Chapter 14 Efficient adjustment
We have seen that the number of valid adjustment sets for the effect of \(A\) on \(Y\) in Figure 13.1 is large; in fact there are 48 possible sets to use. How should we choose which one to use? Logically, it makes sense to choose the one that leads to an estimator of the causal effect that has the lowest possible variance. This is known as the optimal adjustment set (Henckel, Perković, and Maathuis 2022).
Definition 14.1 Suppose that we are interested in the total effect of \(A\) on \(Y\). We define the causal nodes as all vertices on a causal path from \(A\) to \(Y\), other than \(A\) itself. We write this set as \(\mathop{\mathrm{cn}}_\mathcal{G}(A\to Y)\).
We also define the forbidden nodes as consisting of \(A\) or any descendants of causal nodes \[\begin{align*} \mathop{\mathrm{forb}}_\mathcal{G}(A\to Y) &= \mathop{\mathrm{de}}_\mathcal{G}(\mathop{\mathrm{cn}}_\mathcal{G}(A\to Y)) \cup \{A\}. \end{align*}\]
Note that strict descendants of \(A\) are generally not forbidden nodes, as we will see presently.
Figure 13.1: A causal directed graph.
Example 14.1 Consider the graph in Figure 13.1; we have \[\begin{align*} \mathop{\mathrm{cn}}_\mathcal{G}(A\to Y) &= \{M,Y\} & \mathop{\mathrm{cn}}_\mathcal{G}(Z \to Y) &= \{A,M,X,Y\}, \end{align*}\] and \[\begin{align*} \mathop{\mathrm{forb}}_\mathcal{G}(A\to Y) &= \{A,M,Y,K,R\} & \mathop{\mathrm{forb}}_\mathcal{G}(Z \to Y) &= \{Z,A,M,X,Y,K,R\}, \end{align*}\]
14.1 Optimal adjustment set
A result originally given by Henckel, Perković, and Maathuis (2022) for the case of linear graphical models, and later expanded to general distributions by Rotnitzky and Smucler (2020), is that the optimal adjustment set for the causal effect of \(A\) on \(Y\) is given by \[\begin{align*} O_\mathcal{G}(A\to Y) &= \mathop{\mathrm{pa}}_\mathcal{G}(\mathop{\mathrm{cn}}_\mathcal{G}(A\to Y)) \setminus (\mathop{\mathrm{cn}}_\mathcal{G}(A\to Y) \cup \{A\}). \end{align*}\] Considering the graph in Figure 13.1, we obtain that \[\begin{align*} O_\mathcal{G}(A\to Y) &= \{A,L,M,X,S\} \setminus \{A,M,Y\} = \{L,X,S\} \end{align*}\] and \[\begin{align*} O_\mathcal{G}(Z \to Y) &= \{Z,W,A,L,M,X,S\} \setminus \{Z,A,M,X,Y\} = \{W,L,S\}. \end{align*}\] The key is that the optimal adjustment set is the one that (i) ensures there is no confounding and, conditional upon this, (ii) reduces the variation in the outcome as much as possible. Importantly it never includes any instruments—that is, variables that affect the treatment but not the outcome—because this reduces the natural variation that can be used to estimate the causal effect.
Many researchers support the idea that something like an adjustment set should be chosen without looking at the values of the outcome (e.g. Chapter 13 of, Imbens and Rubin 2015). Note that the optimal adjustment set cannot be found in this manner, and there is no guarantee that (without looking at the distribution of the outcome) the covariates one chooses to control for will not completely determine the treatment value.
14.2 Other scenarios
In settings where one had multiple treatments there is not guaranteed to be any valid adjustment set, and we have to use other methods. See the section on time-varying confounding for examples of this. In the case of a graph with some unobserved variables, there is not guaranteed to be an adjustment set that is optimal over all parameter settings. See Rotnitzky and Smucler (2020) for further details.