Pure Indirect Effect in Mediation

• Pure indirect effect is a mediation estimand that quantifies the portion of a treatment's impact transmitted through a mediator instead of alternate pathways.
• It is derived from nested counterfactual contrasts and expressed via formulations like Pearl’s and Lok’s, with implications differing across linear and nonlinear models.
• Alternative estimands such as organic and controlled indirect effects address challenges like unmeasured confounding and non-manipulability of mediators.

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1. Canonical formulations

With binary treatment $A\in\{0,1\}$, mediator $M$, and outcome $Y$, standard potential-outcomes notation uses $Y_{a,m}$ for the outcome under treatment $a$ and mediator value $m$, $M_a$ for the mediator under treatment $E(Y_{1,M_1})-E(Y_{1,M_0})$^^^^1server1^^^^, and $E(Y_{1,M_1})-E(Y_{1,M_0})$1 by consistency. In the formulation emphasized by Lok, the natural direct effect at the unit level is $E(Y_{1,M_1})-E(Y_{1,M_0})$2, while the natural indirect effect—often also treated as the pure indirect effect when paired with the pure/natural direct effect—is $E(Y_{1,M_1})-E(Y_{1,M_0})$3; population versions are obtained by taking expectations, yielding

$E(Y_{1,M_1})-E(Y_{1,M_0})$4

Pearl’s symmetric formulation instead defines the natural/pure indirect effect as

$E(Y_{1,M_1})-E(Y_{1,M_0})$5

with companion natural/pure direct effect

$E(Y_{1,M_1})-E(Y_{1,M_0})$6

These are mathematically different presentations of the same natural-effects program, and they explain why the phrase “pure indirect effect” can point either to a baseline-reference or a treated-reference nested counterfactual contrast (Lok, 2015, &&&^^^^1server1^^^^&&&).

A further subtlety is scale. Pearl shows that in general nonlinear models the exact decomposition is written with a reverse transition,

$E(Y_{1,M_1})-E(Y_{1,M_0})$7

and

$E(Y_{1,M_1})-E(Y_{1,M_0})$8

whereas the familiar additive identity $E(Y_{1,M_1})-E(Y_{1,M_0})$9 is recovered in linear systems. Thus, “pure indirect effect” is conceptually stable as a nested counterfactual object, but its algebraic relation to total effect depends on the model class and effect scale (&&&^^^^1server1^^^^&&&).

2. Cross-world structure and identification

The defining feature of the pure indirect effect is its cross-world structure. Quantities such as $E\{Y(0,M(1))-Y(0,M(0))\}$^^^^1server1^^^^ or $E\{Y(0,M(1))-Y(0,M(0))\}$1 combine an outcome under one treatment regime with a mediator value arising under another. Lok emphasizes that this is conceptually demanding for two reasons: $E\{Y(0,M(1))-Y(0,M(0))\}$2 is unobserved for treated units, and the notation presupposes that $E\{Y(0,M(1))-Y(0,M(0))\}$3 is meaningful for all $E\{Y(0,M(1))-Y(0,M(0))\}$4, which may be unrealistic when the mediator is not itself a manipulable treatment (Lok, 2015).

Standard identification therefore requires more than randomization of treatment. The usual conditions combine consistency and positivity with no unmeasured confounding of the $E\{Y(0,M(1))-Y(0,M(0))\}$5-$E\{Y(0,M(1))-Y(0,M(0))\}$6, $E\{Y(0,M(1))-Y(0,M(0))\}$7-$E\{Y(0,M(1))-Y(0,M(0))\}$8, and $E\{Y(0,M(1))-Y(0,M(0))\}$9-$A\in\{0,1\}$^^^^1server1^^^^ relations, plus a cross-world condition such as

$A\in\{0,1\}$1

or, equivalently in another notation,

$A\in\{0,1\}$2

Under these assumptions, the familiar mediation formula identifies the nested counterfactual mean:

$A\in\{0,1\}$3

An equivalent standard formula appears in front-door-based mediation notation as

$A\in\{0,1\}$4

These formulas are the identification workhorse for the pure/natural indirect effect family (Lok, 2015, Miles, 2022, &&&1^^^^1server1^^^^&&&).

