Path-Specific Effects From Edge Interventions

Why Assign Treatment Values to Edges?

In a mediation problem, treatment can affect an outcome through more than one route. For example, treatment $A$ may affect outcome $Y$ directly and may also affect mediator $M$, which then affects $Y$. A question about the effect along $A \to M \to Y$ asks us to change treatment on that route while holding its value fixed on the direct route $A \to Y$.

A conventional intervention sets a treatment node to one value. But in the route-specific question, the treatment value entering the mediator equation can differ from the value entering the outcome equation. An edge intervention represents this distinction by assigning values to arrows rather than requiring one common value at the treatment node.

Definition for a Simple Mediator Pathway

Consider the directed acyclic graph with arrows $A \to M$, $M \to Y$, and $A \to Y$. Write $M(a)$ for the mediator value that would occur under treatment $a$, and $Y(a,m)$ for the outcome that would occur if treatment were set to $a$ and mediator to $m$. These counterfactuals let us describe an edge intervention on the two arrows leaving $A$.

Definition: The edge intervention assigning $a$ to $A \to M$ and $a^*$ to $A \to Y$ gives the mediator its value $M(a)$, but supplies $a^*$ as the treatment input to the outcome equation. Its counterfactual outcome is $Y(a^*,M(a))$. The effect along the mediated path, relative to assigning $a^*$ on both arrows, is $$ \operatorname{PSE}_{A\to M\to Y}(a,a^*) = E\!\left[Y(a^*,M(a))-Y(a^*,M(a^*))\right]. $$

This contrast changes the treatment value along $A \to M$ from $a^*$ to $a$, while keeping the treatment value on the direct arrow $A \to Y$ at $a^*$. For binary treatment, a common contrast takes $a=1$ and $a^*=0$. A positive value then means that, on the outcome scale used, assigning treatment on the mediated route increases the mean outcome relative to assigning control on that route, with the direct arrow held at control.

The expression $Y(a^*,M(a))$ combines two counterfactual instructions: the mediator is generated under $a$, while the outcome equation receives $a^*$ directly. It is therefore not generally the outcome under one ordinary intervention setting $A$ to a single value. The definition specifies the causal quantity of interest; it does not, by itself, show that the quantity can be identified from observed data.

How the Nested Counterfactual Arises

The edge-specific assignment is especially transparent when written using structural equations. Suppose the mediator and outcome are generated by equations

M=f_M(A,U_M), \qquad Y=f_Y(A,M,U_Y),

where $U_M$ and $U_Y$ represent the factors determining the mediator and outcome in the structural model. Under an ordinary intervention $A=a$, the mediator equation receives $a$, giving $M(a)=f_M(a,U_M)$. For the edge intervention, the mediator equation receives $a$, but the treatment input in the outcome equation is set to $a^*$. Substituting the mediator value into the outcome equation gives

Y_{\text{edge}(a,a^*)} =f_Y\!\left(a^*, f_M(a,U_M), U_Y\right) =Y(a^*,M(a)).

If both arrows instead receive $a^*$, the corresponding outcome is $f_Y(a^*,f_M(a^*,U_M),U_Y)=Y(a^*,M(a^*))$. Subtracting the two outcomes for each unit and averaging yields the path-specific effect in the definition. This substitution explains why the contrast is nested: the mediator counterfactual is an input to the outcome counterfactual.

Worked Example: A Linear Structural Model

Consider a hypothetical population in which treatment $A$ is binary and the mediator and outcome obey the deterministic equations $M(a)=2+0.5a$ and $Y(a,m)=10+3a+4m$. The mediator is measured in hours of activity per week, and the outcome is a symptom score in points. Calculate the effect along $A \to M \to Y$ for $a=1$ and $a^*=0$.

First calculate the mediator values under each treatment:

M(1)=2+0.5(1)=2.5,\qquad M(0)=2+0.5(0)=2.

The edge intervention uses $A=1$ on $A \to M$, so the mediator is 2.5 hours per week. It uses $A=0$ on $A \to Y$, so the outcome equation has a zero in its treatment input. Thus,

Y(0,M(1))=10+3(0)+4(2.5)=20\text{ points}, $$ $$ Y(0,M(0))=10+3(0)+4(2)=18\text{ points}.

The path-specific contrast is $20-18=2$ points. Under these equations, assigning treatment on the mediator route while keeping the direct route at control increases the mean symptom score by 2 points. Because the model is deterministic and has the same equations for every unit, the population mean contrast equals this unit-level contrast.

Reading the Edge Assignment Precisely

The edge values must be stated explicitly. In the simple graph, assigning $a$ to $A \to M$ and $a^*$ to $A \to Y$ does not mean setting $A$ to both values in an ordinary treatment assignment. It means replacing the treatment input separately in each child node’s structural equation. The mediator equation uses $a$; the outcome equation’s direct treatment input uses $a^*$.

The reference assignment matters too. Comparing $Y(a^*,M(a))$ with $Y(a^*,M(a^*))$ isolates the change in the mediator route while leaving the direct treatment input the same in both terms. A different contrast could assign $a$ on the direct arrow in both terms. It would ask a related question at a different direct-edge setting, not represent the same contrast.

