Back-door adjustment identifies a causal effect by blocking all noncausal paths entering treatment while leaving directed causal paths intact. The criterion formalizes confounder adjustment and also explains why adjusting for mediators or colliders can be harmful.
The back-door criterion
Definition. A set \(Z\) satisfies the back-door criterion relative to treatment \(A\) and outcome \(Y\) if:
- no member of \(Z\) is a descendant of \(A\);
- \(Z\) blocks every path between \(A\) and \(Y\) that begins with an arrow into \(A\).
Back-door adjustment theorem
Theorem. If \(Z\) satisfies the back-door criterion and the causal Markov model is valid, then \[ p(y\mid\operatorname{do}(a)) =\int p(y\mid a,z)\,p(z)\,\mathrm{d}z. \]
Proof. Let \(G_{\underline A}\) denote the graph with arrows out of \(A\) removed. The back-door criterion implies \(Y\perp_d A\mid Z\) in \(G_{\underline A}\), so action-observation exchange gives \(p(y\mid\operatorname{do}(a),z)=p(y\mid a,z)\). Because \(Z\) contains no descendants of \(A\), intervention on \(A\) does not change its distribution: \(p(z\mid\operatorname{do}(a))=p(z)\). Apply total probability.
Potential-outcome proof
Theorem. If the graphical criterion entails \(Y(a)\perp\!\!\!\perp A\mid Z\), then under consistency and positivity, \[ \mathbb E Y(a)=\mathbb E\{\mathbb E(Y\mid A=a,Z)\}. \]
Proof. This is iterated expectation, conditional exchangeability, positivity, and consistency in succession: \[ \mathbb EY(a) =\mathbb E[\mathbb E\{Y(a)\mid Z\}] =\mathbb E[\mathbb E\{Y(a)\mid A=a,Z\}] =\mathbb E[\mathbb E(Y\mid A=a,Z)]. \]
Common causes and precision variables
A sufficient adjustment set need not contain every pretreatment variable. Variables causing both \(A\) and \(Y\) must be blocked. A strong predictor of \(Y\) that does not affect \(A\) can improve precision. A variable that affects only \(A\) can amplify finite-sample variance and, under residual unmeasured confounding, may amplify bias. Graphical admissibility and statistical efficiency are separate questions.
Why mediators are forbidden for total effects
Example. In \(A\to M\to Y\), \[ \sum_m p(y\mid a,m)p(m) \] does not equal the total interventional distribution, whose mediator law is \(p(m\mid a)\). Adjusting for \(M\) blocks the mediated path and targets a different, often cross-world, direct-effect estimand.
M-bias from collider adjustment
Example. In \(A\leftarrow U_1\to C\leftarrow U_2\to Y\), the only path between \(A\) and \(Y\) contains collider \(C\) and is blocked marginally. Adjusting for \(C\) opens the path and induces association. A variable can be measured before treatment and still be an inappropriate adjustment variable.
The generalized adjustment criterion
Definition. For total effects, the forbidden set contains descendants, in a graph with incoming treatment arrows removed, of nodes lying on proper directed paths from \(A\) to \(Y\). An adjustment set must avoid this forbidden set and block every proper noncausal path from \(A\) to \(Y\).
This criterion extends the original back-door condition and allows some descendants of treatment that do not lie on or descend from causal pathways, when they do not create new bias. The original criterion remains a transparent sufficient rule.
Multiple treatments
For a joint intervention on treatment vector \(\mathbf A\), a proper back-door path cannot pass through another treatment as part of the intervention's causal effect. The adjustment formula becomes \[ p(y\mid\operatorname{do}(\mathbf a)) =\int p(y\mid\mathbf a,z)p(z)\,\mathrm{d}z \] when \(Z\) satisfies the joint adjustment criterion and positivity holds for the treatment vector.
Minimal versus optimal adjustment sets
Proposition. If both \(Z\) and \(Z\cup W\) are valid adjustment sets, adding \(W\) does not change the population identifying functional when all conditional laws are correct, but it can change semiparametric efficiency and finite-sample stability.
Proof. Both functionals equal the same interventional distribution by the adjustment theorem. Their estimators condition or weight on different covariate spaces, changing nuisance complexity, residual variance, and overlap.
Adjustment cannot repair structural nonidentification
If every path-blocking set includes an unmeasured common cause, no observed adjustment formula exists. Parametric regression does not restore nonparametric identification; it extrapolates through model restrictions. Front-door structure, instruments, proxies, negative controls, sensitivity models, or new data may provide alternatives, but each invokes new assumptions.