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Complete Graphical Characterization and Construction of Adjustment Sets in Markov Equivalence Classes of Ancestral Graphs
Emilija Perković, Johannes Textor, Markus Kalisch, Marloes H. Maathuis
TL;DR
Covariate adjustment in observational studies requires graphical criteria that avoid inappropriate conditioning and can accommodate latent confounding or uncertain structure. This paper gives one sound-and-complete criterion across DAGs, MAGs, CPDAGs, and PAGs, plus constructive algorithms and proofs. It concludes that valid adjustment sets can be tested and constructed across these graph classes, while some identifiable effects require other methods and the framework excludes unobserved selection variables.
Problem
Observational covariate adjustment can induce collider bias, and existing criteria did not provide one complete framework across DAGs, MAGs, CPDAGs, and PAGs.
Method
The paper formulates a generalized graphical adjustment criterion, derives constructive sets and m-separation-based algorithms, and implements them in dagitty and pcalg.
Results
The criterion is sound and complete for DAGs, MAGs, CPDAGs, and PAGs, and the paper supplies procedures to test and construct all valid adjustment sets.
Takeaways & Limitations
The framework supports covariate adjustment when causal structure is incomplete or latent confounding is represented, including validity across Markov equivalence classes.
Takeaways & Limitations
The framework does not generally identify effects identifiable only by other methods and assumes no unobserved selection variables.
Abstract
from arXiv · showhide
We present a graphical criterion for covariate adjustment that is sound and complete for four different classes of causal graphical models: directed acyclic graphs (DAGs), maximum ancestral graphs (MAGs), completed partially directed acyclic graphs (CPDAGs), and partial ancestral graphs (PAGs). Our criterion unifies covariate adjustment for a large set of graph classes. Moreover, we define an explicit set that satisfies our criterion, if there is any set that satisfies our criterion. We also give efficient algorithms for constructing all sets that fulfill our criterion, implemented in the R package dagitty. Finally, we discuss the relationship between our criterion and other criteria for adjustment, and we provide new soundness and completeness proofs for the adjustment criterion for DAGs.
1 Introduction
The paper develops a unified, sound-and-complete adjustment criterion for DAGs, MAGs, CPDAGs, and PAGs, addressing covariate-selection challenges under latent confounding and incomplete structural knowledge. It also provides constructive and algorithmic tools, proofs, and scope boundaries for covariate adjustment.
- Motivation: Covariate adjustment can create collider bias in observational data, so adding pre-exposure variables does not necessarily improve causal estimates.The paper also highlights the Table 2 fallacy, where regression coefficients may represent total effects, direct effects, or no causal effect.
- Unified criterion: The generalized adjustment criterion is sound and complete for DAGs, MAGs, CPDAGs, and PAGs, including settings with latent confounding or structural uncertainty.For CPDAGs and PAGs, validity means the set works across every DAG or MAG in the represented Markov equivalence class.
- Unified criterion: For the illustrated CPDAG, {A, Z} adjusts for the total effect of X on Y across all represented DAGs.The example shows that causal effects can be estimated without knowing the full causal structure.
- Construction and implementation: The paper defines a constructive set whenever any valid set exists and develops procedures to test and construct all valid adjustment sets.The construction reduces adjustment to m-separation and supports implementation in dagitty and pcalg.
- Additional results: The authors provide new soundness and completeness proofs for the revised DAG adjustment criterion and analyze relationships with back-door criteria.They also identify cases where multiple criteria share adjustment sets and cases where only the generalized adjustment criterion applies.
- Scope and limitations: The framework generally does not identify every causally identifiable effect because some effects require methods such as IDA, front-door adjustment, or the ID algorithm.The paper also assumes no unobserved selection variables because selection bias often prevents identification by covariate adjustment alone.
2 Preliminaries
The preliminaries define the graph, path, ancestry, collider, and m-separation concepts used throughout the paper. They also distinguish DAGs, MAGs, CPDAGs, and PAGs, together with the densities and causal interpretations associated with them.
- Graphs and paths: A graph contains nodes and edges with endpoint marks that may be arrowheads, tails, or circles, producing directed, bidirected, nondirected, and partially directed edges.The paper treats simple graphs, with at most one edge between any pair of nodes.
