Confounder selection via iterative graph expansion

1. Problem Statement

In observational studies, estimating the causal effect of a treatment $X$ on an outcome $Y$ requires adjusting for a set of covariates $S$ to satisfy conditional ignorability (also known as conditional exchangeability). The task of identifying such a set is called confounder selection.

• The Challenge: The standard approach relies on Pearl's back-door criterion, which requires a pre-specified Causal Directed Acyclic Graph (DAG) representing all relevant variables and their causal relationships.
• Limitations of Current Methods:
  • Constructing a full DAG is often infeasible in practice because researchers may not know all relevant variables (observed or unobserved) or all causal relations among them.
  • Existing "static" methods (e.g., the Disjunctive Cause Criterion) often require pre-specifying the set of observed covariates, making them unsuitable for the design phase of a study where the goal is to determine what data to collect.
  • Most methods do not allow for an interactive, iterative process where domain knowledge is elicited step-by-step.

2. Methodology: Iterative Graph Expansion

The authors propose an interactive, bottom-up procedure called Iterative Graph Expansion. This method does not require a pre-specified global causal graph. Instead, it constructs the necessary structural information incrementally by querying the user (domain expert) for specific local relationships.

Core Concepts

• Symmetric Reformulation: The authors reformulate the back-door criterion symmetrically. Instead of focusing on directed paths from $X$ to $Y$, they focus on confounding paths (paths with arrowheads at both endpoints) and confounding arcs (confounding paths with no colliders).
• Primary Adjustment Sets: A set $C$ is a primary adjustment set for a pair of variables $(A, B)$ relative to a set $S$ if conditioning on $S \cup C$ blocks all confounding arcs between $A$ and $B$.
  • Crucially, blocking confounding arcs is "monotonic": adding variables to the adjustment set never re-opens a blocked arc (unlike blocking general confounding paths, which can be reopened by conditioning on colliders).
• Graph Expansion Process:
  1. Initialization: Start with a "working graph" containing only $X$ and $Y$, connected by a dashed bidirected edge (representing potential uncontrolled confounding).
  2. Iteration:
     • Select a dashed bidirected edge $(A, B)$ in the current graph.
     • Query the user for primary adjustment sets for $(A, B)$ relative to the current set of covariates.
     • Expansion: If a primary adjustment set $C$ is found, remove the edge $(A, B)$ and add the vertices in $C$ to the graph. Draw new dashed bidirected edges between every new vertex and every existing vertex (and between new vertices) to represent potential unobserved confounding.
     • Termination of Edge: If no primary adjustment set exists (i.e., the confounding is unmeasurable), the dashed edge becomes a solid edge (indicating irreducible bias).
  3. Completion: The process repeats until $X$ and $Y$ are no longer connected by any bidirected edges (dashed or solid). The set of vertices added to the graph (excluding $X$ and $Y$) constitutes a sufficient adjustment set.

Algorithmic Implementation

The procedure is formalized in Algorithm 1 (CONFOUNDERSELECT), which uses a priority queue to manage the search space.

• It maintains three sets of edges: $B_n$ (blocked/absent), $B_y$ (unblockable/solid), and $B_u$ (uncertain/dashed).
• It utilizes a min-cut strategy to prioritize edges that, if expanded, require the fewest additional variables to disconnect $X$ and $Y$, optimizing for efficiency.
• Algorithm 2 (FINDPRIMARY) handles the user interaction, iteratively asking for common causes and mediators to construct primary adjustment sets.

3. Key Contributions

1. Theoretical Reformulation:

   • Introduced the concept of confounding arcs and a symmetric back-door criterion.
   • Proved that a set $S$ is a sufficient adjustment set if and only if $X$ and $Y$ belong to different "districts" (maximal sets connected by bidirected edges) in the marginal graph over $S \cup \{X, Y\}$. This is termed the District Criterion.

2. Novel Procedure (Iterative Graph Expansion):

   • Developed a method that inverts the process of graph marginalization (latent projection). Instead of starting with a full graph and marginalizing out unobserved variables, it starts with the outcome/treatment pair and expands the graph by eliciting local structural knowledge.
   • The procedure is Sound and Complete:
     • Soundness: If the user correctly identifies primary adjustment sets, every set returned by the algorithm is a sufficient adjustment set.
     • Completeness: If the user provides all minimal primary adjustment sets at each step, the algorithm will find all minimal sufficient adjustment sets.

3. Practical Advantages:

   • Minimal Knowledge Requirement: Users only need to identify common causes and mediators. They do not need to understand d-separation, draw full DAGs, or specify directed edges between confounders.
   • Interactive Design: The method is designed for the design phase of a study, helping researchers determine exactly which variables need to be measured to control for confounding.
   • Robustness: The construction naturally guards against collider bias (M-bias) because the algorithm only expands based on blocking confounding arcs, avoiding the introduction of colliders unless necessary.

4. Results and Validation

• Theoretical Proofs: The authors provide rigorous proofs (Theorems 1–4) demonstrating that the iterative expansion preserves m-separation properties and that the District Criterion is equivalent to the standard back-door criterion.
• Examples:
  • Butterfly Bias: The method successfully identifies the minimal sufficient adjustment sets $\{B, C\}$ and $\{B, D\}$ in the classic butterfly bias example, demonstrating its ability to navigate complex confounding structures without a pre-drawn graph.
  • Shrier and Platt Example: Applied to a complex DAG involving exercise and injury, the algorithm successfully identified multiple minimal sufficient adjustment sets (e.g., $\{E, F\}$, $\{T, F\}$) by iteratively expanding the graph based on user queries.
• Software: A Shiny web application is provided to facilitate the interactive use of the procedure.

5. Significance

This paper represents a significant shift in how confounder selection is approached in causal inference:

• From Static to Dynamic: It moves away from the "static" assumption that a full causal model is known beforehand. Instead, it treats confounder selection as an interactive discovery process.
• Bridging Theory and Practice: It lowers the barrier to entry for applying rigorous causal inference methods. Domain experts do not need to be graph theory experts; they only need to answer intuitive questions about causes and mediators.
• Study Design: By determining what variables are necessary to measure before data collection, the method directly addresses the "design trumps analysis" philosophy in observational studies, potentially preventing wasted resources on irrelevant variables or fatal flaws in study design due to unmeasured confounding.

In summary, Guo and Zhao provide a mathematically rigorous yet practically accessible framework for confounder selection that leverages iterative graph expansion to overcome the limitations of pre-specifying complex causal graphs.