Direct and Indirect Effects

article2001UAI2,402 citations

Establishes a formal framework for causal mediation analysis in nonlinear and nonparametric models by defining path-specific direct and indirect effects and deriving the conditions required to estimate them from empirical data.

Listen

Organizations routinely face complex policy, legal, and healthcare challenges where they must separate the direct impact of an action from its indirect consequences. For instance, legal standards in hiring discrimination require evaluating whether decisions depend directly on protected characteristics rather than applicant qualifications, while medical policies must distinguish a drug's direct therapeutic efficacy from behavior triggered by side effects. While traditional structural equation models estimate these pathways under strict linear assumptions, real-world systems are predominantly nonlinear, leaving decision-makers without a rigorous framework to quantify indirect effects or evaluate interventions that alter causal pathways.

The article establishes a formal mathematical and operational foundation to define, measure, and estimate direct, indirect, and path-specific effects in both linear and nonlinear causal models. It aims to determine the exact conditions required to identify these distinct causal pathways from standard experimental and observational data.

To achieve this, the article uses structural counterfactual analysis and graphical causal modeling. Rather than relying on prescriptive interventions that fix intermediate variables to uniform values, the approach introduces a descriptive framework based on natural behaviors. It conceptualizes effects through path-deactivation, evaluating what occurs when specific pathways of influence are modified or severed while intermediate variables follow their naturally occurring distributions across the population.

The analysis yields several key findings. First, it demonstrates that indirect effects and natural direct effects cannot be isolated by simply fixing intermediate variables to uniform values; instead, they require descriptive formulations that track natural baseline variations across individuals. Second, the article proves that direct and indirect effects are fully identifiable in standard Markovian models—acyclic systems without unmeasured confounders—and can be computed consistently from observational data. Third, it establishes formal graphical criteria for non-Markovian systems, showing that natural effects can be identified from experimental or observational data whenever specific sets of non-descendant background variables block confounding paths between mediators and outcomes. Fourth, the article shows that while direct and indirect effects simply add up to the total effect in linear models, nonlinear systems follow a more subtle relationship: the total effect equals the natural indirect effect minus the reverse transition of the natural direct effect. Finally, the analysis introduces a broader definition of path-specific effects, demonstrating that isolating effects along arbitrary intermediate paths involves far more restrictive conditions and is not generally identifiable even in basic Markovian systems.

These findings have major implications for policy evaluation, risk assessment, and decision analysis. Decision-makers often consider nonstandard policy options, such as eliminating an adverse side effect, withholding information from a competitor, or preventing hiring managers from asking about demographic attributes. Because these actions deactivate specific causal links rather than forcing intermediate variables to fixed values, standard experimental metrics often fail to evaluate them. The proposed framework allows leaders to evaluate these targeted structural interventions without introducing artificial model variables or relying on impossible multi-stage experiments on the same individuals.

Organizations evaluating policy or process changes should incorporate these natural direct and indirect effect formulations into their causal analysis pipelines. When planning data collection, analysts should map intermediate variables and proactively gather non-descendant background factors to satisfy the necessary graphical independence criteria. For broader research initiatives, further work is required to systematically characterize which complex path-specific subgraphs can be identified in observational settings.

Confidence in these findings is high regarding the formal mathematical proofs and graphical identification conditions. However, practical application requires caution, as conclusions depend heavily on the validity of the underlying causal diagram and the assumption that all necessary confounding factors have been measured without bias.

Cognitive Systems Laboratory University of California, Los Angeles correlation analysis http standard model path-specific effects direct effects indirect effects

arXiv: 1301.2300

Book: A First Course in Causal Inference, Peng Ding (2024). This book establishes the foundational potential outcomes framework and causal inference principles that underpin the formal definition of direct and indirect effects.
Paper: Counterfactual Fairness, Matt J. Kusner et al. (2017). This single-chapter ebook utilizes structural causal models and counterfactual definitions that directly rely on path-specific effect formulations.

