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A confounder is a pretreatment variable relevant to removing confounding for a specified treatment-outcome question. A common cause $C$ with arrows $A\leftarrow C\to Y$ is a basic example: it creates a backdoor path between treatment $A$ and outcome $Y$.

Association with both variables is not a sufficient definition. In an appropriate causal directed acyclic graph, conditioning on a common effect can instead introduce collider bias. A confounder is a causal adjustment role, rather than any predictor with a small p-value.

One precise definition identifies a confounder as a member of some minimal sufficient adjustment set: there is a set containing $C$ that makes the relevant potential outcomes independent of treatment, and no proper subset of that set does so. The role can therefore depend on which other covariates are available and on the effect being estimated. A variable that merely reduces residual bias need not satisfy this definition.^[ref]

Confounding occurs when the compared treatment groups differ in relevant potential outcomes, so their observed outcome contrast mixes the causal effect with differences in baseline prognosis. In a binary-treatment model, absence of confounding is expressed through ignorability, $Y(a)\perp A$ for each compared $a$, or through conditional exchangeability after a sufficient pretreatment adjustment set is specified.

For a causal directed acyclic graph, $A\leftarrow U\to Y$ is the simplest backdoor path: the common cause changes both assignment and outcome. Confounding by indication is a practical example when baseline severity influences which treatment is received. Outcome standardization, matching and inverse probability weighting can address measured confounding under their identification assumptions.

A changed adjusted estimate does not, by itself, prove that confounding was removed. Noncollapsibility of the odds ratio can produce different marginal and conditional odds ratios even without confounding; adjustment for an inappropriate variable can introduce collider bias. Causal structure and the target effect must guide the adjustment choice.^[ref]

Ilya Shpitser develops foundations of causal inference. With Tyler J. VanderWeele, he compared formal definitions of a confounder by asking whether adjusting for all such variables removes confounding, and whether each variable can help in an appropriate adjustment context.^[ref]

This makes adjustment a problem about causal structure. A variable's observed correlation with treatment and outcome alone cannot determine whether conditioning on it is useful. Their analysis relates a confounder to membership in a minimally sufficient adjustment set.^[ref]

Tyler J. VanderWeele develops methods and foundations for causal inference. With Ilya Shpitser, he examined how to define a confounder so that the definition supports valid adjustment decisions.^[ref]

Their preferred definition concerns a pretreatment variable belonging to some minimally sufficient adjustment set. This connects the variable's role to elimination of confounding, rather than merely to prediction of exposure and outcome. The distinction matters because a predictive variable can be unsuitable for adjustment, and the contribution of one covariate depends on what else is adjusted for.^[ref]