Supplement to:
Conditional Adjustment in a Markov Equivalence Class

Many scientific disciplines have an interest in identifying and estimating causal effects for specific subgroups of a population. For instance, researchers may want to know if a medical treatment is beneficial for people with heart disease or if the treatment will harm older patients (Brand and Xie, 2010; Health, 2010). Such causal effects are referred to as conditional causal effects or heterogeneous causal effects. The identification of these conditional causal effects from observational data is the subject of this work.

Much of the literature on estimating conditional causal effects from observational data focuses on the conditional average treatment effect (CATE; Athey and Imbens, 2016; Wager and Athey, 2018; Künzel et al., 2019; Nie and Wager, 2021; Kennedy et al., 2022). The CATE is represented as a contrast of means for a response $Y$ under different do-interventions (see Section 2 for definition) of a treatment $X$ when conditioning on a set of covariate values $\mathbf{z}$. These means take the form $\operatorname{\mathbb{E}}[Y|do(X=x),\mathbf{Z}=\mathbf{z}]$.

Some results on CATE estimation assume that the conditioning set $\mathbf{Z}$ is rich enough to capture all relevant common causes of $X$ and $Y$ – meaning that $X$ and $Y$ are unconfounded given $\mathbf{Z}$. This implies

which allows the CATE to be estimated as a difference of means from observational data.

However, this assumption does not hold in all applications. Consider, for example, the setting depicted in the causal directed acyclic graph (DAG) of Figure 1, where we want to compute a causal effect of $X$ on $Y$ given some set $\mathbf{Z}$. In this setting, age and smoking status are common causes of $X$ and $Y$, and therefore, $X$ and $Y$ are confounded unless we condition on both age and smoking status ($\mathbf{Z}=\{Age,Smoking\}$). But we may want to know the causal effect of $X$ on $Y$ conditional on age alone ($\mathbf{Z}=\{Age\}$).

To allow for estimation of the CATE in such cases, various recent works (Abrevaya et al., 2015; Fan et al., 2022; Chernozhukov et al., 2023; Smucler et al., 2020) have proposed estimation methods that rely on knowing an additional set of covariates $\mathbf{S}$ that – together with $\mathbf{Z}$ – leads to $X$ and $Y$ being unconfounded. We refer to this set of variables as a conditional adjustment set (Definition 1). For such a set $\mathbf{S}$,

In the example above, if $\mathbf{Z}=\{Smoking\}$, then $\mathbf{S}=\{Age\}$.

Of course, not all conditional causal effect research focuses on estimation through the functional in Equation (2). Notably, other work has explored identifiability without limiting focus to a particular functional. For example, Shpitser and Pearl (2008) and Jaber et al. (2019, 2022) focus on the conditions under which the interventional distribution $f(\mathbf{y}|do(\mathbf{x}),\mathbf{z})$ is identifiable given a causal graph. Though these results broaden the options for identification, estimators based on these results would have to rely on functionals that may prove difficult to estimate, such as $\frac{f(\mathbf{y},\mathbf{z}|do(\mathbf{x}))}{f(\mathbf{z}|do(\mathbf{x}))}$ (Shpitser and Pearl, 2008; Jaber et al., 2019, 2022). Our work addresses this by focusing on identification of the same interventional distribution given a causal graph – but through the use of conditional adjustment sets, which may lead to more desirable estimators. To the best of our knowledge, this area of research is largely unexplored.

Our main contribution is the conditional adjustment criterion (Definitions 2 and 7), a graphical criterion that we show is necessary and sufficient for identifying a conditional adjustment set (Theorems 3 and 9). We additionally provide explicit sets that satisfy this criterion when any such set exists. We note, however, that these results are restricted to a setting where the conditioning set $\mathbf{Z}$ consists of variables known to be unaffected by treatment. While this restricted setting produces limitations (see the second example in the discussion, Section 5), our results are broadly applicable to a variety of research questions. For example, the restriction is met when the conditioning set includes exclusively pre-treatment variables.