Paper: Design and Analysis of Switchback Experiments, Iavor Bojinov et al. (2023). This single-chapter ebook extends causal effect estimation methods to switchback experimental designs while managing interference over time.

Abstract

The direct effect of one eventon another can be defined and measured byholding constant all intermediate variables between the this http URL effects present conceptual andpractical difficulties (in nonlinear models), because they cannot be isolated by holding certain variablesconstant. This paper shows a way of defining any path-specific effectthat does not invoke blocking the this http URL permits the assessment of a more naturaltype of direct and indirect effects, one thatis applicable in both linear and nonlinear models. The paper establishesconditions under which such assessments can be estimated consistentlyfrom experimental and nonexperimental data,and thus extends path-analytic techniques tononlinear and nonparametric models.

Topics

correlation analysis http standard model stochastic processes

Keywords

path-specific effects direct effects indirect effects total effects back-door criterion causal graph latent variables

1 INTRODUCTION
2 CONCEPTUAL ANALYSIS
2.1 Direct versus Total Effects
2.2 Descriptive versus prescriptive interpretation
2.3 Policy implications of the Descriptive interpretation
2.4 Descriptive interpretation of indirect effects
3 FORMAL ANALYSIS
3.1 Notation
3.2 Controlled Direct Effects (review)
3.3 Natural Direct Effects: Formulation
3.4 Natural Direct Effects: Identification
3.5 Natural Indirect Effects: Formulation
3.6 Natural Indirect Effects: Identification
3.7 General Path-specific Effects
4 Conclusions
Acknowledgment
References

Knowls

Knowl 1 — Natural Direct Effect
definition
In a structural causal model with treatment variable $X$ , response variable $Y$ , and intermediate mediator variables $Z$ , the unit-level natural direct effect of a transition from reference value $X = x^*$ to $X = x$ in unit situation $U = u$ is defined as:

$NDE(x, x^*; Y, u) = Y_{x, Z_{x^*}(u)}(u) - Y_{x^*}(u)$

where $Y_{x, z}(u)$ denotes the potential outcome of $Y$ under external interventions $do(X = x)$ and $do(Z = z)$ , $Z_{x^*}(u)$ is the value that $Z$ would attain in unit $u$ under $do(X = x^*)$ , and $Y_{x^*}(u) = Y_{x^*, Z_{x^*}(u)}(u)$ . Qualitatively, $X = x$ has a natural direct effect on $Y$ in situation $u$ if $Y_{x, Z_{x^*}(u)}(u) \neq Y_{x^*}(u)$ .

The average natural direct effect across a population of units is the expectation over $U$ :

$NDE(x, x^*; Y) = \mathbb{E}[Y_{x, Z_{x^*}}] - \mathbb{E}[Y_{x^*}]$

This quantity measures the expected change in $Y$ when $X$ changes from $x^*$ to $x$ while holding the mediator $Z$ constant at whatever value $Z$ would have naturally attained for each individual under the baseline condition $X = x^*$ .
Knowl 2 — Natural Indirect Effect
definition
In a structural causal model with treatment variable $X$ , response variable $Y$ , and mediating variables $Z$ , the unit-level natural indirect effect of a transition from baseline $X = x^*$ to $X = x$ in unit situation $U = u$ is defined as:

$NIE(x, x^*; Y, u) = Y_{x^*, Z_x(u)}(u) - Y_{x^*}(u)$

where $Y_{x^*, Z_x(u)}(u)$ denotes the value that $Y$ would attain under intervention $do(X = x^*)$ combined with setting $Z$ to the value $Z_x(u)$ that $Z$ would have attained in unit $u$ under $do(X = x)$ , and $Y_{x^*}(u) = Y_{x^*, Z_{x^*}(u)}(u)$ . Qualitatively, $X = x$ has an indirect effect on $Y$ in situation $u$ if $Y_{x^*, Z_x(u)}(u) \neq Y_{x^*}(u)$ .

