Worked Path-Specific Effects in a DAG

Why Assign Treatment Separately to Different Paths?

Treatment may affect an outcome through more than one route. For example, a program could change a biological mediator that affects health, while also changing access to care through a different pathway. A total treatment effect combines these routes. A path-specific effect instead asks what would happen if treatment were changed along one selected route while its value along a competing route were held fixed.

The key operation is an edge intervention: treatment values are assigned to particular arrows leaving treatment, rather than assigning one treatment value everywhere. This is a counterfactual contrast, not a comparison of regression coefficients. Its interpretation depends on which paths are selected and which treatment value is assigned to the other paths.

A DAG and Its Edge Assignments

Consider a simple directed acyclic graph with binary treatment \(A\), target mediator \(M\), competing-pathway variable \(L\), and outcome \(Y\). The arrows are \(A\to M\), \(A\to L\), \(M\to Y\), and \(L\to Y\). There is no direct arrow from \(A\) to \(Y\), and there are no arrows between \(M\) and \(L\). The two treatment-to-outcome routes are \(A\to M\to Y\) and \(A\to L\to Y\).

For example, \(\psi(1,0)-\psi(0,0)\) changes treatment only along the target path, with the competing path held at its control value. In contrast, \(\psi(1,1)-\psi(0,1)\) changes the target path while the competing path is held at its treated value. These are distinct estimands when the outcome response to \(M\) depends on \(L\).

This definition uses a mixed assignment: the two arrows leaving \(A\) can receive different values. It does not describe a treatment that a person literally received. It defines a counterfactual intervention on the causal mechanisms represented by the DAG.

Identification Formula for the Parallel-Path DAG

Write \(\mu(m,l,c)=E[Y\mid M=m,L=l,C=c]\), where \(C\) denotes measured baseline covariates; the simple graph above is the special case with no \(C\). Under the assumptions below, the edge-intervention mean is identified by

For continuous variables, the corresponding sums are integrals. The formula averages the outcome response over the mediator distribution under treatment value \(a\), the competing-pathway distribution under value \(b\), and the baseline covariate distribution in the target population.

The formula requires more than observing an association between each variable and the outcome. For this graph, sufficient assumptions include consistency; no unmeasured confounding of treatment with the mediators or outcome conditional on \(C\); no unmeasured confounding of either mediator-outcome relation conditional on \(A,C\) and the other mediator; and positivity for the treatment and mediator values used in the formula. The graph also has to represent the causal structure: in particular, there must not be an unaccounted-for common cause of \(M\) and \(L\).

For the mixed counterfactual, the factorization into separate \(M\) and \(L\) distributions also relies on an appropriate cross-world condition. One sufficient structural formulation is that the mediator mechanisms have independent errors conditional on \(C\), with no shared unmeasured causes linking their counterfactual values. This is not guaranteed by an observed-data regression or by randomizing \(A\). If that condition or the other assumptions fail, the displayed expression need not identify the path-specific mean.

Worked Calculation With Two Binary Pathways

Why the Reference Value Matters When Paths Interact

A path-specific contrast is defined relative to the treatment values assigned to the other paths. In the example, the interaction term \(3ml\) means that the effect through \(M\) depends on the value assigned along the \(L\) path. The same formula gives all four edge-intervention means:

Standardizing Over Baseline Covariates

When baseline covariates affect the mediator distributions or outcome, calculate the edge-intervention means within covariate strata and then average those means using the covariate distribution in the target population. This standardization is part of the identification formula, not an optional adjustment after calculating an unadjusted contrast.

Deriving the Edge G-Formula

The formula follows by averaging the outcome response over the variables generated under the specified edge assignment. Under the graph and assumptions, the joint distribution of \(M(a)\) and \(L(b)\), conditional on \(C=c\), factors into their respective conditional distributions under \(A=a\) and \(A=b\). The conditional mean outcome given their values is identified by the observed outcome response, and consistency links observed responses to the corresponding counterfactuals. Therefore:

This justification depends on the stated causal structure and cross-world condition. In particular, if \(M\) and \(L\) share an unmeasured cause, their mixed counterfactual values may be dependent even after conditioning on \(C\). Multiplying the two observed conditional distributions would then impose a joint distribution that the data do not establish. The formula should not be used merely because the observed DAG has arrows from \(A\) to both variables.

Common Mistake: Treating a Path-Specific Effect as a Regression Coefficient

A common error is to call the coefficient of \(M\) in a regression of \(Y\) on \(A,M,L\) “the effect through \(M\).” That coefficient is a conditional model parameter; it is not generally the contrast \(\psi(1,b)-\psi(0,b)\). The path-specific contrast averages the outcome response over the mediator distributions induced by the assigned treatment values and can depend on the competing-path assignment, as the interaction example showed.

A second error is to set \(A=1\) for one mediator and \(A=0\) for another without stating which counterfactual estimand this defines. The correction is to name the edge assignments explicitly, define the resulting mean, and only then calculate the contrast. Also state the identifying assumptions; a numerical edge-g-formula calculation is not evidence that those assumptions hold.