In considering the problem of identifying a conditional adjustment set, we assume that the underlying causal system can be represented by a causal DAG. When we collect observational data on all variables in the system, we can attempt to learn this causal DAG by relying on the constraints present in the data (Spirtes et al., 1999; Chickering, 2002; Zhang, 2008b; Hauser and Bühlmann, 2012; Mooij et al., 2020; Squires and Uhler, 2022). However, this task is often impossible from observational data alone, regardless of the available sample size. And further, we cannot always observe every variable.

Thus, our work focuses on causal models that represent Markov equivalence classes of graphs that can be learned from observational data: a maximally oriented partially directed acyclic graph (MPDAG; Meek, 1995) and a maximally oriented partial ancestral graph (PAG; Richardson and Spirtes, 2002). An MPDAG represents a restriction of the Markov equivalence class of DAGs that can be learned from observational data and background knowledge when all variables are observed (Andersson et al., 1997; Meek, 1995; Chickering, 2002). A PAG represents a Markov equivalence class of maximal ancestral graphs (MAGs; Richardson and Spirtes, 2002), which can be learned from observational data and which allows for unobserved variables (Spirtes et al., 2000; Zhang, 2008b; Ali et al., 2009). A MAG, in turn, can be seen as a marginalization of a DAG containing only the observed variables (Richardson and Spirtes, 2002). See Section 2 and Supp. A for further definitions.

The structure of this paper is as follows: Section 2 provides preliminary definitions, with the remaining definitions given in Supp. A. Section 3 contains all results for the MPDAG setting. In particular, we introduce our conditional adjustment criterion in Section 3.1; Section 3.2 illustrates applications of our criterion with examples; Section 3.3 provides several methods for constructing conditional adjustment sets; and Section 3.4 includes a discussion of the similarities of our conditional adjustment criterion with both the adjustment criterion of Perković et al. (2017) and the $\mathbf{Z}$-dependent dynamic adjustment criterion of Smucler et al. (2020). We present some analogous results for PAGs in Section 4, and we discuss some limitations of our results and areas for future work in Section 5.

We use capital letters (e.g. $X$) to denote nodes in a graph as well as random variables that these nodes represent. Similarly, bold capital letters (e.g. $\mathbf{X}$) are used to denote node sets and random vectors.

Nodes, Edges, and Subgraphs. A graph $\mathcal{G}=(\mathbf{V},\mathbf{E})$ consists of a set of nodes (variables) $\mathbf{V}=\left\{V_{1},\dots,V_{p}\right\},p\geq 1$, and a set of edges $\mathbf{E}$. Edges can be directed ($\rightarrow$), bi-directed ($\leftrightarrow$), undirected ( or $-$), or partially directed (). We use $\bullet$ as a stand in for any of the allowed edge marks. An edge is into (out of) a node $X$ if the edge has an arrowhead (tail) at $X$. An induced subgraph $\mathcal{G}_{\mathbf{V^{\prime}}}=(\mathbf{V^{\prime}},\mathbf{E^{\prime}})$ of $\mathcal{G}$ consists of $\mathbf{V^{\prime}}\subseteq\mathbf{V}$ and $\mathbf{E^{\prime}}\subseteq\mathbf{E}$ where $\mathbf{E^{\prime}}$ are all edges in $\mathbf{E}$ between nodes in $\mathbf{V^{\prime}}$.

Directed and Partially Directed Graphs. A directed graph contains only directed edges ($\to$). A partially directed graph may contain undirected edges ($-$) and directed edges ($\to$).

Mixed and Partially Directed Mixed Graphs. A mixed graph may contain directed and bi-directed edges. The partially directed mixed graphs we consider can contain any of the following edge types: , , $\to$, and $\leftrightarrow$. Hence, an edge in a partially directed graph can only refer to edge $\to$, whereas in a partially directed mixed graph, can represent $\to$, $\leftrightarrow$, or .

