Additive Decomposition of Natural Effects

Why Compatibility Matters

A mediation analysis can describe a total treatment effect by separating a contrast associated with changing the mediator from one associated with changing treatment while holding the mediator reference fixed. On the additive scale, these component contrasts can sum exactly to the total effect—but only when their reference values fit together. The reason is algebraic: the intermediate counterfactual outcome at the end of one component must be the same outcome at the start of the next.

The result is an identity among nested counterfactuals. It does not, by itself, show that the effects can be identified or estimated from observed data. The focus here is the decomposition and the compatibility condition that makes it hold.

Counterfactuals and Effect Definitions

Let treatment \(A\) take the values 0 and 1, let \(M(a)\) denote the mediator value that would occur under treatment \(a\), and let \(Y(a,m)\) denote the outcome that would occur if treatment were set to \(a\) and the mediator to \(m\). The nested outcome \(Y(a,M(a'))\) is the outcome under treatment value \(a\) with the mediator set to the value it would take under treatment value \(a'\). All contrasts below compare treatment 1 with treatment 0 in the same target population.

Definition: The total natural effect on the additive scale is \(E[Y(1,M(1))-Y(0,M(0))]\). The natural direct effect compares treatment values while fixing the mediator to a natural value. The natural indirect effect compares natural mediator values while fixing treatment in the outcome counterfactual. The subscript on each effect below records the reference value held fixed.

There are two compatible ways to pair the components. In the first, the mediator is referenced to its value under treatment 0 in the direct effect, and treatment is fixed at 1 in the indirect effect:

\begin{aligned} \mathrm{NDE}(0) &= E[Y(1,M(0))-Y(0,M(0))],\\ \mathrm{NIE}(1) &= E[Y(1,M(1))-Y(1,M(0))]. \end{aligned}

In the second, the mediator is referenced to its value under treatment 1 in the direct effect, and treatment is fixed at 0 in the indirect effect:

\begin{aligned} \mathrm{NDE}(1) &= E[Y(1,M(1))-Y(0,M(1))],\\ \mathrm{NIE}(0) &= E[Y(0,M(1))-Y(0,M(0))]. \end{aligned}

The labels 0 and 1 indicate the relevant reference value; they do not mean that the effect is evaluated only among people who received that treatment. Each expectation is over the same target population. Since these are differences in outcomes, both components and the total effect have the outcome’s units.

The Additive Decomposition

For the first pairing, add and subtract the intermediate outcome \(Y(1,M(0))\) inside the total-effect contrast. The resulting two differences are exactly the definitions of \(\mathrm{NDE}(0)\) and \(\mathrm{NIE}(1)\):

\begin{aligned} Y(1,M(1))-Y(0,M(0)) &= \{Y(1,M(1))-Y(1,M(0))\}\\ &\quad + \{Y(1,M(0))-Y(0,M(0))\}. \end{aligned}

Taking expectations and using linearity of expectation gives the population identity. The second pairing follows by inserting \(Y(0,M(1))\) instead:

\begin{aligned} E[Y(1,M(1))-Y(0,M(0))] &= \mathrm{NIE}(1)+\mathrm{NDE}(0),\\ E[Y(1,M(1))-Y(0,M(0))] &= \mathrm{NDE}(1)+\mathrm{NIE}(0). \end{aligned}

Theorem: If the nested counterfactual outcomes are defined in the same population, then on the additive scale \(\mathrm{TE}=\mathrm{NDE}(0)+\mathrm{NIE}(1)=\mathrm{NDE}(1)+\mathrm{NIE}(0)\), where \(\mathrm{TE}=E[Y(1,M(1))-Y(0,M(0))]\). The first equality uses the common intermediate outcome \(Y(1,M(0))\); the second uses \(Y(0,M(1))\).

This is an algebraic result, not a model-based approximation. It does not require a linear outcome model, a constant effect, or absence of treatment–mediator interaction. It does require that the terms refer to the same outcome definition, population, treatment contrast, and nested counterfactual system. The identity itself does not establish that any of these counterfactual quantities can be recovered from observed data.

Worked Example: Decomposing Effects in a Two-Person Population

Consider a hypothetical population of two people and an outcome measured in score points. The following table gives each person’s mediator values under treatment 0 and 1, and the potential outcome \(Y(a,m)\) at each treatment–mediator combination.

Person	\(M(0)\)	\(M(1)\)	\(Y(0,0)\)	\(Y(0,1)\)	\(Y(1,0)\)	\(Y(1,1)\)
A	0	1	10	13	15	21
B	1	0	8	12	11	14

For person A, the total effect is \(Y(1,M(1))-Y(0,M(0))=Y(1,1)-Y(0,0)=21-10=11\) points. The compatible components are \(\mathrm{NDE}(0)=Y(1,0)-Y(0,0)=15-10=5\) points and \(\mathrm{NIE}(1)=Y(1,1)-Y(1,0)=21-15=6\) points. Their sum is \(5+6=11\).

For person B, the total effect is \(Y(1,0)-Y(0,1)=11-12=-1\) point. Here \(\mathrm{NDE}(0)=Y(1,1)-Y(0,1)=14-12=2\) points and \(\mathrm{NIE}(1)=Y(1,0)-Y(1,1)=11-14=-3\) points. Their sum is \(2+(-3)=-1\).

