I have made the following model in DAGitty:
Where X2 is controlled for.
DAGitty says:
The total effect cannot be estimated due to adjustment for an intermediate or a descendant of an intermediate.
I asked here whether it would be possible to obtain the treatment effect after controlling for x2.
But I guess my fundamental question is: if I control for both x2 and x1, why, from the theory of DAGs, that doesn't identify the treatment effect?
I mean, controlling for x2, which is a collider, open a backdoor path x1-y. But controlling for x1 should close this backdoor path again.
Why, from a theoretical perspective, that doesn't happen?
I will reinforce my understanding of how that should work with another example.
Consider the following DAG:
x is the treatment, and y is the outcome.
In this DAG I can control for nothing.
Or I can control for {m,a}, or for {m,b}, or for {m,a,b}.
In fact, m is a collider, and controlling for it induces a backdoor path x-a-m-b-y.
But I can close the backdoor path so opened by controlling for also a, or for also b, or for also a and b together.
Why that doesn't happen with the first DAG I posted?
If I control for x2 (the collider), and so open the backdoor path t-x1-x2-y, why I can't close the backdoor path so opened by controlling for x1?
I would like an answer from a theoretical point of view.