  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.