Worked Example: Binary Mediator and Unit-Level Counterfactuals

Suppose a hypothetical population consists of four equally weighted units. The table gives each unit’s mediator under both treatment values and its outcome under control for each possible mediator value. Outcomes are binary. These entries are stipulated counterfactual values for the example, not observed outcomes from a study.

Unit	$M(0)$	$M(1)$	$Y(0,0)$	$Y(0,1)$
1	0	0	0	0
2	0	1	0	1
3	1	1	1	1
4	0	1	0	0

For the edge intervention with $a=1$ on $A \to M$ and $a^*=0$ on $A \to Y$, each unit’s outcome is $Y(0,M(1))$. The four values are $0,1,1,0$, so their mean is

E[Y(0,M(1))]=\frac{0+1+1+0}{4}=0.50.

Under the reference assignment, both arrows receive 0, and the outcomes are $Y(0,M(0))$. The four values are $0,0,1,0$, giving

E[Y(0,M(0))]=\frac{0+0+1+0}{4}=0.25.

The path-specific effect is $0.50-0.25=0.25$, or a 25-percentage-point increase in the mean binary outcome in this hypothetical population. The calculation follows the edge assignment unit by unit: first select $M(1)$ or $M(0)$, then use that mediator value together with the direct treatment input 0.

Separate Edge Values in a Nonlinear Model

When the outcome equation includes an interaction, the effect through the mediator can depend on the value assigned to the direct edge. This is one reason the reference value belongs in the definition. The following calculation uses a deterministic model to show how the edge inputs determine the counterfactual outcome.

Worked Example: Keeping Track of Both Treatment Inputs

In a hypothetical setting, let $M(a)=a$, where the mediator is binary, and let the outcome score be $Y(a,m)=5+2a+3m+4am$. Calculate the mediated-path contrast for $a=1,a^*=0$, and compare it with the contrast obtained when the direct-edge input is 1 in both terms.

For the first contrast, the mediator values are $M(1)=1$ and $M(0)=0$. With the direct-edge input fixed at 0, the outcomes are

$$ Y(0,M(1))=5+2(0)+3(1)+4(0)(1)=8, $$ $$ Y(0,M(0))=5+2(0)+3(0)+4(0)(0)=5. $$

Thus the effect along the mediator path at direct-edge value 0 is $8-5=3$ score points. Now hold the direct-edge input at 1 in both terms. The comparison is $Y(1,M(1))-Y(1,M(0))$:

$$ Y(1,M(1))=5+2(1)+3(1)+4(1)(1)=14, $$ $$ Y(1,M(0))=5+2(1)+3(0)+4(1)(0)=7. $$

This second contrast is $14-7=7$ score points. The two results differ because the interaction term $4am$ makes the mediator’s contribution depend on the treatment input on the direct edge. Both calculations are valid edge-specific comparisons, but they answer different questions because they use different reference values on $A \to Y$.

Definition Is Not Identification

A causal quantity can be precisely defined even when the observed data do not determine its value without additional assumptions. The edge-intervention expression specifies which treatment input is sent along each arrow and which counterfactual outcomes are compared. It does not establish that the required nested counterfactual means can be recovered from a particular data set.

Accordingly, the numerical examples above use structural equations or complete hypothetical counterfactual profiles. They demonstrate how to calculate the defined contrast when the relevant counterfactual information is supplied. In an applied analysis, one must separately justify the assumptions and identification strategy that connect observed data to that contrast.

Common Mistake

Wrong reasoning: “The treatment is set to 1, so every equation should receive 1; this is just the outcome under treatment.”
Correction: A path-specific edge intervention may send 1 to $A \to M$ and 0 to $A \to Y$. In the simple mediator graph, its outcome is $Y(0,M(1))$, not the ordinary outcome $Y(1)$. State the value assigned to each relevant arrow before writing the counterfactual.
Another pitfall: Reporting a path-specific contrast as if it were automatically estimable from observed treatment groups. The contrast defines a causal target; identification from observed data is a separate question.

Key takeaway: For the simple pathway $A \to M \to Y$, assign $a$ to $A \to M$ and $a^*$ to $A \to Y$. The resulting outcome is $Y(a^*,M(a))$; comparing it with $Y(a^*,M(a^*))$ changes the mediator route while holding the direct-edge input fixed.

Check Your Understanding

Use the edge assignments and counterfactual notation precisely in each answer.

In the graph $A \to M$, $M \to Y$, and $A \to Y$, what treatment values are supplied to the mediator and direct outcome inputs in $Y(0,M(1))$?
Explain why $Y(0,M(1))$ is not generally the outcome under an ordinary intervention setting $A$ to one value.
If $E[Y(0,M(1))]=12$ points and $E[Y(0,M(0))]=9$ points, what is the mediated-path contrast, and how should it be interpreted?
In a model with an interaction between the direct treatment input and mediator, why might the mediated-path contrast differ according to the direct-edge reference value?
Does defining an edge-intervention contrast establish that it is identified from observed data? Explain briefly.

Unit	\(M(0)\)	\(M(1)\)	\(Y(0,0)\)	\(Y(0,1)\)
1	0	0	0	0
2	0	1	0	1
3	1	1	1	1
4	0	1	0	0