This identification burden is also the main weakness of the classical pure indirect effect. Fulcher and coauthors make the negative point explicitly: the classic natural indirect effect is not robust to unmeasured exposure–outcome confounding because its identification requires their assumption $A\in\{0,1\}$5, in addition to assumptions ruling out unmeasured exposure–mediator and mediator–outcome confounding and exposure-induced mediator–outcome confounding (&&&1^^^^1server1^^^^&&&).

3. Mediational interpretation, null criteria, and interaction

Because the pure indirect effect follows each unit’s own mediator response across treatment worlds, it is often treated as the benchmark estimand for genuine mediation. This benchmark status is formalized by the criteria introduced in the analysis of randomized interventional indirect effects. There, the natural indirect effect satisfies the sharp null criterion and the monotonicity criterion: if no individual-level indirect effect exists, then the population-level natural indirect effect must be zero; by contrast, randomized interventional indirect effects can be nonzero even when no individual in the population exhibits an indirect effect (Miles, 2022).

A later comparison of natural, interventional, and separable indirect effects sharpens this point. That paper defines the natural indirect effect as

$A\in\{0,1\}$6

and argues that it is the most fundamental of the three because it satisfies the mediation null criterion by construction. In the same analysis, separable indirect effects and interventional indirect effects acquire a proper mediational interpretation only when additional assumptions force them to coincide with the natural indirect effect (Chen et al., 5 Jul 2025).

The benchmark role of the pure/natural indirect effect does not end the controversy. A separate line of work argues that standard two-way decompositions are path-dependent and bury treatment–mediator interaction inside the indirect effect. The proposed path-free decomposition

$A\in\{0,1\}$7

splits total effect into direct, indirect, and treatment–mediator interaction components. In that view, the standard pure/natural indirect effect is not rejected as formally invalid, but it is criticized for conflating mediation with interaction (Lee, 2021).

4. Alternatives to the classical pure indirect effect

Several alternative estimands have been proposed because the pure indirect effect may be undefined, hard to identify, or poorly aligned with particular intervention questions.

Estimand family	Defining contrast	Relation to pure indirect effect
Organic indirect effect	$A\in\{0,1\}$8 or, relative to no treatment, $A\in\{0,1\}$9	Replaces unit-specific mediator setting by a distribution-matching intervention
Population intervention indirect effect	$M$^^^^1server1^^^^	Alternative mediated component of the population intervention effect
Randomized interventional indirect effect	$M$1	Uses random draws from mediator distributions rather than each unit’s own $M$2
Controlled indirect effect	$M$3	Single-mediator intervention effect, not a natural/pure effect

Organic effects are motivated by the claim that setting the mediator to exact unit-specific cross-world values is often not meaningful. Lok defines an intervention $M$4 as organic with respect to $M$5 if

$M$6

and

$M$7

The resulting organic indirect effect is $M$8. Under the usual cross-world assumptions, natural effects are special cases of organic effects; without those assumptions, the organic formulation remains meaningful when exact mediator setting does not (Lok, 2015).

The population intervention indirect effect,

$M$9

is introduced as a different mediated-effect target that remains identifiable under unmeasured exposure–outcome confounding, provided the exposure–mediator and mediator–outcome relations are sufficiently unconfounded. It is therefore a weaker-assumption alternative to the classical pure/natural indirect effect, not a mere relabeling of it (&&&1^^^^1server1^^^^&&&).

The controlled indirect effect is a further departure. In clinically oriented multi-mediator work, the estimand

$Y$^^^^1server1^^^^

measures the effect of intervening on one manipulable mediator while allowing the rest of the mediator system to evolve naturally under treatment $Y$1. That framework is explicitly presented as an alternative to pure/natural mediation because it avoids cross-world quantities and focuses on single-mediator interventions; the companion scaled controlled indirect effect mixes mediation and interaction rather than isolating a pure natural pathway (Sun et al., 2020).

