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Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis

This paper introduces cumulative natural direct and indirect effects (CNDE and CNIE) as a new framework for causal mediation analysis that resolves the interpretational paradoxes of traditional measures by ensuring the decomposition of total effects satisfies both skew-symmetry and additivity for continuous and ordinal treatments.

Original authors: Yuta Kawakami, Jin Tian

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuta Kawakami, Jin Tian

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why a car is speeding up. You know the driver pressed the gas pedal (the Treatment), and the car went faster (the Outcome). But there's a middleman: the engine revs up first (the Mediator), which then pushes the car.

Scientists have long used a method called "Causal Mediation Analysis" to split the total speed-up into two parts:

Direct Effect: How much the gas pedal made the car go faster without the engine revving (maybe through a direct mechanical link).
Indirect Effect: How much the gas pedal made the car go faster because it revved the engine first.

For a long time, researchers used a standard tool to measure these two parts, called NDE (Natural Direct Effect) and NIE (Natural Indirect Effect).

The Problem: The "Paradoxical" Ruler

The authors of this paper discovered that the standard tool (NDE/NIE) is like a broken ruler. It has two major flaws that make the results confusing and sometimes contradictory:

The "Flip-Flop" Problem (Lack of Skew-Symmetry):
Imagine you measure the distance from your house to the park. If you walk from House $\to$ Park, the distance is 5 miles. If you walk Park $\to$ House, the distance should be -5 miles (the same magnitude, just the opposite direction).
With the old tool, if you asked, "How much did smoking 1 extra cigarette increase heart disease risk?" you might get an answer saying, "It's 100% caused by high blood pressure."
But if you flipped the question to, "How much did reducing smoking by 1 cigarette decrease heart disease risk?" the same tool might say, "It's 0% caused by blood pressure; it's all direct!"
The Paradox: The mechanism of the disease shouldn't change just because you asked the question in reverse. The old tool gives you two completely different stories for the same reality.
The "Broken Sum" Problem (Lack of Additivity):
Imagine you walk from your house to the park (5 miles), and then from the park to the library (3 miles). The total distance should be 8 miles.
With the old tool, if you measured the trip from House $\to$ Park and then Park $\to$ Library separately, the numbers might add up to something weird, like 6 miles or 10 miles, instead of the true 8 miles.
The Paradox: You can't trust the parts to tell you about the whole. If you break a long journey into small steps, the old tool says the "indirect" cause changes depending on how you slice the journey.

The Solution: A New "Cumulative" Ruler

To fix this, the authors invented new tools called CNDE (Cumulative Natural Direct Effect) and CNIE (Cumulative Natural Indirect Effect).

Think of the old method as taking a snapshot of the journey at the very start and the very end, ignoring the middle.
The new method is like walking the entire path step-by-step.

How it works: Instead of looking at the big jump from "0 cigarettes" to "40 cigarettes" all at once, the new tool looks at the tiny, infinitesimal steps: "What happens when you go from 0 to 0.1? Then 0.1 to 0.2?"
It calculates the direct and indirect effects for every single tiny step along the way and then adds them all up (cumulates them).

Why This New Ruler is Better

It's Consistent (Skew-Symmetry): If you walk House $\to$ Park, you get +5. If you walk Park $\to$ House, you get -5. The story is the same; only the direction changes. The new tool ensures that the proportion of the effect caused by the engine (mediator) stays the same, no matter which way you look at it.
It Adds Up (Additivity): If you walk House $\to$ Park and then Park $\to$ Library, the new tool guarantees that the sum of the two small trips equals the big trip. You can trust that the "indirect" cause for the whole journey is just the sum of the "indirect" causes of the little steps.

Real-World Test: The Smoking Study

The authors tested this new tool using real data about smoking, body weight (BMI), and heart rate.

Old Tool Results: When they looked at increasing smoking from 20 to 40 cigarettes, the tool said the effect was mostly direct. But when they looked at decreasing from 40 to 20, it said the effect was mostly indirect. It was a confusing mess.
New Tool Results: The new tool gave a consistent story. Whether smoking went up or down, it showed that about two-thirds of the effect was direct and one-third was indirect (mediated by BMI). The numbers added up perfectly: the effect of going from 20 to 40 was exactly the sum of going 20 to 30 and 30 to 40.

Summary

The paper argues that the old way of measuring "direct" and "indirect" causes is mathematically flawed because it produces contradictory results depending on how you ask the question. The authors propose a new "cumulative" method that treats the treatment (like smoking or dosage) as a continuous path. By adding up tiny steps along that path, the new method provides a stable, logical, and non-contradictory way to understand how treatments affect outcomes through mediators.

Technical Summary: Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis

Problem Statement
Causal mediation analysis aims to decompose the total effect (TE) of a treatment $X$ on an outcome $Y$ into direct and indirect pathways through a mediator $M$ . The standard framework relies on Natural Direct Effects (NDE) and Natural Indirect Effects (NIE). While the TE satisfies two fundamental algebraic properties desirable for interpretable causal effects—skew-symmetry (reversing the treatment contrast reverses the sign of the effect) and additivity (effects over adjacent intervals sum consistently)—the standard NDE and NIE generally fail to satisfy these properties.

