indirect path. A back-door path is an indirect path with an arrow into $X$ .

"a back-door path is defined as any path between the causal variable and the outcome variable that begins with an arrow that points to the causal variable" (p. 30).^[1] So this definition doesn't mention indirect paths, but I guess it's implied, because a direct path from the outcome variable to the causal variable would mean that the causal variable isn't really a causal variable?

Examples

$X\leftarrow Z\to Y$ is a back-door path from $X$ to $Y$ : it is indirect (involves $Z$ ) and has an arrow into $X$ (from $Z$ ).
$X\to Z\leftarrow W\to Y$ is not a back-door path from $X$ to $Y$ : it is an indirect path but there is no arrow into $X$ .

References

↑ Stephen L. Morgan; Christopher Winship. Counterfactual and Causal Inference: Methods and Principles for Social Research. 2nd ed. Cambridge University Press. 2015.

[1] Stephen L. Morgan; Christopher Winship. Counterfactual and Causal Inference: Methods and Principles for Social Research. 2nd ed. Cambridge University Press. 2015.

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