Paths and Cycles. For disjoint node sets $\mathbf{X}$ and $\mathbf{Y}$, a path from $\mathbf{X}$ to $\mathbf{Y}$ is a sequence of distinct nodes $\langle X,\dots,Y\rangle$ from some $X\in\mathbf{X}$ to some $Y\in\mathbf{Y}$ for which every pair of successive nodes is adjacent. A path consisting of undirected edges ($-$ or ) is an undirected path. A directed path from $X$ to $Y$ is a path of the form $X\to\dots\to Y$. A directed path from $X$ to $Y$ and the edge $Y\to X$ form a directed cycle. A directed path from $X$ to $Y$ and the edge $X\to Y$ form an almost directed cycle. A path $\langle V_{1},\dots,V_{k}\rangle$, $k>1$, in a graph $\mathcal{G}$ is a possibly directed path if no edge $V_{i}\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\leftarrow$} \put(3.0,0.0){$\bullet$} \end{picture}V_{j},1\leq i<j\leq k$, is in $\mathcal{G}$ (Perković et al., 2017, Zhang, 2008a).

A path from $\mathbf{X}$ to $\mathbf{Y}$ is proper (w.r.t. $\mathbf{X}$) if only its first node is in $\mathbf{X}$. A path from $X$ to $Y$ is a back-door path if does not begin with a visible edge out of $X$ (see definition of visible below; Pearl, 2009, Maathuis and Colombo, 2015). For a path $p=\langle X_{1},X_{2},\dots,X_{k}\rangle$ and $i,j,k$ such that $1\leq i<j\leq k$, we define the subpath of $p$ from $X_{i}$ to $X_{j}$ as the path $p(X_{i},X_{j})=\langle X_{i},X_{i+1},\dots,X_{j}\rangle$.

Colliders, Shields, and Definite Status Paths. If a path $p$ contains $X_{i}\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\bullet$} \put(1.0,0.0){$\rightarrow$} \end{picture}X_{j}\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\leftarrow$} \put(3.0,0.0){$\bullet$} \end{picture}X_{k}$ as a subpath, then $X_{j}$ is a collider on $p$. A path $\langle X_{i},X_{j},X_{k}\rangle$ is an unshielded triple if $X_{i}$ and $X_{k}$ are not adjacent. A path is unshielded if all successive triples on the path are unshielded. A node $X_{j}$ is a definite non-collider on a path $p$ if the edge $X_{i}\leftarrow X_{j}$ or $X_{j}\rightarrow X_{k}$ is on $p$, or if $\langle X_{i},X_{j},X_{k}\rangle$ is an undirected subpath of $p$ and $X_{i}$ is not adjacent to $X_{k}$. A node is of definite status on a path if it is a collider, a definite non-collider, or an endpoint on the path. A path $p$ is of definite status if every node on $p$ is of definite status.

Blocking, D-separation, and M-separation. Let $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ be pairwise disjoint node sets in a directed or partially directed graph $\mathcal{G}$. A definite-status path $p$ from $\mathbf{X}$ to $\mathbf{Y}$ is d-connecting given $\mathbf{Z}$ if every definite non-collider on $p$ is not in $\mathbf{Z}$ and every collider on $p$ has a descendant in $\mathbf{Z}$. Otherwise, $\mathbf{Z}$ blocks $p$. If $\mathbf{Z}$ blocks all definite status paths between $\mathbf{X}$ and $\mathbf{Y}$ in $\mathcal{G}$, then $\mathbf{X}$ is d-separated from $\mathbf{Y}$ given $\mathbf{Z}$ in $\mathcal{G}$ and we write $(\mathbf{X}\perp_{d}\mathbf{Y}|\mathbf{Z})_{\mathcal{G}}$ (Pearl, 2009).

If $\mathcal{G}$ is a mixed or partially directed mixed graph, the analogous terms to d-connection and d-separation are called m-connection and m-separation (Richardson and Spirtes, 2002). If a path is not m-connecting in such a graph $\mathcal{G}$ we will also call it blocked. We will also use the same notation $\perp_{d}$ to denote m-separation in a mixed or partially directed mixed graph $\mathcal{G}$.