A related criticism concerns reference-level dependence. One recent proposal argues that the average natural/pure indirect effect can be unsuitable for path-deactivation policy questions because the answer depends on which exposure level is chosen as the reference world. The paper therefore introduces a front-door-based alternative indirect-effect functional $Y$2 intended for settings where the target question is the effect of deactivating the direct path rather than the effect relative to a chosen baseline exposure (Peña, 2023).

5. Extensions, estimation frameworks, and specialized settings

In high-dimensional linear mediation, the principal target is often a product/path coefficient rather than a nonparametric nested counterfactual. The indirect effect is written

$Y$3

and the supplement shows that under the linear model with no interactions and the usual no-unmeasured-confounding assumptions,

$Y$4

In that setting, $Y$5 is the average natural indirect effect per unit exposure change, so high-dimensional de-biasing methods can be interpreted as inference for a pure/natural indirect effect only under the linear causal model and its assumptions (Zhou et al., 2019).

Distributional refinements of the pure indirect effect also exist. A quantile-based decomposition defines $Y$6-specific indirect effects and shows that

$Y$7

Under the common convention used there, $Y$8 is the classical pure indirect effect, so the average PIE is decomposed into mediator-quantile-specific contributions. This makes explicit where in the mediator distribution the indirect pathway operates (Geraci et al., 2017).

Binary recursive path analysis provides another exact but specialized extension. For a binary treatment, binary mediator, and binary outcome under logistic models, the marginal log-odds effect decomposes as

$Y$9

where the residual term reflects nonlinearity and non-collapsibility. In that framework the indirect effect is first defined by path deletion, but under structural causal interpretation and no unmeasured confounding it can be interpreted as the pure natural indirect effect. The presence of the residual term shows that even when a pure indirect effect is well defined, simple additive mediation algebra may fail on the log-odds scale (Raggi et al., 2021).

More recent work extends pure indirect effects to settings that are not standard single-time mediation problems. In semi-competing risks, indirect effects on a terminal event are defined through cross-world event-process counterfactuals and can be decomposed either by holding fixed the prevalence of the non-terminal event among survivors or by holding fixed the hazard of the non-terminal event; both are natural indirect effects in the paper’s terminology, but they answer different causal questions (Deng et al., 2023). In HIV cure-screening applications, the pure indirect effect relative to no treatment,

$Y_{a,m}$^^^^1server1^^^^

or its organic analogue

$Y_{a,m}$1

is used because treatment enters only through the mediator distribution. Under a probit outcome model and normal mediator model, the mediation formula remains valid regardless of whether there is treatment–mediator interaction in the outcome model, and the effect can be estimated from a hypothesized treatment-induced shift in the mediator plus untreated outcome data, with explicit corrections for assay limits and measurement error (Herath et al., 15 Jul 2025).

6. Distinction from interference-based “indirect effects”

The phrase “indirect effect” is also used outside mediation, most notably in the causal inference literature on interference. There the average indirect effect is

$Y_{a,m}$2

which measures the effect of one unit’s treatment on other units’ outcomes. This estimand is explicitly introduced as an indirect effect under interference, not as a pure/natural indirect effect in the mediation-analysis sense. It involves no mediator, no path-specific decomposition, and no same-unit cross-world contrast such as $Y_{a,m}$3 or $Y_{a,m}$4 (Hu et al., 2021).

Accordingly, the term “pure indirect effect” is most precise when reserved for mediation estimands in the natural-effects family, or for alternatives explicitly designed in relation to that family. In adjacent literatures the adjective “indirect” may instead refer to stochastic intervention contrasts, controlled single-mediator intervention effects, or cross-unit spillovers, each corresponding to a distinct causal question rather than a variant of the classical pure indirect effect.