CIS Primer Question 3.3.1

Here are my solutions to question 3.3.1 of Causal Inference in Statistics: a Primer (CISP).

Part a and b

For the causal effect of \(X\) on \(Y\), every backdoor path must pass via \(Z\). Since \(Z\) is adjacent to \(X\), we must condition on \(Z\). Since \(Z\) is a collider for \(B \rightarrow Z \rightarrow C\), we must also condition on either \(A\), \(B\), \(C\), or \(D\). Thus, the sets of variables that satisfy the backdoor criteria are arbitrary unions of the following minimal sets:

• \(\{ Z, A \}\),
• \(\{ Z, B \}\),
• \(\{ Z, C \}\), and
• \(\{ Z, D \}\).

Part c

All backdoor paths from \(D\) to \(Y\) must pass both \(C\) and \(Z\). We can block all backdoor paths by conditioning on \(C\). If we don’t condition on \(C\), then we must condition on \(Z\). Since \(Z\) is a collider, conditioning on it requires us to also condition on one of \(B\), \(A\), \(X\), or \(W\) (the nodes on the only backdoor path). The minimal sets satisfying the backdoor criteria are:

• \(\{ C \}\),
• \(\{ Z, B \}\),
• \(\{ Z, A \}\),
• \(\{ Z, X \}\), and
• \(\{ Z, W \}\).

Note that \(\{C, Z\}\) also satisfies the backdoor criteria but is not a union of any minimal sets.

All backdoor paths from \(\{D, W\}\) to \(Y\) must pass \(Z\) and must pass either \(C\) or \(X\). The node \(Z\) is sufficient to block all backdoor paths after intervening on \(D\) and \(W\). If we don’t condition on \(Z\), then we must condition on \(X\) and \(C\). The minimal sets satisfying the backdoor criteria are:

• \(\{ C, X \}\), and
• \(\{ Z \}\) .