Ancestral Relationships. If $X\to Y$, then $X$ is a parent of $Y$. If $X-Y$, $X\begin{picture}(5.0,1.0)(0.0,0.0)\put(1.0,1.0){\circle{1.0}} \put(1.5,1.0){\line(1,0){2.0}} \put(4.0,1.0){\circle{1.0}} \end{picture}Y$, $X\begin{picture}(5.0,1.0)(0.0,0.0)\put(1.0,1.0){\circle{1.0}} \put(1.2,0.0){$\rightarrow$} \end{picture}Y$, or $X\to Y$, then $X$ is a possible parent of $Y$. If there is a directed path from $X$ to $Y$, such as $X\to M_{1}\to\dots\to M_{k}$, $M_{k}=Y$, $k\geq 1$, then $X$ is an ancestor of $Y$, $Y$ is a descendant of $X$, and $M_{1},\dots,M_{k}$ are mediators for $X$ and $Y$. We use the convention that if $Y$ is a descendant of $X$, then $Y$ is also a mediator for $X$ and $Y$. If there is a possibly directed path from $X$ to $Y$, then $X$ is a possible ancestor of $Y$, $Y$ is a possible descendant of $X$, and any node on this path that is not $X$ is a possible mediator of $X$ and $Y$. We use the convention that if $Y$ is a possible descendant of $X$, then $Y$ is also a possible mediator for $X$ and $Y$. We also use the convention that every node is an ancestor, descendant, possible ancestor, and possible descendant of itself. The sets of parents, possible parents, ancestors, descendants, possible ancestors, and possible descendants of $X$ in $\mathcal{G}$ are denoted by $\operatorname{Pa}(X,\mathcal{G})$, $\operatorname{PossPa}(X,\mathcal{G})$, $\operatorname{An}(X,\mathcal{G})$, $\operatorname{De}(X,\mathcal{G})$, $\operatorname{PossAn}(X,\mathcal{G})$, and $\operatorname{PossDe}(X,\mathcal{G})$, respectively. Similarly, we denote the sets of mediators and possible mediators for $X$ and $Y$ in $\mathcal{G}$ by $\operatorname{Med}({X,Y},\mathcal{G})$ and $\operatorname{PossMed}({X,Y},\mathcal{G})$.

We let $\operatorname{An}(\mathbf{X},\mathcal{G})=\cup_{X\in\mathbf{X}}\operatorname{An}(X,\mathcal{G})$, with analogous definitions for $\operatorname{De}(\mathbf{X},\mathcal{G})$, $\operatorname{PossAn}(\mathbf{X},\mathcal{G})$, and $\operatorname{PossDe}(\mathbf{X},\mathcal{G})$. For disjoint node sets $\mathbf{X}$ and $\mathbf{Y}$, we let $\operatorname{Med}(\mathbf{X,Y},\mathcal{G})$ be the union of all mediators of $X\in\mathbf{X}$ and $Y\in\mathbf{Y}$ that lie on a proper causal path from $\mathbf{X}$ to $\mathbf{Y}$, with an analogous definition for $\operatorname{PossMed}(\mathbf{X,Y},\mathcal{G})$. Unconventionally, we define $\operatorname{Pa}(\mathbf{X},\mathcal{G})=(\cup_{X\in\mathbf{X}}\operatorname{Pa}(X,\mathcal{G}))\setminus\mathbf{X}$. We denote that $X$ is adjacent to $Y$ in $\mathcal{G}$ by $X\in\operatorname{Adj}(Y,\mathcal{G})$.

DAGs and PDAGs. A directed graph without directed cycles is a directed acyclic graph (DAG). A partially directed acyclic graph (PDAG) is a partially directed graph without directed cycles.