The average natural indirect effect across a population of units is given by:

$NIE(x, x^*; Y) = \mathbb{E}[Y_{x^*, Z_x}] - \mathbb{E}[Y_{x^*}]$

This measures the expected change in outcome $Y$ when the treatment variable is kept fixed at its reference level $x^*$ while the mediator $Z$ changes to the value it would have attained under treatment $X = x$ .
Knowl 3 — Total Effect Decomposition for Nonlinear Models
theoretical result
In general nonlinear, nonparametric structural causal models, the total causal effect $TE(x, x^*; Y) = \mathbb{E}[Y_x] - \mathbb{E}[Y_{x^*}]$ relates to the average natural direct effect ( $NDE$ ) and average natural indirect effect ( $NIE$ ) through the exact identities:

$TE(x, x^*; Y) = NIE(x, x^*; Y) - NDE(x^*, x; Y)$

$TE(x, x^*; Y) = NDE(x, x^*; Y) - NIE(x^*, x; Y)$

In linear structural equation models, where the effect of a transition from $x^*$ to $x$ is proportional to $x - x^*$ and thus satisfies $NDE(x^*, x; Y) = -NDE(x, x^*; Y)$ and $NIE(x^*, x; Y) = -NIE(x, x^*; Y)$ , these formulas reduce to the standard additive decomposition:

$TE(x, x^*; Y) = NDE(x, x^*; Y) + NIE(x, x^*; Y)$

In nonlinear models, total effect does not generally equal $NDE(x, x^*; Y) + NIE(x, x^*; Y)$ due to interaction effects between the treatment and the mediator.
Knowl 4 — Experimental Identification of the Natural Direct Effect
theoretical result
Let $X$ be the treatment variable, $Y$ the response variable, and $Z$ the mediating variables. If there exists a set $W$ of covariates that are nondescendants of $X$ or $Z$ such that the potential outcomes satisfy conditional counterfactual independence:

$Y_{xz} \perp\!\!\perp Z_{x^*} \mid W \quad \text{for all } z \text{ and } x$

then the average natural direct effect $NDE(x, x^*; Y)$ is experimentally identifiable from randomized interventions on $(X, Z)$ and $X$ , and is given by:

$NDE(x, x^*; Y) = \sum_{w, z} \left[ \mathbb{E}[Y_{xz} \mid W = w] - \mathbb{E}[Y_{x^*z} \mid W = w] \right] P(Z_{x^*} = z \mid W = w) P(W = w)$

A sufficient graphical condition for this independence is that $W$ d-separates $Y$ from $Z$ in the graph $G_{\underline{X}\overline{Z}}$ obtained by removing all arrows emerging from $X$ and all arrows entering $Z$ : $(Y \perp\!\!\perp Z \mid W)_{G_{\underline{X}\overline{Z}}}$ .
Knowl 5 — Nonexperimental Identification and Mediation Formula for Natural Direct Effects
theoretical result
The average natural direct effect $NDE(x, x^*; Y)$ is identifiable from nonexperimental (observational) data if there exists a covariate set $W$ of nondescendants of $X$ or $Z$ such that $Y_{xz} \perp\!\!\perp Z_{x^*} \mid W$ , and both $P(Y_{xz} = y \mid W = w)$ and $P(Z_{x^*} = z \mid W = w)$ are identifiable from observational data.

In Markovian causal models (acyclic structural equation models with mutually independent error terms), the conditional independence holds with $W = \emptyset$ . If $S$ is any set of covariates satisfying the back-door criterion between $X$ and $Z$ , the average natural direct effect is identified by:

$NDE(x, x^*; Y) = \sum_{s, z} \left[ \mathbb{E}[Y \mid X = x, Z = z] - \mathbb{E}[Y \mid X = x^*, Z = z] \right] P(Z = z \mid X = x^*, S = s) P(S = s)$

When the relationship between $X$ and $Z$ is unconfounded ( $P(Z_{x^*} = z) = P(Z = z \mid X = x^*)$ ), this simplifies to the Mediation Formula:

