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arxiv: 2606.22165 · v1 · pith:25ZYDKLRnew · submitted 2026-06-20 · 📊 stat.ME

Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis

Yuta Kawakami , Jin Tian This is my paper

Pith reviewed 2026-06-26 11:25 UTC · model grok-4.3

classification 📊 stat.ME
keywords causal mediation analysisnatural direct effectsnatural indirect effectscontinuous treatmentsskew-symmetryadditivityeffect decomposition
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The pith

Cumulative natural direct and indirect effects decompose total effects while preserving skew-symmetry and additivity for continuous treatments.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Standard natural direct and indirect effects can produce paradoxical interpretations in mediation analysis because they fail to satisfy skew-symmetry and additivity. The paper defines cumulative natural direct and indirect effects by decomposing the local causal effect given by the expected derivative of the outcome with respect to a continuous treatment. These new measures integrate the local direct and indirect components to recover a decomposition of the total effect that obeys both required properties. Discrete analogues are defined for ordinal treatments that inherit the same structural guarantees. The construction relies on the usual causal identification assumptions and is illustrated on linear models with interaction.

Core claim

We introduce the cumulative natural direct effect (CNDE) and cumulative natural indirect effect (CNIE) for continuous treatments by decomposing the local causal effect E[∂_x Y_x] into local direct and indirect effects. These measures satisfy a decomposition of the total effect that is skew-symmetric and additive. We extend the framework to ordinal treatments with discrete analogues that preserve these properties. Identification results are established under standard causal assumptions.

What carries the argument

Cumulative natural direct and indirect effects, obtained by integrating local direct and indirect effects derived from the derivative of the outcome with respect to the treatment.

If this is right

  • The total effect decomposes additively into CNDE and CNIE without sign or ordering paradoxes.
  • The decomposition remains valid when treatment is reversed, satisfying skew-symmetry.
  • Discrete cumulative effects defined over ordered treatment levels inherit the same decomposition properties.
  • Standard identification formulas apply directly to the new measures under the usual no-confounding conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction could be applied to other continuous-treatment effect decompositions where symmetry properties are required.
  • In settings with treatment-mediator interactions, the cumulative measures may produce more stable policy interpretations than conventional natural effects.
  • Extensions to time-varying treatments might follow by replacing the derivative with an appropriate infinitesimal operator.

Load-bearing premise

Identification of the cumulative effects from observed data requires the standard assumptions of consistency, positivity, and no unmeasured confounding for the treatment-mediator, treatment-outcome, and mediator-outcome relations.

What would settle it

A numerical example or dataset in which the sum of the estimated CNDE and CNIE differs from the total effect, or in which reversing the treatment direction fails to produce the negative of the original decomposition, would falsify the claimed preservation of additivity and skew-symmetry.

Figures

Figures reproduced from arXiv: 2606.22165 by Jin Tian, Yuta Kawakami.

Figure 1
Figure 1. Figure 1: The causal graph representing SCM MN . Structural Causal Models. We use the language of Structural Causal Models (SCM) as our basic semantic and inferential framework [Pearl, 2009]. An SCM M is a tuple ⟨V , U, F, PU ⟩, where U is a set of exogenous (unobserved) variables following a joint distribution PU , and V is a set of endogenous (observable) variables whose values are determined by structural functio… view at source ↗
Figure 2
Figure 2. Figure 2: The causal graph representing SCM M1 (Baron and Kenny’s model). Local Total Effect. Average partial causal effect (APCE) is another useful measure to evaluate the total effect of X on Y for a continuous treatment [Chamberlain, 1984, Kawakami et al., 2023]. In this paper, we call it the local total effect (LTE), which captures the infinitesimal total effect of X at X = x: Definition 3 (LTE). The local total… view at source ↗
Figure 3
Figure 3. Figure 3: The causal graph representing SCM MC . If i > j, then S-CNDE-O(x ′′, x′ ; Y ) − S-CNIE-O(x ′ , x′′; Y ) (145) = S-CNDE-O(xj , xi ; Y ) − S-CNIE-O(xi , xj ; Y ) (146) = X i−1 k=j S-NDE(xk, xk+1; Y ) − X i−1 k=j S-NIE(xk+1, xk; Y ) (147) = X i−1 k=j {S-NDE(xk, xk+1; Y ) + S-NIE(xk, xk+1; Y )} (148) = X i−1 k=j TE(xk, xk+1; Y ) = TE(xj , xi ; Y ) = TE(x ′′, x′ ; Y ). (149) Theorem 10. [Identification of direc… view at source ↗
read the original abstract

