According to DAG theory, why controlling for this variable doesn't close the backdoor path opened by controlling for the collider?

Question

I have made the following model in DAGitty:

Where $X_2$ 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 $x_2$.

But I guess my fundamental question is: if I control for both $x_2$ and $x_1$, why, from the theory of DAGs, that doesn't identify the treatment effect?

I mean, controlling for $x_2$, which is a collider, open a backdoor path $x_1 \leftrightarrow y$. But controlling for $x_1$ 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:

where $x$ 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 $x \leftrightarrow a \leftrightarrow b \leftrightarrow y$.

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 $x_2$ (the collider), and so open the backdoor path $t \leftrightarrow x1 \leftrightarrow y$, why I can't close the backdoor path so opened by controlling for $x_1$?

I would like an answer from a theoretical point of view.

Answer 1

The difference is in the fact that $X_2$ is an effect of the target

This is a really cool question that I hadn't thought enough about!

A linear regression of the sort of $Y\sim T+X_1+X_2$ will aim to find the coefficients that explain the most variation an the target. In your first example, controlling for $X_2$ not only opens an otherwise closed path, but it also includes an effect of the target variable $Y$. In an additive noise model (as is often assumed for theoretical proofs and in simulations), $Y$ would be given as $ Y := \beta T + N_Y$ where $N_Y$ is an independent noise component. Once you include an effect of $Y$, in this case $X_2$, the $N_Y$ component can be partially explained. For this reason, $X_2$ is given a non-zero weight. Since $X_2$ is also an indirect effect of $T$, the weight given to $T$ in the regression becomes a biased estimator for $\beta$. Graphically speaking, you can draw an additional arrow for $N_Y$ pointing into $Y$ (you can do the same for all additive noise components). Conditioning on $X_2$ creates a dependency between $T$ and $N_Y$ (because $X_2$ is a descendant of the collider $Y$ between them) which biases the estimate of $\beta$ (See also Figure 11.5 in Pearl 2009). In your second example, none of the control variables contain any information about $Y$ other than that already contained in your treatment, so all of the adjustment sets work.

Controlling for $X_2$ therefore introduces two separate problems: It opens a confounding path between $T$ and $Y$, and it opens another path between $T$ and $N_Y$. Controlling for $X_1$ solves the first problem, but not the second, which is why the estimation is still biased.

Answer 2

Our target estimand is the effect of $T$ on $Y$ in the population. Let me give you two examples to show what is happening:

Consider the DAG: $T \rightarrow Y \rightarrow X_2$ where I've removed $X_1$. If you put this into DAGitty and condition on $X_2$, you get the same message as your first DAG, i.e., "The total effect cannot be estimated due to adjustment for an intermediate or a descendant of an intermediate". The reason for this is that if you condition on $X_2$, a descendant of $Y$, you cannot estimate the total effect of $T \rightarrow Y$ in the population. This is simply because conditioning on $X_2$ means that you are estimating $T \rightarrow Y$ (unbiasedly) in a subset of the population where $X_2 = x_2'$, which is not necessarily the causal effect in the population. I.e., this is a potentially biased estimate of your target estimand. Blocking the non-causal path through $X_1$ by adjusting for $X_1$ in your actual DAG does serve to mitigate bias however, but not all potential bias.

Now consider the DAG: $T \rightarrow X_1 \rightarrow X_2 \rightarrow Y$. This is the same as your first DAG except that I've removed the arrow from $T$ to $Y$ implying that the causal effect of $T$ on $Y$ is zero for every individual in the population (i.e., sharp causal null). Here, if you condition on $X_2$, it clearly opens the non-causal path $T \rightarrow X_1 \rightarrow X_2 \rightarrow Y$ and you create a non-causal association between $T$ and $Y$. This is classic selection bias. BUT if you then also condition on $X_1$, you can block this non-causal path AND recover the unbiased causal effect which is zero because in the subset of individuals with $X_2 = x_2'$, as for every individual in the population, the causal effect is zero. This is an unbiased estimate of your target estimand.

You can check that this works in DAGitty.