The authors demonstrate that this failure leads to "paradoxical interpretations." Specifically:

Lack of Skew-Symmetry: The relative contributions of direct and indirect effects change depending on the direction of the treatment contrast (e.g., $X=x''$ vs. $X=x'$ versus $X=x'$ vs. $X=x''$ ). This can lead to contradictory conclusions about whether an effect is mediated or direct simply by flipping the reference points.
Lack of Additivity: The decomposition of a total effect over a large interval does not necessarily equal the sum of decompositions over sub-intervals. This makes it difficult to interpret mediation mechanisms coherently across different treatment levels.

These issues are particularly pronounced in models with treatment-mediator interactions (e.g., linear models with interaction terms), where the standard decomposition fails to align with intuitive path-based interpretations.

Methodology
To address these structural limitations, the paper proposes new measures of direct and indirect effects that preserve skew-symmetry and additivity. The methodology is divided into two settings: continuous treatments and ordinal treatments.

Continuous Treatments (CNDE and CNIE):
- Local Decomposition: Instead of decomposing global contrasts directly, the authors introduce a local decomposition of the Local Total Effect (LTE), defined as the derivative $E[\partial_x Y_x]$ . Under assumptions of differentiability and continuity, the LTE is additively decomposed into a Local Natural Direct Effect (LNDE) and a Local Natural Indirect Effect (LNIE).
- Cumulative Integration: The new measures, Cumulative Natural Direct Effect (CNDE) and Cumulative Natural Indirect Effect (CNIE), are defined as the integrals of LNDE and LNIE along the treatment path from $x'$ to $x''$ .
- Key Distinction: Unlike NDE/NIE, which integrate derivatives while holding one argument fixed at an endpoint, CNDE/CNIE integrate local infinitesimal effects where both the treatment level and the reference mediator value evolve along the path.
- Identification: Under standard sequential ignorability and regularity conditions, CNDE and CNIE are identified via integrals of conditional expectations and densities derived from observational data.
Ordinal Treatments (Discrete Analogues):
- Since derivatives are undefined for discrete variables, the authors define discrete analogues by summing finite-step contrasts along ordered treatment levels.
- CNDE-O and CNIE-O: These are constructed by summing standard NDE and NIE over adjacent levels. While they satisfy additivity over monotonic sequences, they do not satisfy skew-symmetry because the underlying NDE/NIE do not.
- Skew-Symmetric Versions (S-NDE, S-NIE, S-CNDE-O, S-CNIE-O): To recover skew-symmetry for ordinal treatments, the authors propose averaging the forward and backward endpoint evaluations of NDE and NIE. The resulting S-CNDE-O and S-CNIE-O satisfy both skew-symmetry and restricted additivity (holding for monotonic triplets).

Key Contributions

Identification of Structural Flaws: The paper formally identifies that the failure of NDE and NIE to satisfy skew-symmetry and additivity leads to paradoxical interpretations in mediation analysis, particularly in the presence of interactions.
New Effect Measures:
- For continuous treatments, the introduction of CNDE and CNIE, which provide a pathwise decomposition of the TE that satisfies both skew-symmetry and additivity for any nonparametric model.
- For ordinal treatments, the development of S-CNDE-O and S-CNIE-O, which utilize skew-symmetric versions of NDE/NIE to preserve both structural properties.
Theoretical Guarantees: The paper establishes decomposition theorems showing that the TE can be expressed as the sum of the proposed direct and indirect effects (e.g., $TE = CNDE + CNIE$) and provides identification results under standard causal assumptions (sequential ignorability).
Empirical Illustration: Using a real-world dataset (Framingham Heart Study) examining the effect of cigarette smoking on heart rate mediated by BMI, the authors demonstrate that standard NDE/NIE estimates exhibit the predicted paradoxes (e.g., drastically different effect sizes and proportions when reversing treatment contrasts). In contrast, the proposed CNDE and S-CNDE-O measures yield coherent, additive, and direction-invariant decompositions.

Results
In the empirical application, the standard NDE and NIE estimates showed significant asymmetry; for instance, reversing the treatment contrast from (20 to 40 cigarettes) to (40 to 20) resulted in NDE estimates of 0.955 and -0.363, respectively, and NIE estimates of 0.161 and 0.431. This violated the intuitive expectation that the proportion of mediated effect should remain consistent regardless of direction. Furthermore, the sum of indirect effects over sub-intervals (20→30 and 30→40) did not equal the indirect effect over the full interval (20→40), violating additivity.

The proposed measures (CNDE, CNIE, S-CNDE-O, S-CNIE-O) resolved these issues. The estimates for the cumulative measures were nearly identical in magnitude but opposite in sign when the contrast was reversed (satisfying skew-symmetry), and the effects over sub-intervals summed exactly to the effect over the full interval (satisfying additivity). This provided a consistent interpretation: for the 20–40 contrast, the effect was decomposed into a moderate positive direct component and a small negative indirect component, a pattern that remained stable across sub-intervals.

Significance
The paper argues that while NDE and NIE are valid causal quantities, their lack of fundamental algebraic properties complicates interpretation and can lead to misleading conclusions in practice. The proposed framework offers a principled alternative that ensures structural coherence in effect decomposition. By preserving skew-symmetry and additivity, the new measures (CNDE/CNIE and their ordinal counterparts) provide a more intuitive and stable basis for understanding mediation mechanisms, particularly in settings with continuous or ordinal treatments and interaction effects. The authors note that this work emphasizes the importance of structural properties in effect decomposition and suggests that these measures should be preferred when path-based interpretations are desired. The paper concludes by identifying future directions, including extending these measures to settings with unmeasured confounding (using instrumental variables or sensitivity analysis) and handling multiple mediators.