$NDE(x, x^*; Y) = \sum_z \left[ \mathbb{E}[Y \mid X = x, Z = z] - \mathbb{E}[Y \mid X = x^*, Z = z] \right] P(Z = z \mid X = x^*)$
Knowl 6 — Identification of Natural Indirect Effects
theoretical result
If there exists a set $W$ of covariates, nondescendants of $X$ or $Z$ , such that $Y_{x^*z} \perp\!\!\perp Z_x \mid W$ for all $x$ and $z$ , the average natural indirect effect $NIE(x, x^*; Y)$ is experimentally identifiable and given by:

$NIE(x, x^*; Y) = \sum_{w, z} \mathbb{E}[Y_{x^*z} \mid W = w] \left[ P(Z_x = z \mid W = w) - P(Z_{x^*} = z \mid W = w) \right] P(W = w)$

In observational studies, $NIE(x, x^*; Y)$ is identifiable whenever $\mathbb{E}[Y_{x^*z} \mid W = w]$ , $P(Z_x = z \mid W = w)$ , and $P(Z_{x^*} = z \mid W = w)$ are identifiable for all $z, w$ .

In unconfounded Markovian models, this reduces to:

$NIE(x, x^*; Y) = \sum_z \mathbb{E}[Y \mid X = x^*, Z = z] \left[ P(Z = z \mid X = x) - P(Z = z \mid X = x^*) \right]$
Knowl 7 — Controlled Direct Effect
definition
In a structural causal model with causal graph $G$ , treatment variable $X$ , response variable $Y$ , and intermediate variables $Z$ (specifically the parents of $Y$ in $G$ excluding $X$ ), the unit-level controlled direct effect of transition $X = x^*$ to $X = x$ under setting $Z = z$ for unit $U = u$ is defined as:

$CDE_z(x, x^*; Y, u) = Y_{xz}(u) - Y_{x^*z}(u)$

The average controlled direct effect across the population is:

$CDE_z(x, x^*; Y) = \mathbb{E}[Y_{xz} - Y_{x^*z}]$

Qualitatively, $X$ has a controlled direct effect on $Y$ in unit $u$ if there exist $z, x, x^*$ such that $Y_{xz}(u) \neq Y_{x^*z}(u)$ . Unlike natural direct effects, controlled direct effects represent prescriptive policies where the mediator $Z$ is fixed uniformly across all individuals to a specified value $z$ via external intervention $do(Z = z)$ .
Knowl 8 — Path-Specific Effects via Structural Path Deactivation
model/method
Let $G$ be the causal graph associated with structural causal model $M$ , and let $g$ be an edge-subgraph of $G$ containing the paths selected for effect analysis. The $g$ -specific effect of transitioning treatment $X$ from baseline $x^*$ to $x$ is defined through a modified model $M_g^*$ .

For each variable $X_i$ with parent set $PA_i$ in $G$ , partition $PA_i$ into $PA_i(g)$ (parents linked to $X_i$ in $g$ ) and $PA_i(\overline{g})$ (parents not linked to $X_i$ in $g$ ). Replace each structural equation $f_i(pa_i, u)$ in $M$ with:

$f_i^*(pa_i, u; g) = f_i\left(pa_i(g), pa_i^*(g), u\right)$

where $pa_i^*(g) = PA_i(\overline{g})_{x^*}(u)$ is the value that variables in $PA_i(\overline{g})$ would attain in $M$ under $X = x^*$ .

The unit-level $g$ -specific effect of $x$ on $Y$ relative to reference $x^*$ is defined as the total effect in $M_g^*$ :

$SE_g(x, x^*; Y, u)_M = TE(x, x^*; Y, u)_{M_g^*}$

This formulation evaluates effects along specific path subsets by freezing non-selected pathways to their reference baseline values rather than fixing nodes to constant values.
Knowl 9 — Non-Identifiability of General Path-Specific Effects in Markovian Models
theoretical result
While direct and indirect effects are identifiable from observational data in Markovian models (acyclic structural models with mutually independent error terms), general path-specific effects transmitted through arbitrary subgraphs $g$ are not generally identifiable in Markovian models.

