Additive Decomposition of Interventional Effects

Why Additive Decomposition Requires Compatible Effects

A total effect compares outcomes under two exposure levels. Mediation asks how much of that contrast is associated with changing the mediator distribution and how much remains when the mediator intervention is held fixed. For interventional effects, this division is defined through stochastic mediator interventions: rather than assigning each person a mediator value tied to their own counterfactual history, a policy draws a mediator value from a specified distribution.

On an additive scale, the decomposition is a difference of mean outcomes. Its arithmetic is simple, but the definitions must line up: the direct and indirect effects have to share an intermediate intervention and the same outcome scale, population, and covariate standardization. Otherwise, their sum may not be the total contrast of interest.

Definition: Let $A$ be an exposure with levels $a_0$ and $a_1$, $M$ a mediator, and $Y$ an outcome. Let $g_0$ and $g_1$ be two specified stochastic interventions on $M$, such as policies that draw mediator values from distributions associated with the two exposure levels. Write $\mu(a,g)$ for the mean outcome under an intervention setting $A=a$ and applying mediator policy $g$, in a specified target population. The interventional total effect for this pair of policies is $$ TE_{\mathrm{int}}=\mu(a_1,g_1)-\mu(a_0,g_0). $$

A policy may draw $M$ from a distribution conditional on baseline covariates $C$. If so, the same target population and the same covariate standardization must be used in every mean being compared. The notation $\mu(a,g)$ suppresses those details, but it does not make them optional. If $Y$ is measured in days, for example, every contrast below is also measured in days; if $Y$ is binary and the means are risks, the contrasts are risk differences.

The definition is causal: it refers to outcomes under interventions, not merely observed conditional means. The intervention must be sufficiently well specified to give $\mu(a,g)$ a meaning. Whether observed data identify those means requires additional assumptions; the decomposition below is an algebraic result and does not establish identification by itself.

The Two Standard Additive Decompositions

One way to decompose the total effect is to hold the mediator policy at $g_0$ while changing the exposure, then change the mediator policy while holding the exposure at $a_1$. This defines an interventional direct effect relative to $g_0$ and an interventional indirect effect at $a_1$:

IDE(g_0)=\mu(a_1,g_0)-\mu(a_0,g_0), \qquad IIE(a_1)=\mu(a_1,g_1)-\mu(a_1,g_0).

Their sum is the interventional total effect because the two middle terms cancel:

\begin{aligned} IDE(g_0)+IIE(a_1) &=\{\mu(a_1,g_0)-\mu(a_0,g_0)\} +\{\mu(a_1,g_1)-\mu(a_1,g_0)\}\\ &=\mu(a_1,g_1)-\mu(a_0,g_0)\\ &=TE_{\mathrm{int}}. \end{aligned}

There is a second valid decomposition. Hold the mediator policy at $g_1$ while changing the exposure, then change the mediator policy while holding the exposure at $a_0$:

IDE(g_1)=\mu(a_1,g_1)-\mu(a_0,g_1), \qquad IIE(a_0)=\mu(a_0,g_1)-\mu(a_0,g_0).

This pair also sums to $TE_{\mathrm{int}}$: the terms $\mu(a_0,g_1)$ cancel. The two decompositions are both correct, but their component effects need not be equal. When the outcome response to changing the mediator policy differs across exposure levels, the allocation of the total contrast between direct and indirect components depends on which exposure level or mediator policy is held fixed.

Theorem: For any well-defined means $\mu(a,g)$, the following additive decompositions hold: $$ TE_{\mathrm{int}}=IDE(g_0)+IIE(a_1)=IDE(g_1)+IIE(a_0). $$ Each equality follows by adding contrasts that share and cancel the same intermediate mean. No assumption of no exposure–mediator interaction is needed for these equalities. Such interaction can, however, make the component effects in the two decompositions different.

Worked Additive Decompositions

Worked Example: Decomposing a Difference in Mean Recovery Time

Suppose a hypothetical analysis defines two exposure levels, $a_0$ and $a_1$, and two mediator policies, $g_0$ and $g_1$. The outcome is mean recovery time in days in one fixed target population. The four intervention means are:

Exposure	Policy $g_0$	Policy $g_1$
$a_0$	20 days	23 days
$a_1$	26 days	32 days

The total interventional effect is the contrast between the two matched endpoints: $$ TE_{\mathrm{int}}=32-20=12\text{ days}. $$ Using $g_0$ as the fixed mediator policy for the direct effect, and $a_1$ for the indirect effect, gives: $$ IDE(g_0)=26-20=6\text{ days}, \qquad IIE(a_1)=32-26=6\text{ days}. $$ Thus $6+6=12$ days, which equals the total interventional effect.

The other valid decomposition is: $$ IDE(g_1)=32-23=9\text{ days}, \qquad IIE(a_0)=23-20=3\text{ days}. $$ Again, $9+3=12$ days. The two decompositions agree on the total but not on the allocation: the direct component is 6 days in the first decomposition and 9 days in the second. All four means refer to the same outcome, target population, and intervention definitions, so both calculations are coherent.