Causal mediation analysis provides a fundamental framework for quantifying the contributions of different pathways from a treatment $X$ to an outcome $Y$ through a mediator. The natural direct and indirect effects (NDE and NIE) are widely used to decompose the total effect. In this paper, we observe that NDE and NIE can give rise to paradoxical interpretations due to their failure to satisfy two desirable properties of interpretable causal effects: skew-symmetry and additivity. To address these limitations, we introduce new measures of direct and indirect effects for continuous treatments, termed the cumulative natural direct and indirect effects (CNDE and CNIE), constructed by decomposing local causal effects $\mathbb{E}[\partial_xY_{x}]$ into local direct and indirect effects. CNDE and CNIE yield a decomposition of the total effect that preserves both skew-symmetry and additivity. We further extend this framework to ordinal treatments by defining discrete analogues of the cumulative effects over ordered treatment levels that preserve these structural properties. We establish decomposition and identification results for the proposed measures under standard causal assumptions. We illustrate their behavior, in comparison with NDE and NIE, using canonical linear mediation models with interaction and a real-world dataset.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper introduces cumulative natural direct and indirect effects (CNDE and CNIE) for continuous treatments in causal mediation analysis. These are constructed by first decomposing the local total effect E[∂_x Y_x] into local direct and indirect components and then integrating to obtain cumulative versions. The authors claim that CNDE and CNIE provide a decomposition of the total effect that satisfies skew-symmetry and additivity (unlike standard NDE/NIE), establish decomposition and identification results under standard causal assumptions, extend the framework to ordinal treatments via discrete analogues, and illustrate the measures in linear mediation models with interaction and on a real dataset.

Significance. If the decomposition, identification, and algebraic properties hold, the proposed measures would supply a coherent alternative for mediation analysis with continuous or ordinal treatments that avoids certain paradoxical interpretations arising from failures of skew-symmetry and additivity. The explicit construction from local effects, the extension to ordered discrete treatments, and the comparison with existing linear-model results constitute a targeted methodological contribution in causal inference.

major comments (3)
  1. [§3] §3 (definition of CNDE/CNIE): the construction decomposes and integrates E[∂_x Y_x], which presupposes that the potential-outcome map x ↦ Y_x is differentiable (at least almost everywhere). The identification results later invoke only the standard quartet of consistency, positivity, and no unmeasured confounding for the three relations; none of these conditions entails differentiability or even continuity of Y_x. Consequently the claimed skew-symmetry and additivity hold only conditionally on an extra regularity assumption not listed among the “standard causal assumptions.”
  2. [§4] §4 (identification and decomposition theorems): the paper states that decomposition and identification results are established, yet the provided text does not display the explicit integral expressions for CNDE and CNIE nor the step-by-step verification that integration preserves skew-symmetry and additivity for arbitrary (sufficiently regular) response surfaces. Without these derivations the central claim that the cumulative effects “yield a decomposition … that preserves both skew-symmetry and additivity” cannot be fully assessed.
  3. [§5] §5 (linear-model illustration): the comparison with NDE/NIE is performed only inside canonical linear models with interaction. It remains unclear whether the reported numerical differences between CNDE/CNIE and NDE/NIE persist, or reverse, under non-linear data-generating processes that still satisfy the differentiability requirement.
minor comments (2)
  1. [§3] Notation for the local total effect E[∂_x Y_x] is introduced without an explicit statement of the measure with respect to which the derivative is taken or the domain on which it is assumed to exist.
  2. [Abstract] The abstract refers to “canonical linear mediation models with interaction” but does not specify the precise interaction term; the main text should state the model equation explicitly before presenting numerical results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [§3] §3 (definition of CNDE/CNIE): the construction decomposes and integrates E[∂_x Y_x], which presupposes that the potential-outcome map x ↦ Y_x is differentiable (at least almost everywhere). The identification results later invoke only the standard quartet of consistency, positivity, and no unmeasured confounding for the three relations; none of these conditions entails differentiability or even continuity of Y_x. Consequently the claimed skew-symmetry and additivity hold only conditionally on an extra regularity assumption not listed among the “standard causal assumptions.”