- Graphs and paths: A path is a sequence of distinct adjacent nodes; directed paths point consistently toward their endpoint, and subpaths and concatenations preserve path structure.Path length is the number of edges.
- Ancestry and separation: An ancestor has a directed path to a node, while a possible ancestor has a possibly directed path; parent, descendant, and ancestor sets are defined accordingly.Each node is conventionally its own ancestor, descendant, possible ancestor, and possible descendant.
- Ancestry and separation: A node is a collider when adjacent path edges have arrowheads pointing into it, while definite non-colliders and collider paths determine path status.These distinctions are used in the definition of m-connection.
- Ancestry and separation: A path is m-connecting given Z when definite non-colliders are outside Z and every collider has a descendant in Z; otherwise Z blocks the path.m-separation requires all paths between the relevant node sets to be blocked.
- Causal Bayesian networks: DAG Bayesian networks factorize a joint density by parent sets, and causal DAGs interpret directed edges as direct causal effects under interventions do(X = x).The corresponding truncated factorization is also called the g-formula or manipulated density.
- Graph classes: MAGs represent observed variables from DAGs with unobserved variables while preserving ancestral and m-separation relationships; without selection bias, they contain directed and bidirected edges.A MAG is ancestral and maximal when every nonadjacent pair can be m-separated by a suitable set.
- Graph classes: CPDAGs and PAGs uniquely represent Markov equivalence classes of DAGs and MAGs, respectively, preserving common adjacencies and invariant edge marks.A density is consistent with a CPDAG or PAG when it is consistent with at least one represented causal DAG or MAG.
3 The Generalized Adjustment Criterion
The generalized adjustment criterion characterizes valid covariate adjustment across DAGs, CPDAGs, MAGs, and PAGs using amenability, forbidden-set, and path-blocking conditions. An equivalent m-separation formulation in a proper back-door graph supports systematic construction and verification of adjustment sets.
- Adjustment sets: Adjustment sets identify post-intervention distributions from observational conditional densities, enabling causal-effect estimation.The adjustment formula links f(y | do(x)) to observed conditional densities through Z.
- Criterion components: Amenability requires every proper possibly directed path from X to Y to begin with a visible edge out of X.This condition addresses uncertainty and latent confounding represented by CPDAGs and PAGs.
- Criterion components: The forbidden set contains possible descendants of nodes on proper possibly directed causal paths, excluding X, and cannot be used for adjustment.Adjusting for such descendants can open non-causal walks through collider structures.
- Criterion components: The generalized adjustment criterion requires amenability, exclusion of forbidden nodes, and blocking all proper definite-status non-causal paths.For DAGs and MAGs, it reduces to previously established adjustment criteria.
- Soundness and completeness: Theorem 5 states that the generalized criterion is necessary and sufficient for adjustment in DAGs, CPDAGs, MAGs, and PAGs.Thus, every set satisfying the graphical conditions is valid, and every valid adjustment set satisfies them.
- Efficient verification: Replacing path blocking with m-separation in the proper back-door graph yields an equivalent criterion.The graph is formed by removing visible edges out of X that lie on proper possibly directed paths to Y.
4 Constructing Adjustment Sets
The paper develops construction results for generalized adjustment sets across DAGs, CPDAGs, MAGs, and PAGs. It provides a constructive set whenever adjustment is possible and reduces efficient verification to m-separation in a subgraph.
- Pre-processing: Pre-processing can remove exposures without a proper possibly directed path to the responses, while preserving adjustment for the remaining exposures.If X′ is the retained exposure subset, an adjustment set for (X,Y) is also valid for (X′,Y).
- Constructive sets: Adjust(X, Y, G) is defined from possible ancestors of X ∪ Y after excluding exposures, responses, and the forbidden set.For DAGs and MAGs, possible ancestors reduce to ancestors.
- Constructive sets: A constructive-set theorem states that, when any valid set avoids a descendral set I containing the forbidden set, Adjust(X,Y,G) \ I is valid exactly when such a set exists.This supports constructing adjustment sets that exclude specified nodes.
