# Questions on estimating causal influence from Directed Acyclic Graphs

I am reading McElreath's Statistical Rethinking book and I'm wondering if anyone can help clarify my doubts on confronting confounding with DAGs. I'll specify 2 examples:

1. Taken from link. The DAG below is an example on p.186 of the book. $$U$$ is an unobserved variable. To find the causal influence of $$X$$ on $$Y$$, we find all paths from $$X$$ to $$Y$$, classify them as open/closed, and do conditioning if necessary. The path with the node $$B$$ has a collider so we want to exclude it. The paths that we care about are then given by $$X \leftarrow U \leftarrow A \rightarrow C \rightarrow Y$$ and $$X \rightarrow Y$$. Am I therefore correct in saying that to estimate the total causal influence of $$X$$ on $$Y$$, I regress $$Y$$ on the variables $$X,A,C$$?

1. Taken from link. The DAG below is from Problem 6M3 on p.189. Here, the two non-direct paths from $$X$$ to $$Y$$ contain $$Z$$ which is a collider. In this case, how do we then estimate the total causal influence of $$X$$ on $$Y$$? Including $$Z$$ as predictor in the regression would confuse the model since it's a collider, i.e. all coefficients of the model except that for $$Z$$ would be conditioned on $$Z$$ (and some other variables). Do we just regress $$Y$$ on $$X$$?

Thanks for reading and I hope to get some clarifications!

For example 1, one of $$(A, C)$$ would suffice to close the open path, but controlling for both is also fine. In example 2 there are no open paths, so there is nothing to control for (and indeed controlling for $$Z$$ would be wrong), just as you say.
• Thanks for the response but I'm confused by "control". For 1), I know that it suffices to condition on $C$ or $A$ only. So does that mean you only regress $Y$ on $C,X$? For 2) When you say "there is nothing to control", does that mean we just regress $Y$ on $X$? Aug 7 at 14:35
• @AdrianKeister I included the point on how controlling for $Z$ in example 2 would be wrong. Thanks for pointing this out! Sep 14 at 18:19