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On the definition of a confounder
Tyler J. VanderWeele, Ilya Shpitser
TL;DR
The paper addresses the lack of consensus on a formal definition of a confounder despite a formal counterfactual definition of confounding. It evaluates candidate definitions against confounding-control and bias-reduction properties, finding one definition that satisfies both and extending it conditionally.
Problem
The literature lacks consensus on a formal definition of a confounder, even though confounding itself is formally defined using counterfactual independence.
Method
The paper compares candidate confounder definitions using properties concerning control of confounding and elimination or reduction of confounding bias.
Results
Only one considered candidate definition satisfies both that controlling all confounders controls confounding and that each confounder helps eliminate or reduce bias in some context.
Takeaways & Limitations
The paper proposes defining a confounder through membership in a minimally sufficient adjustment set and gives a conditional analogue, with bias-reducing variables termed surrogate confounders.
Takeaways & Limitations
The bias-based candidate definition is inherently scale-dependent, and confounder status may differ across effect scales.
Abstract
from arXiv · showhide
The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all "confounders" suffices to control for "confounding" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a "confounder" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X,C) but such that for no proper subset of (X,C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a "surrogate confounder." These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.
1. Introduction.
Traditional definitions treated confounders as pre-exposure variables associated with exposure and outcome, but causal inference formalized confounding through counterfactual independence. The paper evaluates candidate confounder definitions against whether they support confounding control and bias reduction.
- Traditional conceptions defined confounders as pre-exposure variables associated with exposure and outcome, possibly conditional on other covariates.
- Such associations can identify variables whose control introduces rather than eliminates bias, making the traditional definition inadequate.
- The causal inference literature instead formally defines confounding through dependence between exposure and counterfactual outcomes.
- The paper assesses candidate definitions by asking whether controlling for all confounders controls confounding and whether each confounder can reduce or eliminate bias.
2. Notation and framework.
The paper uses potential outcomes and causal diagrams to formalize exposure, outcome, covariates, and unconfoundedness. Sufficient adjustment sets identify causal effects, while minimally sufficient sets contain no unnecessary coordinates.
- A denotes exposure, Y outcome, and C, S, and X pre-exposure covariates or covariate sets.
- For binary exposure, the average causal effect is E(Y1) − E(Y0), under a potential-outcomes framework that invokes no interference.
- No confounding conditional on S means Ya ⊥⊥ A | S, and any such S is a sufficient adjustment set.
- A minimally sufficient adjustment set achieves conditional independence while no strict coordinate subset does.
- Causal diagrams are structural-equation models encoding counterfactual relationships through recursive substitution.
- The framework permits measured and unmeasured variables, and whether a variable qualifies depends on the underlying causal diagram.
3. Candidate definitions for a confounder.
The paper examines six candidate definitions of a confounder, ranging from associations and backdoor-path blocking to membership in minimally sufficient sets and bias reduction. Several definitions are scale-dependent or difficult to test empirically.
- Definition 1 identifies a confounder through conditional associations of C with exposure and outcome.
- Definition 2 identifies a confounder as a pre-exposure covariate that blocks a backdoor path from exposure to outcome.
- Definition 3 requires C to belong to every minimally sufficient adjustment set, whereas Definition 4 requires membership in some minimally sufficient adjustment set.
- Definition 4 is equivalent to requiring some set X such that (X,C) is minimally sufficient for unconfoundedness.
- Definitions 5 and 6 characterize confounders through lower bias or a changed estimate after controlling for C with X.
- Definition 5 is inherently scale-dependent, so C may qualify for Y but not for log(Y), and most definitions are not empirically testable without experimental data or strong assumptions.
4. Properties of a confounder.
The paper formalizes two requirements for a confounder: controlling all confounders should remove confounding, and each confounder should help eliminate or reduce bias in some context. These requirements distinguish candidate definitions.
- Property 1 requires that conditioning on all variables classified as confounders yields Ya ⊥⊥ A | S.
- The domain of “all confounders” is defined relative to pre-exposure variables on a particular causal diagram.
- Properties 2A and 2B require a confounder to help eliminate or reduce bias, respectively, when controlled alongside some covariate set X.
- The paper argues that an adequate confounder definition should satisfy Property 1 and at least one of Properties 2A or 2B.
- Figure 1 shows that Definition 1 fails Properties 2A and 2B.
5. Properties of the candidate definitions.
The candidate definitions differ in whether adjusting for all identified confounders controls confounding and whether each confounder can eliminate or reduce bias. Among the candidates considered, Definition 4 satisfies both core properties across causal diagrams, while other definitions fail at least one.
- Definition 1 satisfies Property 1 only under faithfulness but fails both bias-reduction properties because associations can identify variables whose adjustment induces or increases bias.In Figure 1, C3 qualifies under Definition 1, yet controlling for it either leaves bias unchanged or increases bias.
