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A backdoor adjustment set contains no descendant of the exposure and blocks every backdoor path from the exposure to the outcome. Averaging the conditional outcome distribution over such a set identifies the total causal effect.

A backdoor path creates noncausal association by entering the exposure through an arrowhead rather than beginning with a causal arrow leaving it.

A backdoor path from exposure $X$ to outcome $Y$ begins with an arrow entering $X$. An adjustment set identifies the total effect when it blocks every such path without conditioning on descendants of $X$.

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]

Front-door adjustment identifies an exposure effect through an observed mediator when the mediator intercepts every directed exposure-outcome path, the exposure-mediator relation has no unblocked backdoor path, and exposure blocks every backdoor path from mediator to outcome.

A backdoor path from $A$ to $Y$ is a path whose first edge has an arrowhead at $A$. The set $X_I$ satisfies the backdoor criterion when it contains no descendant of $A$ and blocks every backdoor path from $A$ to $Y$. Under this criterion, consistency, and positivity,so adjustment for $X_I$ identifies the intervention mean.

WriteSince $X_1$ has no parents, its total effect has no backdoor path, and

For $X_2$, $X_3$ intercepts every directed path to $X_4$. Conditional on $X_1$, there is no unblocked backdoor path from $X_2$ to $X_3$, and $(X_1,X_2)$ blocks every backdoor path from $X_3$ to $X_4$. The conditional front-door adjustment therefore givesandPart ii shows that the inner sum, after also averaging $X_4$, does not actually depend on $x_1$.

For $X_3$, $(X_1,X_2)$ is a sufficient backdoor adjustment set. HenceThese three formulas identify the requested average treatment effects from the observed joint distribution.

The $X_1$ coefficient is the total causal effect of $X_1$ on $Y$. The node $X_1$ has no parents, so there is no backdoor path into it, and the unadjusted regression sums all directed paths from $X_1$ to $Y$.