- Constructive sets: Corollary 15 makes Adjust(X,Y,G) a complete existence test: an adjustment set exists if and only if this set satisfies amenability and the blocking condition.The criterion applies uniformly to DAGs, CPDAGs, MAGs, and PAGs.
- Examples: In the PAG examples, the same candidate set works for P1 but no adjustment set exists for P2 because it fails to block X ↔ V3 ↔ V4 → Y.Corollary 15 turns the blocking failure into a nonexistence conclusion for P2.
- Implementation: Direct path-based verification can be exponential, whereas Theorem 7 replaces blocking with m-separation in a proper back-door graph, checkable by graph traversal.For adjacency-list representations, the Bayes-Ball implementation runs in O(|p| + |E|).
5 Relationship to (Generalized) Back-door Criteria
The paper compares its generalized adjustment criterion with Pearl’s and the generalized back-door criteria, deriving constructive sets and characterizing when these criteria differ or coincide. It also identifies graphical conditions for nonexistence and gives equivalence results under restricted graph conditions.
- Back-door criteria: Pearl’s back-door criterion requires excluding descendants of X and blocking every path between X and Y that contains an arrow into X.
- Back-door criteria: The generalized back-door criterion extends these requirements to DAGs, CPDAGs, MAGs, and PAGs using possible descendants and definite-status back-door paths.
- Constructive sets: For multiple exposures, constructing a Pearl back-door set is less straightforward because an adjustment set may exist even when no back-door set exists.
- Constructive sets: Corollary 22 gives a constructive Pearl back-door set whenever one exists, while Corollary 24 gives an analogous constructive generalized back-door set.
- Differences between criteria: Theorem 26 characterizes four graphical patterns associated with failure of the generalized adjustment, generalized back-door, or Pearl back-door criteria.
- Differences between criteria: When the graph is amenable, condition (2) is equivalent to nonexistence of an adjustment set, and forbidden exposure nodes provide a sufficient condition for that failure.
- Equivalence conditions: With one exposure, generalized adjustment and generalized back-door sets exist under exactly the same circumstances; further equivalences hold under stated restrictions on directed or possibly directed paths among exposures.
6 Discussion
The paper establishes a sound and complete generalized adjustment criterion for DAGs, MAGs, CPDAGs, and PAGs, with algorithmic procedures for testing and constructing valid sets. It identifies remaining scope for selection variables and open questions about estimator efficiency.
- The generalized adjustment criterion is sound and complete for DAGs, MAGs, CPDAGs, and PAGs.
- The paper provides ingredients for testing the criterion, constructing all valid sets, or determining that none exists.
- The framework generalizes the algorithmic treatment of covariate adjustment from DAGs and MAGs to CPDAGs and PAGs.
- Extending graphical adjustment conditions to MAGs and PAGs with selection variables remains future work.
- Choosing the most efficient adjustment set remains an open question because valid sets can induce estimators with different efficiencies.
A Preliminaries
The preliminaries define graph structures, path and separation concepts, and foundational results used to reason about adjustment across DAGs, MAGs, CPDAGs, and PAGs. They also connect graphical separation to node cuts, path properties, and generalized back-door construction.
- Moralization adds edges between unconnected parents of a common child and then makes all graph edges undirected.
- In a DAG, d-separation can be reduced to finding node cuts in a moralized ancestral graph.
- A shortest m-connecting path in a DAG or MAG corresponds to a definite-status path in the associated CPDAG or PAG.
- The construction of R_X removes directed edges out of X that are visible in the represented graph, supporting generalized back-door analysis.
- A generalized back-door set exists exactly when Y is not adjacent to X in R_X and DSEP(X,Y,R_X) avoids possible descendants of X.
A.1 Rules of the Do-calculus (Pearl, 2009, Chapter 3.4)
The do-calculus rules characterize when observations or interventions can be inserted, deleted, or exchanged using d-separation in appropriately modified DAGs. Together, they provide formal transformations for causal-effect expressions.
- The rules apply to disjoint node sets in a causal DAG and are valid for every density function consistent with that DAG.
- Rule 1 permits insertion or deletion of observations when the relevant variables are d-separated after deleting incoming edges into X′.
- Rule 2 permits exchanging an action for an observation when d-separation holds in a graph with incoming edges into X′ and outgoing edges from Z′ removed.