- Definition 2 satisfies Property 1 but fails Properties 2A and 2B: a variable can block a backdoor path without helping eliminate or reduce bias.In Figure 2, C2 qualifies but no covariate context makes its control eliminate bias, and examples show it can increase bias.
- Definition 3 satisfies Property 2A but fails Property 1 because distinct minimally sufficient adjustment sets can leave no variable classified as a confounder.In Figure 3, either C1 or C2 alone forms a minimally sufficient set, so neither belongs to every such set, although unadjusted confounding remains.
- Definition 4 satisfies Property 1 and Property 2A for every causal diagram and generally satisfies Property 2B.Its confounders form the union of minimally sufficient adjustment sets, which is itself a sufficient adjustment set.
- Definition 5 satisfies Property 2B but not Properties 1 or 2A, while Definition 6 fails Property 1 and both bias properties.Definition 5 can rely on scale-dependent cancellation, whereas Definition 6 can misclassify variables in M-bias or collider-stratification structures.
- The paper therefore defines confounders using Definition 4 and calls variables that reduce but do not eliminate bias surrogate confounders.Such variables can reduce bias by serving as proxies for variables satisfying Definition 4, although the distinction is scale-dependent for Definition 5.
6. Some extensions, implications and further results.
The paper extends the confounder framework to conditional and parameter-specific settings, examines robustness across expanded causal diagrams, and maps logical relationships among candidate definitions. It also identifies scale dependence and representation choices as important qualifications.
- Conditional and parameter-specific extensions: A conditional confounder is defined relative to covariates L that will be controlled regardless of whether C is included.This extends the unconditional notion to an investigator’s prespecified adjustment context.
- Conditional and parameter-specific extensions: For a particular causal parameter, C is a confounder when adjusting for (X,C) identifies the parameter but no proper subset of (X,C) does.The parameter-specific definition applies to quantities such as causal risk differences or risk ratios.
- Further results and limitations: Definition 4 treats alternative representations such as C, log(C), measured C*, measurement error, and BMI as confounders in specified minimally sufficient settings.These variables need not remain confounders conditional on the corresponding original or alternative covariates.
- Logical relationships among definitions: Definitions 3, 4, 5, and 6 imply selected weaker definitions, but other implications generally fail or can fail because of scale-dependent cancellations.Definition 3 implies Definitions 4, 2, and 1; Definition 4 implies 2 and 1; Definition 5 implies 6 and 1; and Definition 6 implies 1.
- Further results and limitations: Using Property 2A as the definition can satisfy Property 1 on causal diagrams, but this does not hold generally in the counterfactual framework.The same definition can classify a variable as a confounder even when the exposure effect is already unconfounded without adjustment.
7. Concluding remarks.
The paper argues that only one of the considered confounder definitions satisfies both controlling confounding and helping remove confounding in some context. It therefore proposes a counterfactual-based definition for pre-exposure covariates.
- Only one candidate definition satisfies both proposed properties: controlling all confounders controls confounding, and each confounder can remove confounding in some context.
- A confounder is a pre-exposure covariate C whose inclusion in some covariate set makes the exposure effect unconfounded, while no proper subset does.
Review of causal diagrams.
The paper presents directed acyclic graphs as representations of causal relationships and defines paths, graph structure, and backdoor adjustment. The backdoor path criterion guarantees an unconfounded exposure effect when its conditions hold.
- A directed graph contains nodes and directed edges; paths connect distinct nodes, while directed paths follow arrow directions.
- A directed acyclic graph has no directed cycle, and graph terminology includes parents, children, ancestors, descendants, and ancestor sets.
- A path is blocked by conditioning when a noncollider on it is conditioned on, or when a collider and its relevant descendants are not conditioned on.
- Causal directed acyclic graphs associate variables with nonparametric structural equations whose error terms are mutually independent.
- A backdoor path begins with an edge into exposure A; a set X satisfies the criterion when it contains no exposure descendant and blocks every such path.
Empirical testing for confounders and confounding.
Empirical identification of confounders generally requires untestable assumptions or subject-matter knowledge because counterfactual no-confounding conditions and causal-graph structure cannot be tested directly. Under a known sufficient adjustment set, some variables can nevertheless be removed as unnecessary.
- No-confounding conditional independence involving potential outcomes cannot be tested empirically with observed data.
- Definitions based on observed associations may identify a confounder, but nonconfounder status can remain unverifiable when relevant covariates are unmeasured.
- Backdoor-path definitions cannot be empirically verified without assumptions about the underlying causal diagram.
- Definitions based on minimally sufficient adjustment sets require checking a no-confounding condition that is itself not empirically testable.
- Determining whether a covariate is a confounder requires untestable assumptions, although assuming a sufficient adjustment set can sometimes support removing unnecessary variables.