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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.
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.
