On the definition of a confounder

The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The causal inference literature has not, however, produced a clear 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. A variable that helps reduce bias but not eliminate bias we propose referring to as a surrogate confounder