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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]

A minimal sufficient adjustment set is a sufficient adjustment set none of whose proper subsets is sufficient. Minimality is by set inclusion and does not imply minimum cardinality or uniqueness.

Definition 1 satisfies Property 1 but not Property 2. Let $S_1$ contain every non-descendant that blocks some backdoor path. Every backdoor path begins $A\leftarrow P$ for a parent $P$ of $A$; whenever that path can transmit confounding, $P\in S_1$. Conditioning on $S_1$ therefore blocks every such path at its first nonendpoint vertex, so $S_1$ is sufficient.

For failure of Property 2, consider the faithful graph with arrowsThe variable $X_i$ blocks the backdoor path $A\leftarrow C\to X_i\to Y$, so Definition 1 calls it a confounder. Every sufficient set containing $X_i$ must nevertheless contain $C$ to block $A\leftarrow C\to Y$. Once $C$ is included, deleting $X_i$ leaves a sufficient set. Thus $X_i$ can never be essential as Property 2 demands.

Definition 2 satisfies Property 2 but not Property 1. If $X_i$ belongs to every minimal sufficient adjustment set, choose one such set $I$ and put $J=I\setminus \{i\}$. By minimality, $J\cup \{i\}$ is sufficient and $J$ is not, proving Property 2.

For failure of Property 1, use the faithful chain-shaped backdoor pathBoth $\{C_1\}$ and $\{C_2\}$ are minimal sufficient adjustment sets. No variable belongs to every minimal sufficient set, so Definition 2 labels no variable a confounder, but the empty set is not sufficient.

Definition 3 satisfies Property 1 but not Property 2. Under faithfulness, its associational criterion contains enough non-descendants to block every open backdoor path. Indeed, if such a path remained open, its first unconditioned parent of $A$ would be D-connected to $A$ and, after a suitable conditioning set, to $Y$ given $A$; faithfulness would place that parent in the Definition 3 set, a contradiction. Thus adjusting for all variables selected by Definition 3 is sufficient.

For failure of Property 2, considerwith both $Z$ and $U$ observed and a faithful distribution. The instrumental variable $Z$ is associated with $A$. Conditioning on the collider $A$ opens $Z\to A\leftarrow U\to Y$, so $Z$ is associated with $Y$ given $A$ and Definition 3 calls it a confounder. Any sufficient set containing $Z$ must also contain $U$ to block $A\leftarrow U\to Y$, but $\{U\}$ is already sufficient. Hence removing $Z$ never destroys sufficiency, violating Property 2.