The population mean total effect is \((11+(-1))/2=5\) points. The population mean components are \((5+2)/2=3.5\) points and \((6+(-3))/2=1.5\) points, respectively. Thus \(3.5+1.5=5\) points, matching the total effect. The calculation illustrates the identity both person by person and after averaging.

Using the Other Compatible Pair

The alternative decomposition inserts \(Y(0,M(1))\) between the two endpoints. It changes which treatment level is held fixed for the indirect effect and which natural mediator value is used for the direct effect. The algebra remains exact because the shared intermediate outcome cancels.

Worked Example: Checking the Alternative Reference Values

Use the same hypothetical two-person population. For person A, the direct effect with mediator reference 1 is \(\mathrm{NDE}(1)=Y(1,1)-Y(0,1)=21-13=8\) points. The indirect effect with treatment fixed at 0 is \(\mathrm{NIE}(0)=Y(0,1)-Y(0,0)=13-10=3\) points. Their sum, \(8+3=11\), equals A’s total effect.

For person B, \(\mathrm{NDE}(1)=Y(1,0)-Y(0,0)=11-8=3\) points, and \(\mathrm{NIE}(0)=Y(0,0)-Y(0,1)=8-12=-4\) points. Their sum is \(3+(-4)=-1\) point, B’s total effect.

Averaging gives \(\mathrm{NDE}(1)=(8+3)/2=5.5\) points and \(\mathrm{NIE}(0)=(3+(-4))/2=-0.5\) point. The sum is \(5.5+(-0.5)=5\) points. The intermediate outcome in each person’s calculation appears once with a positive sign and once with a negative sign, so it cancels.

A Useful Special Case

If treatment does not change the mediator for any person, then \(M(1)=M(0)\) for everyone. Each natural indirect effect is consequently zero: its two nested outcomes have the same mediator value and the same treatment value. The total effect then equals either compatible natural direct effect. The following hypothetical example checks this implication directly.

Worked Example: No Change in the Mediator

Suppose three hypothetical people have \(M(0)=M(1)\), with common mediator values 2, 1, and 3, respectively. Their outcomes under treatment 0 at those mediator values are 30, 24, and 40 points; under treatment 1 they are 35, 26, and 43 points.

The individual total effects are \(35-30=5\), \(26-24=2\), and \(43-40=3\) points. Their mean is \((5+2+3)/3=10/3=3.333\) points. Since the mediator is unchanged for each person, each \(\mathrm{NIE}(1)\) is zero. The corresponding \(\mathrm{NDE}(0)\) values are 5, 2, and 3 points, giving a mean direct effect of \(10/3=3.333\) points. Thus the mean total effect equals the mean direct effect plus zero indirect effect.

Common Mistake: Pairing Mismatched Components

A tempting but generally incorrect calculation is to add \(\mathrm{NDE}(0)\) to \(\mathrm{NIE}(0)\). These terms do not share the intermediate outcome needed for cancellation: \(\mathrm{NDE}(0)\) compares outcomes at \(M(0)\), while \(\mathrm{NIE}(0)\) compares mediator values under an outcome treatment of 0. Their sum need not be the total effect.

In the two-person example, the mean \(\mathrm{NDE}(0)\) is 3.5 points, and the mean \(\mathrm{NIE}(0)\) is \((-1)/2=-0.5\) point. Their sum is 3 points, not the 5-point mean total effect. The correction is to pair \(\mathrm{NDE}(0)\) with \(\mathrm{NIE}(1)\), or \(\mathrm{NDE}(1)\) with \(\mathrm{NIE}(0)\). In each valid pair, the adjacent nested counterfactual outcomes match exactly.

Write the total-effect endpoints.
Start from \(Y(1,M(1))-Y(0,M(0))\).

Choose one intermediate outcome.
Insert either \(Y(1,M(0))\) or \(Y(0,M(1))\).

Match the two resulting contrasts.
Identify the direct and indirect effects that use the chosen intermediate outcome on opposite sides.

Average if the target is a population effect.
By linearity of expectation, the individual algebraic identity yields the corresponding population identity.

The main practical lesson is to check the counterfactual endpoints before adding reported components. Compatible reference values give an exact additive decomposition; mismatched values do not. This bookkeeping result clarifies what the component effects sum to, while leaving separate questions of identification and estimation to the assumptions and methods used in a particular analysis.

Check Your Understanding

For each question, track the nested counterfactual outcomes and the reference values.

Write the additive decomposition that uses \(\mathrm{NDE}(0)\), and identify the intermediate outcome that cancels.
Write the alternative decomposition that uses \(\mathrm{NIE}(0)\). Which mediator reference is used in its direct effect?
Why does the algebraic decomposition not, by itself, establish that the natural effects can be identified from observed data?
If \(M(1)=M(0)\) for every person, what happens to the natural indirect effects and the total effect decomposition?
Explain why adding \(\mathrm{NDE}(0)\) and \(\mathrm{NIE}(0)\) is not generally guaranteed to recover the total effect.