MAGs. A mixed graph without directed or almost directed cycles is called ancestral. Note that we do not consider ancestral graphs that represent selection bias (see Zhang, 2008a, for details). A maximal ancestral graph (MAG) is an ancestral graph $\mathcal{M}=(\mathbf{V,E})$ where every pair of non-adjacent nodes $X$ and $Y$ in $\mathcal{M}$ can be m-separated by a set $\mathbf{Z}\subseteq\mathbf{V}\setminus\{X,Y\}$. A DAG $\mathcal{D}=(\mathbf{V,E})$ with unobserved variables $\mathbf{U}\subseteq\mathbf{V}$ can be uniquely represented by a MAG $\mathcal{M}=(\mathbf{V}\setminus\mathbf{U},\mathbf{E^{\prime}})$, which preserves the ancestry and m-separations among the observed variables (Richardson and Spirtes, 2002).

MPDAGs and Markov Equivalence. All DAGs over a node set $\mathbf{V}$ with the same adjacencies and unshielded colliders can be uniquely represented by a completed PDAG (CPDAG). These DAGs form a Markov equivalence class with the same set of d-separations. A maximally oriented PDAG (MPDAG) is formed by taking a CPDAG, adding background knowledge (by directing undirected edges), and completing Meek (1995)’s orientation rules. We say a DAG is represented by an MPDAG $\mathcal{G}$ if it has the same nodes, adjacencies, and directed edges as $\mathcal{G}$. The set of such DAGs – denoted by $[\mathcal{G}]$ – forms a restriction of the Markov equivalence class so that all DAGs in $[\mathcal{G}]$ have same set of d-separations. Note that if $\mathcal{G}$ has the edge $A-B$, then $[\mathcal{G}]$ contains at least one DAG with $A\to B$ and one DAG with $A\leftarrow B$ (Meek, 1995). Further, note that all DAGs and CPDAGs are MPDAGs.

PAGs and Markov Equivalence. All MAGs that encode the same set of m-separations form a Markov equivalence class, which can be uniquely represented by a partial ancestral graph (PAG; Richardson and Spirtes, 2002; Ali et al., 2009). $[\mathcal{G}]$ denotes all MAGs represented by a PAG $\mathcal{G}$. We say a DAG $\mathcal{D}$ is represented by a PAG $\mathcal{G}$ if there is a MAG $\mathcal{M}\in[\mathcal{G}]$ such that $\mathcal{D}$ is represented by $\mathcal{M}$.

We do not consider PAGs that represent selection bias (see Zhang, 2008b). Further, we only consider maximally informative PAGs (Zhang, 2008b). That is, if a PAG $\mathcal{G}$ has the edge $A\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\bullet$} \put(1.1,1.0){\line(1,0){2.4}} \put(4.0,1.0){\circle{1.0}} \end{picture}B$, then $[\mathcal{G}]$ contains a MAG with $A\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\bullet$} \put(1.0,0.0){$\rightarrow$} \end{picture}B$ and a MAG with $A\leftarrow B$. (We preclude MAGs with $A-B$ by assuming no selection bias.) Any arrowhead or tail edge mark in a PAG $\mathcal{G}$ corresponds to that same arrowhead or tail edge mark in every MAG in $[\mathcal{G}]$. The edge orientations in every PAG we consider are completed with respect to orientation rules $R1-R4$ and $R8-R10$ of Zhang (2008b).

Visible and Invisible Edges. Given a MAG or PAG $\mathcal{G}$, a directed edge $X\rightarrow Y$ is visible in $\mathcal{G}$ if there is a node $V\notin\operatorname{Adj}(Y,\mathcal{G})$ such that $\mathcal{G}$ contains either $V\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\bullet$} \put(1.0,0.0){$\rightarrow$} \end{picture}X$ or $V\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\bullet$} \put(1.0,0.0){$\rightarrow$} \end{picture}V_{1}\leftrightarrow\dots\leftrightarrow V_{k}\leftrightarrow X$, where $k\geq 1$ and $V_{1},\dots,V_{k}\in\operatorname{Pa}(Y,\mathcal{G})\setminus\{V,X,Y\}$ (Zhang, 2006). A directed edge that is not visible in a MAG or PAG is said to be invisible.

