# CIS Primer Question 3.3.1

February 13, 2019
By

(This article was first published on Brian Callander, and kindly contributed to R-bloggers)

# 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
• $$\{ Z \}$$ .

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