    Authors: We agree that the definition of the local effects and their integration requires differentiability of the potential-outcome map (at least almost everywhere). The identification results rely on the standard assumptions, but the construction of CNDE and CNIE additionally presupposes this regularity condition. We will explicitly add differentiability (a.e.) to the list of assumptions in the revised manuscript and clarify that skew-symmetry and additivity hold under the standard causal assumptions together with this regularity condition. revision: yes

  2. Referee: [§4] §4 (identification and decomposition theorems): the paper states that decomposition and identification results are established, yet the provided text does not display the explicit integral expressions for CNDE and CNIE nor the step-by-step verification that integration preserves skew-symmetry and additivity for arbitrary (sufficiently regular) response surfaces. Without these derivations the central claim that the cumulative effects “yield a decomposition … that preserves both skew-symmetry and additivity” cannot be fully assessed.

    Authors: We will insert the explicit integral expressions for CNDE and CNIE in §4. We will also add a step-by-step verification showing that, for sufficiently regular response surfaces, the integration operation preserves skew-symmetry and additivity. These additions will make the central claim fully verifiable from the text. revision: yes

  3. Referee: [§5] §5 (linear-model illustration): the comparison with NDE/NIE is performed only inside canonical linear models with interaction. It remains unclear whether the reported numerical differences between CNDE/CNIE and NDE/NIE persist, or reverse, under non-linear data-generating processes that still satisfy the differentiability requirement.

    Authors: The linear-model section is intended as an illustration in a setting where NDE/NIE are commonly studied and where their failure of skew-symmetry and additivity is transparent. The algebraic properties of CNDE/CNIE are established generally under the stated assumptions and do not rely on linearity. We will add a clarifying sentence noting that the magnitude and direction of numerical differences are model-dependent and that the linear case is chosen to highlight the interpretative advantage of the new measures. A full exploration of non-linear DGPs is left for future work, as it would require specifying particular non-linear surfaces while preserving differentiability. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit construction

full rationale

The paper defines CNDE and CNIE explicitly by decomposing and integrating the local effect E[∂_x Y_x] into direct and indirect components, with skew-symmetry and additivity following immediately from the integral properties and the definition itself. Identification and decomposition results are stated to hold under the standard quartet of consistency/positivity/no-unmeasured-confounding assumptions, without any reduction to fitted parameters, self-citations, or prior uniqueness theorems. No load-bearing step equates a claimed result to its own inputs by construction; the measures are new objects introduced to satisfy the listed properties, and the derivation chain remains independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper relies on standard domain assumptions for causal identification but introduces no free parameters fitted to data and no new physical entities beyond the defined effect measures.

axioms (1)
  • domain assumption Standard causal assumptions: consistency, positivity, and no unmeasured confounding for the relevant relationships.
    Invoked to establish identification results for CNDE and CNIE as stated in the abstract.
invented entities (2)
  • Cumulative natural direct effect (CNDE) no independent evidence
    purpose: Direct effect measure satisfying skew-symmetry and additivity
    Newly introduced measure; no independent evidence outside the paper.
  • Cumulative natural indirect effect (CNIE) no independent evidence
    purpose: Indirect effect measure satisfying skew-symmetry and additivity
    Newly introduced measure; no independent evidence outside the paper.

pith-pipeline@v0.9.1-grok · 5733 in / 1337 out tokens · 26668 ms · 2026-06-26T11:25:17.520303+00:00 · methodology

discussion (0)

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Reference graph

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