# Blocking on descendants in a givenDAG

I was curious about the following causal problem

Given the following DAG and asked to list the minimum conditioning sets that satisfy the back-door criterion for identifying the effect of X on Y, should I only condition on Z4, as Z1 is caused by Z4, condition on both Z1 and Z4 as they appear on different non-causal paths from X to Y or only on Z1, as this also blocks Z4.

Sorry if this seems a basic question, but I have some troubles understanding the intuition behind the backdoor criteria.

## 1 Answer

There are two valid sets to close the backdoor paths $$\{Z_4, Z_1\}$$ and $$\{Z_2, Z_1\}$$

To understand why, I will walk through my thought process. First, $$Z_1$$ requires being conditioned on, since it points to both $$X$$ and $$Y$$ directly. However, by conditioning on $$Z_1$$, a path between $$Z_4$$ and $$Z_2$$ is opened (because $$Z_1$$ is a collider on the $$X \leftarrow Z_4 \rightarrow Z_1 \leftarrow Z_2 \rightarrow Y$$ pathway). The now open $$Z_4$$ and $$Z_2$$ pathway now is an open backdoor path between $$X$$ and $$Y$$. As a result, either $$Z_2$$ or $$Z_4$$ must be conditioned on to close that path.

Either is a valid option to choose. You may want to consider which variable has less measurement error potential as a criteria. If you are using standardization to obtain the effect estimate, then $$Z_2$$ would lead to a more precise (i.e. smaller standard error) estimate because it is more predictive of the outcome.

Another easy way to find a sufficient set is to condition on everything that directly points into $$X$$. If all variables with arrows pointing into $$X$$ are conditioned on, that is a sufficient adjustment set.