Chapter 15 Forbidden projection

An alternative approach to finding the optinal adjustment set is to use a method known as forbidden projection. The idea is that we should remove the forbidden variables from the graph, so that we are no longer in any danger of adding them to the adjustment set; we then simply pick our adjustment set to be the parents of the outcome, other than the treatment itself. We first need to define an operation that can remove variables while preserving the causal structure.

15.1 Latent projection

The idea of latent projection is to remove variables in the system that are unobserved. However, we can equally well perform it in the case of variables that we do not wish to use for some other reason, such as them being guaranteed to induce bias if we use them for adjustment.

Definition 15.1 Let $\mathcal{G}$ be a DAG with vertices $V \cup L$ for disjoint $V$ and $L$, and suppose we wish to remove the variables in $L$, but preserve the causal structure. Then we can perform a latent projection. Form a new graph $\tilde{\mathcal{G}}$ with vertices $V$, and add edges:

$v \to w$ if there exist a directed path in $\mathcal{G}$ from $v$ to $w$ with all internal vertices in $L$;
$v \leftrightarrow w$ if there is a two-sided trek from $v$ to $w$ with all internal vertices in $L$.

Definition 15.2 Let $\mathcal{G}$ be a DAG and suppose we are interested in the causal effect of $T$ on $Y$. The forbidden projection is the latent projection of $\mathcal{G}$ over $(V \setminus \mathop{\mathrm{forb}}_\mathcal{G}(A\to Y)) \cup \{A, Y\}$. In other words, we remove all forbidden nodes from the graph except for the outcome and treatment of interest.

In fact, we do not need the second condition of latent projection for the forbidden projection, as we now prove.

Lemma 15.1 Performing the forbidden projection for any causal effect in a DAG will not induce any bidirected ($\leftrightarrow$) edges.

Proof. The only descendant of forbidden nodes that is not also projected out is $Y$. Therefore there is nothing for $Y$ to become bidirected-connected to.

15.2 Adjustment sets

Our main result states that the forbidden projection gives an easy characterization of the optimal adjustment set.

Theorem 15.1 Let $\mathcal{G}$ be a DAG, and let $\tilde{\mathcal{G}}$ be its forbidden projection with respect to $(A,Y)$. Then \[\begin{align*} O_\mathcal{G}(A\to Y) = \mathop{\mathrm{pa}}_{\tilde{\mathcal{G}}}(Y) \setminus \{A\}. \end{align*}\]

Example 15.1 Consider again the graph in Figure 13.1, and consider the forbidden projection with respect to $(A,Y)$. Then

$(a) Graph from Figure \@ref(fig:adj2be) with $\forb_\G(\T \to Y) \setminus \{\T,Y\}$ made latent. (b) Latent projection of graoh in (a).$

Figure 15.1: (a) Graph from Figure 13.1 with $\mathop{\mathrm{forb}}_\mathcal{G}(A\to Y) \setminus \{A,Y\}$ made latent. (b) Latent projection of graoh in (a).

There is a nice duality here: one can show that if we are trying to estimate the causal effect of $A$ on $Y$ then using $\mathop{\mathrm{pa}}_\mathcal{G}(A)$ is a valid adjustment set, but of the possible back-door sets it is the least efficient. On the other hand, using $\mathop{\mathrm{pa}}_{\tilde{\mathcal{G}}}(Y) \setminus \{A\}$ is the _most_efficient.