Worked Example: Standardizing Over a Fixed Covariate Population

Suppose a hypothetical target population consists of 60% low-risk and 40% high-risk individuals. The following are intervention-specific mean outcomes, in units of a continuous health score, within each covariate group:

Exposure and mediator policy	Low-risk mean	High-risk mean
$(a_0,g_0)$	10	20
$(a_1,g_0)$	13	24
$(a_1,g_1)$	16	28
$(a_0,g_1)$	12	25

To compare the intervention means, standardize every row to the same target population. For example, $$ \mu(a_0,g_0)=0.60(10)+0.40(20)=6+8=14. $$ The remaining standardized means are: $$ \begin{aligned} \mu(a_1,g_0)&=0.60(13)+0.40(24)=7.8+9.6=17.4,\\ \mu(a_1,g_1)&=0.60(16)+0.40(28)=9.6+11.2=20.8,\\ \mu(a_0,g_1)&=0.60(12)+0.40(25)=7.2+10=17.2. \end{aligned} $$

The total interventional effect is $20.8-14=6.8$ score units. The decomposition using $g_0$ is: $$ IDE(g_0)=17.4-14=3.4, \qquad IIE(a_1)=20.8-17.4=3.4, $$ and $3.4+3.4=6.8$ score units. If, instead, one used different covariate weights for different means, the cancellation would not represent a contrast in one fixed target population. The arithmetic might still produce numbers, but they would not be the stated decomposition.

Why the Components Can Differ

The two valid decompositions differ whenever the effect of switching mediator policies depends on exposure, or equivalently when the effect of changing exposure depends on the mediator policy. For the means above, compare the direct contrasts:

IDE(g_1)-IDE(g_0) = \{\mu(a_1,g_1)-\mu(a_1,g_0)\} - \{\mu(a_0,g_1)-\mu(a_0,g_0)\}.

The first bracket is the mediator-policy contrast at $a_1$; the second is that contrast at $a_0$. If they differ, the direct effects differ across reference policies, and the indirect effects must differ correspondingly so that their sums remain the same. This is an additive-scale interaction between exposure and mediator policy in the mean outcome. A zero interaction would make the two versions of each component equal, but it is not required for either decomposition to hold.

Worked Example: A Mismatched Pair Does Not Recover the Total Effect

Use the four means from the first example. A direct effect at $g_0$ is $IDE(g_0)=26-20=6$ days. A tempting but mismatched indirect contrast is the change in mediator policy at $a_0$, $IIE(a_0)=23-20=3$ days. Adding them gives $6+3=9$ days, not the total effect of 12 days.

The reason is visible by writing out the terms: $$ \begin{aligned} IDE(g_0)+IIE(a_0) &=\{\mu(a_1,g_0)-\mu(a_0,g_0)\} +\{\mu(a_0,g_1)-\mu(a_0,g_0)\}\\ &=26-20+23-20\\ &=9\text{ days}. \end{aligned} $$ There is no cancellation that leaves $\mu(a_1,g_1)-\mu(a_0,g_0)$. The indirect effect at $a_0$ pairs with $IDE(g_1)$, not $IDE(g_0)$: $9+3=12$ days. Alternatively, $IDE(g_0)$ pairs with $IIE(a_1)$: $6+6=12$ days.

The mismatch does not mean that either component is inherently invalid. It means the selected pair of contrasts has incompatible intermediate means for the stated total-effect target.

When the Decomposition Does and Does Not Apply

The telescoping identity applies when the components and total effect are defined using the same four means, on the same additive outcome scale and for the same target population. The direct and indirect contrasts must pass through a shared intermediate regime. For example, $IDE(g_0)$ and $IIE(a_1)$ both use $\mu(a_1,g_0)$, so that term cancels. Changing the mediator policy definition, the covariate standardization, or the outcome scale partway through the calculation can destroy that compatibility.

A second distinction is between the interventional total effect defined here and another total-effect estimand. The pair $\mu(a_1,g_1)-\mu(a_0,g_0)$ is, by definition, the total contrast for the specified exposure and mediator-policy interventions. It need not equal a total effect defined under a different joint intervention on exposure and mediator. Equality requires that the endpoint regimes represent the target total-effect regimes. The algebra alone cannot establish that they do.

Likewise, algebraic additivity is not proof that the intervention means can be estimated from observed data. Causal interpretation requires well-defined exposure and mediator interventions and consistency between those interventions and the outcomes being considered. Identification from observational data requires further conditions. Keep these questions separate: first define and decompose the causal estimand; then assess whether available data identify its terms.

Common Mistakes

Adding component effects with different intermediate regimes. A direct effect at $g_0$ pairs with an indirect effect at $a_1$; an indirect effect at $a_0$ instead pairs with a direct effect at $g_1$.
Assuming the components are unique. The total contrast can have more than one valid additive decomposition. The chosen reference policy or exposure level determines how the total is allocated.
Changing the population across means. If covariate-specific results are standardized, use the same target-population weights for every intervention mean in the decomposition.
Treating the identity as an identification result. Cancellation proves an equality among defined means; it does not show that those means are recoverable from observed data.
Interpreting an additive decomposition on another scale. The displayed components are differences in means. A ratio-scale contrast, such as a risk ratio, does not generally split into these same additive pieces.

Key takeaway: On the additive scale, a compatible interventional direct effect and indirect effect sum to their interventional total effect because their shared intermediate mean cancels. The reference choice affects the component sizes, while incompatible definitions or targets can make the sum miss the intended total contrast.

Check Your Understanding

Use the definitions and cancellation argument above to answer these questions.

For which indirect effect does $IDE(g_0)$ form a valid decomposition of $TE_{\mathrm{int}}$, and what intermediate mean cancels?
Why can $IDE(g_0)$ and $IDE(g_1)$ differ even though both can be used in a valid decomposition?
If the outcome is a binary event and $\mu(a,g)$ is a risk, what units does the additive decomposition use?
What can go wrong if the four intervention means are standardized to different covariate distributions?
Does the telescoping identity establish that the intervention means are identified from observational data? Explain briefly.

Exposure and mediator policy	Low-risk mean	High-risk mean
\((a_0,g_0)\)	10	20
\((a_1,g_0)\)	13	24
\((a_1,g_1)\)	16	28
\((a_0,g_1)\)	12	25