Markov Compatibility and Positivity. An observational density $f(\mathbf{v})$ is Markov compatible with a DAG $\mathcal{D}=(\mathbf{V},\mathbf{E})$ if $f(\mathbf{v})=\prod_{V_{i}\in\mathbf{V}}f(v_{i}|\operatorname{pa}(v_{i},\mathcal{D}))$. If $f(\mathbf{v})$ is Markov compatible with a DAG $\mathcal{D}$, then it is Markov compatible with every DAG that is Markov equivalent to $\mathcal{D}$ (Pearl, 2009). Hence, we say that a density is Markov compatible with an MPDAG, MAG, or PAG $\mathcal{G}$ if it is Markov compatible with a DAG represented by $\mathcal{G}$. Throughout, we assume positivity. That is, we only consider distributions that satisfy $f(\mathbf{v})>0$ for all valid values of $\mathbf{V}$ (Kivva et al., 2023).

Probabilistic Implications of Graph Separation. Let $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ be pairwise disjoint node sets in a DAG, MPDAG, MAG, or PAG $\mathcal{G}$. If $\mathbf{X}$ and $\mathbf{Y}$ are d-separated or m-separated given $\mathbf{Z}$ in $\mathcal{G}$, then $\mathbf{X}$ and $\mathbf{Y}$ are conditionally independent given $\mathbf{Z}$ in any observational density that is Markov compatible with $\mathcal{G}$ (Lauritzen et al., 1990; Zhang, 2008a; Henckel et al., 2022).

Causal Graphs. Let $\mathcal{G}$ be a graph with nodes $V_{i}$ and $V_{j}$. When $\mathcal{G}$ is an MPDAG, it is a causal MPDAG if every edge $V_{i}\to V_{j}$ represents a direct causal effect of $V_{i}$ on $V_{j}$ and if every edge $V_{i}-V_{j}$ represents a direct causal effect of unknown direction (either $V_{i}$ affects $V_{j}$ or $V_{j}$ affects $V_{i}$). Note that all DAGs are MPDAGs.

When $\mathcal{G}$ is a MAG or PAG, it is a causal MAG or causal PAG, respectively, if every edge $V_{i}\to V_{j}$ represents the presence of a causal path from $V_{i}$ to $V_{j}$; every edge $V_{i}\begin{picture}(5.0,1.0)(0.0,0.0)\put(0.2,0.0){$\leftarrow$} \put(3.0,0.0){$\bullet$} \end{picture}V_{j}$ represents the absence of a causal path from $V_{i}$ to $V_{j}$; and every edge $V_{i}\begin{picture}(5.0,1.0)(0.0,0.0)\put(1.0,1.0){\circle{1.0}} \put(1.5,1.0){\line(1,0){2.0}} \put(4.0,1.0){\circle{1.0}} \end{picture}V_{j}$ represents the presence of a causal path of unknown direction or a common cause in the underlying causal DAG.

Causal and Non-causal Paths. Note that any directed or possibly directed path in a causal graph is causal or possibly causal, respectively. However, since we focus on causal graphs, we will use this causal terminology for paths in any of our graphs. We will say a path is non-causal if it is not possibly causal.

Consistency. Let $f(\mathbf{v})$ be an observational density over $\mathbf{V}$. The notation $do(\mathbf{X}=\mathbf{x})$, or $do(\mathbf{x})$ for short, represents an outside intervention that sets $\mathbf{X}\subseteq\mathbf{V}$ to fixed values $\mathbf{x}$. An interventional density $f(\mathbf{v}|do(\mathbf{x}))$ is a density resulting from such an intervention.