- Rule 3 permits insertion or deletion of actions under d-separation after removing incoming edges into X′ and selected outgoing edges from Z′.
A.2 FCI Orientation Rules (Spirtes et al., 2000, p183)
The FCI orientation rules use local edge patterns, nonadjacency, and discriminating paths to orient partially directed edges in a PAG. These rules progressively replace ambiguous marks with directed or otherwise constrained orientations.
- R1 orients A•→B b•C as A•→B→C when A and C are not adjacent.
- R2 orients A•bC as A•→C when directed paths through B and the relevant edge pattern are present.
- R3 orients D•bB as D•→B using an unshielded collider at B, a connecting path through D, and nonadjacency of A and C.
- R4 uses a discriminating path to orient B b•C as B→C.
B Proofs for Section 3
The proofs establish how amenability characterizes adjustment across CPDAGs, PAGs, and their represented DAGs or MAGs. They also show that violating amenability prevents adjustment by constructing incompatible causal structures or an invisible causal path.
- PAG-to-MAG transfer: Lemma 48 transfers certain invisible-edge configurations from a PAG to a compatible MAG.
- Figure 9 organizes the lemmas used to prove Theorem 5.
- PAGs: A PAG that violates amenability yields a represented MAG containing a directed path beginning with an invisible edge.
- Adjustment consequence: Amenability is preserved across represented DAGs or MAGs, while its violation implies that no adjustment set exists.
- CPDAGs: For CPDAGs, non-amenability produces two represented DAGs in which the same path is causal in one and non-causal in the other, so the causal effect is not identifiable.
C Proofs for Section 4
These proofs show that adjustment-set properties persist when exposure sets are reduced and establish the path-construction lemmas needed for completeness. The arguments rely on replacing problematic paths with shorter, definite-status non-causal paths.
- Exposure subsets: The proof transfers a generalized adjustment criterion from an exposure set X to any subset X′.
- Minimal-path argument: A shortest m-connecting non-causal path is selected to derive structural properties of its colliders and definite non-colliders.
- Path construction: When a collider is not an ancestor of the conditioning set, the proof constructs a proper definite-status non-causal path that remains m-connecting.
- Path construction: The construction uses directed paths from selected nodes to X and Y and concatenates them with a path segment to preserve non-causality and m-connection.
D Proofs for Section 5
The proofs characterize when generalized back-door sets exist and how DSEP sets relate to adjustment sets. They also construct forbidden paths to show when no adjustment set avoiding exposure descendants can exist.
- DAG back-door sets: For DAGs, the proof relates generalized adjustment and back-door criteria through proper back-door paths and their blocking behavior.
- Generalized back-door sets: If a generalized back-door set exists, DSEP(X, Y, RX) is contained in the adjustment set after excluding possible descendants of X.
- Existence conditions: When no adjustment set can avoid possible descendants of X, a proper definite-status non-causal path remains m-connecting after those descendants are removed.
- Forbidden paths: The proof identifies a closest blocking non-collider and shows that nodes farther toward Y lie in the forbidden set.
- Forbidden paths: A path assembled from a shortest route and a possibly directed subsequence can consist entirely of forbidden nodes, establishing condition (2) of Theorem 26.
E Adjustment Criterion for DAGs
The DAG-specific appendix proves that the adjustment criterion is both sound and complete. Its counterexamples use linear Gaussian structural equation models to show that every criterion violation can produce a distribution where adjustment fails.
- Definitions: The appendix defines adjustment sets and the DAG adjustment criterion, including forbidden-set and blocking conditions.
- Main result: Theorem 56 states that satisfying the adjustment criterion is equivalent to being an adjustment set in a causal DAG.
- Completeness: Theorem 57 establishes completeness by showing that every criterion violation admits a density consistent with the DAG for which intervention and adjustment disagree.
- Counterexamples: The counterexamples use linear Gaussian SEMs and path-specific edge coefficients to make adjustment differ from the causal effect.
- Soundness: Theorem 58 establishes soundness: every set satisfying the adjustment criterion is an adjustment set.
- Set enlargement: Adding suitable ancestors of X and Y to an existing adjustment set preserves the adjustment criterion under the stated conditions.