For example, in a Markovian model with structural equations $Z = f_z(X, U_z)$ , $W = f_w(Z, X, U_w)$ , and $Y = f_y(Z, W, U_y)$ , the path-specific effect along the path $g: X \to Z \to W \to Y$ is given at the unit level by:

$TE(x, x^*; Y, u)_{M_g^*} = f_y\left(f_z(x^*, U_z), f_w(f_z(x, U_z), x^*, U_w), U_y\right) - Y_{x^*}(u)$

This expression cannot be identified from nonexperimental distributions or standard experimental manipulations because it requires evaluating downstream equations under conflicting hypothetical states of the mediator $Z$ simultaneously ( $Z$ evaluated under $X = x^*$ for the direct link $Z \to Y$ , but under $X = x$ for the mediated link $Z \to W$ ).

Coverage note — None was omitted; all key definitions, identification theorems, decomposition formulas, graphical criteria, and path-specific generalizations contributed by the paper are fully captured.

References

1.J.L. Gastwirth. Statistical evidence in discrimination cases. Journal of the Royal Statical Society, Series A, 160(Part 2):289–303, 1997.
2.A.S. Goldberger. Structural equation models in the social sciences. Econometrica: Journal of the Econometric Society, 40:979–1001, 1972.
3.J. Hagenaars. Loglinear Models with Latent Variables. Sage Publications, Newbury Park, CA, 1993.
4.G. Hesslow. Discussion: Two notes on the probabilistic approach to causality. Philosophy of Science, 43:290–292, 1976.
5.D. Lewis. Counterfactuals and comparative probability. Journal of Philosophical Logic, 2, 1973.
6.J. Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, New York, 2000.
7.J.M. Robins and S. Greenland. Identifiability and exchangeability for direct and indirect effects. Epidemiology, 3(2):143–155, 1992.
8.D.B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66:688–701, 1974.

Citation

MLA

Pearl, J. “Direct and Indirect Effects”. Probabilistic and Causal Inference, ACM, 2022, pp. 373–92, https://doi.org/10.1145/3501714.3501736.

APA

Pearl, J. (2022). Direct and Indirect Effects. In Probabilistic and Causal Inference (pp. 373–392). ACM. https://doi.org/10.1145/3501714.3501736

Chicago

Pearl, J. 2022. “Direct and Indirect Effects”. In Probabilistic and Causal Inference. ACM. https://doi.org/10.1145/3501714.3501736.

Harvard

Pearl, J. (2022) “Direct and Indirect Effects”, Probabilistic and Causal Inference. ACM, pp. 373–392. Available at: https://doi.org/10.1145/3501714.3501736.

Vancouver

1. Pearl J (2022) Direct and Indirect Effects. In: Probabilistic and Causal Inference. ACM, pp 373–392

BibTeX

@inbook{Pearl_2022, title={Direct and Indirect Effects}, ISBN={9781450395861}, url={http://dx.doi.org/10.1145/3501714.3501736}, DOI={10.1145/3501714.3501736}, booktitle={Probabilistic and Causal Inference}, publisher={ACM}, author={Pearl, Judea}, year={2022}, month=Feb, pages={373–392} }

Metadata:Crossref

Access the Paper

This paper is available from its original source. Click below to access the PDF.

Open PDF

License: https://creativecommons.org/licenses/by/4.0/

Direct and Indirect Effects

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Natural Direct Effect

Knowl 2 — Natural Indirect Effect

Knowl 3 — Total Effect Decomposition for Nonlinear Models

Knowl 4 — Experimental Identification of the Natural Direct Effect

Knowl 5 — Nonexperimental Identification and Mediation Formula for Natural Direct Effects

Knowl 6 — Identification of Natural Indirect Effects

Knowl 7 — Controlled Direct Effect

Knowl 8 — Path-Specific Effects via Structural Path Deactivation

Knowl 9 — Non-Identifiability of General Path-Specific Effects in Markovian Models

References

Citation

BibTeX

Access the Paper