Let $\mathbf{F^{*}}$ denote the set of all interventional densities $f(\mathbf{v}|do(\mathbf{x}))$ such that $\mathbf{X}\subseteq\mathbf{V}$ (including $\mathbf{X}=\emptyset$). A causal DAG $\mathcal{D}=(\mathbf{V,E})$ is a causal Bayesian network compatible with $\mathbf{F^{*}}$ if and only if for all $f(\mathbf{v}|do(\mathbf{x}))\in\mathbf{F^{*}}$, the following truncated factorization holds:

(Pearl, 2009; Bareinboim et al., 2012). We say an interventional density is consistent with a causal DAG $\mathcal{D}$ if it belongs to a set of interventional densities $\mathbf{F^{*}}$ such that $\mathcal{D}$ is compatible with $\mathbf{F^{*}}$. Note that any observational density that is Markov compatible with $\mathcal{D}$ is consistent with $\mathcal{D}$. We say an interventional density is consistent with a causal MPDAG, MAG, or PAG $\mathcal{G}$ if it is consistent with each DAG represented by $\mathcal{G}$ – were the DAG to be causal.

Identifiability. Let $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ be pairwise disjoint node sets in a causal MPDAG or PAG $\mathcal{G}=(\mathbf{V,E})$, and let $\mathbf{F^{*}_{i}}=\{f_{i}(\mathbf{v}|do(\mathbf{x^{\prime}})):\mathbf{X^{\prime}}\subseteq\mathbf{V}\}$ be a set with which a DAG $\mathcal{D}_{i}$ represented by $\mathcal{G}$ is compatible – were $\mathcal{D}_{i}$ to be causal. We say the conditional causal effect of $\mathbf{X}$ on $\mathbf{Y}$ given $\mathbf{Z}$ is identifiable in $\mathcal{G}$ if for any $\mathbf{F^{*}_{1}},\mathbf{F^{*}_{2}}$ where $f_{1}(\mathbf{v})=f_{2}(\mathbf{v})$, we have $f_{1}(\mathbf{y}|do(\mathbf{x}),\mathbf{z})=f_{2}(\mathbf{y}|do(\mathbf{x}),\mathbf{z})$ (Pearl, 2009).

Forbidden Set. Let $\mathbf{X}$ and $\mathbf{Y}$ be disjoint node sets in an MPDAG or PAG $\mathcal{G}$. Then the forbidden set relative to $(\mathbf{X,Y})$ in $\mathcal{G}$ is

In this section, we present our results on identifying a conditional causal effect via our conditional adjustment criterion in the setting of an MPDAG (Definition 2). Examples of how to use our criterion and explicit conditional adjustment sets based on our criterion follow these results. We remark here that our criterion shares similarities with the adjustment criterion for total effect identification of Perković et al. (2017) and with the $\mathbf{Z}$-dependent dynamic adjustment criterion of Smucler et al. (2020), but we save these results and reflections for Section 3.4.

Note that the results of this section hold when a fully oriented DAG is known, since all DAGs are MPDAGs. Throughout, our goal is to identify the conditional causal effect of treatments $\mathbf{X}$ on responses $\mathbf{Y}$ conditional on covariates $\mathbf{Z}$ and given a known graph $\mathcal{G}$.

We now extend our results on conditional adjustment to the setting of a PAG.

This paper defines a conditional adjustment set that can be used to identify a causal effect in a setting where a causal MPDAG or PAG is known (Definition 1). We give necessary and sufficient graphical conditions for identifying such a set when $\mathbf{Z}\cap\operatorname{PossDe}(\mathbf{X},\mathcal{G})=\emptyset$ (Theorems 3 and 9). Further, we provide multiple methods for constructing these sets (Sections 3.3 and 4.2). While our results can be used to identify a broad class of conditional causal effects, we discuss some limitations below.

One such limitation is that there are conditional causal effects that can be identified but cannot be identified using conditional adjustment sets. As an example, consider the causal DAG (and therefore, MPDAG) $\mathcal{G}$ in Figure 4, and let $\mathbf{X}=\{X_{1},X_{2}\}$, $\mathbf{Y}=\{Y\}$, and $\mathbf{Z}=\{V_{2}\}$. Note that the conditional causal effect of $\mathbf{X}$ on $\mathbf{Y}$ given $\mathbf{Z}$ is identifiable using do calculus rules (Pearl, 2009, see Equations (14)-(16) in Supp. B):