CIS Primer Question 3.3.1
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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→Z→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} .
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