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Back-door identification

If one or more back-door paths connect the causal variable to the outcome variable, the causal effect is identified by conditioning on a set of variables \(Z\) if:

Condition 1 All back-door paths between the causal variable and the outcome variable are blocked after conditioning on \(Z\), which will always be the case if each back-door path

  • contains a chain of mediation \(A\rightarrow C \rightarrow B\) where the middle variable \(C\) is in \(Z\)

  • contains a fork of mutual dependence \(A \leftarrow C \rightarrow B\), where the middle variable \(C\) is in \(Z\)

  • contains an inverted fork of mutual causation \(A \rightarrow C \leftarrow B\), where the middle variable \(C\) and all of \(C\)’s decendents are not in \(Z\)

and:

Condition 2 No variables in \(Z\) are decendents of the causal variable that lies on (or decend from other variables that lie on) any of the directed paths that begin at the causal variable and reach the outcome variable.