crumble: A comprehensive framework for modern causal mediation analysis with intermediate confounding

Ivan Diaz; Kara E. Rudolph; Nicholas T. Williams; Richard Liu

arxiv: 2604.09902 · v1 · submitted 2026-04-10 · 📊 stat.ME · stat.AP

crumble: A comprehensive framework for modern causal mediation analysis with intermediate confounding

Richard Liu , Nicholas T. Williams , Kara E. Rudolph , Ivan Diaz This is my paper

Pith reviewed 2026-05-10 16:38 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords causal mediation analysisintermediate confoundingnonparametric estimationnatural direct effectsrandomized interventional effectsrecanting-twin effectsmodified treatment policies

0 comments

The pith

Crumble package gives practical nonparametric tools for estimating direct and indirect effects while handling intermediate confounding.

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

The paper serves as an accessible tutorial on the crumble package for causal mediation analysis. It reviews the definitions, interpretations, and identification assumptions for natural direct and indirect effects, randomized interventional effects, and recanting-twin effects. The guide explains how to apply the package to continuous or multidimensional mediators and non-binary treatments, using modified treatment policies and two case studies from the Job Search Intervention Study. A sympathetic reader would care because these methods allow investigation of causal pathways in settings where traditional approaches fail due to confounding or complex data structures.

Core claim

Crumble enables nonparametric estimation of several mediation parameters, even when mediators are continuous and/or multi-dimensional or when treatments are non-binary, while accommodating intermediate confounding.

What carries the argument

The crumble R package, which implements estimators for the reviewed mediation parameters under standard identification assumptions and modified treatment policies for non-binary treatments.

If this is right

Applied researchers can estimate natural direct and indirect effects with continuous mediators without relying on strong parametric models.
The package supports analysis of non-binary treatments through modified treatment policies.
Suitable mediation parameters can be chosen to account for intermediate confounding in the data.
Demonstrations with real data lower the barrier for implementing these estimators in practice.

Where Pith is reading between the lines

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

This could encourage wider adoption of mediation analysis in observational studies across health and social sciences.
The tutorial approach suggests potential for similar guides on related causal inference methods for longitudinal or time-varying settings.
Users might extend the framework by combining it with other tools for sensitivity analysis when positivity assumptions are borderline.

Load-bearing premise

The standard causal identification assumptions hold, including no unmeasured confounding, consistency, positivity, and correct specification of the nuisance functions.

What would settle it

Applying crumble to the Job Search Intervention Study data produces estimates that contradict results from established parametric methods without clear justification from the identification assumptions.

Figures

Figures reproduced from arXiv: 2604.09902 by Ivan Diaz, Kara E. Rudolph, Nicholas T. Williams, Richard Liu.

**Figure 2.** Figure 2: Three DAGs with “no intermediate confounding” due to the fact that in either of the three [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

read the original abstract

Causal mediation analysis is widely used to investigate how causal effects operate through specific pathways linking treatments or exposures to outcomes. Recently, \texttt{crumble} was developed to enable nonparametric estimation of several mediation parameters, even when mediators are continuous and/or multi-dimensional or when treatments are non-binary. But a practical and accessible guide to using \texttt{crumble} -- one that does not require deep familiarity with mediation analysis or semiparametric theory -- is currently lacking. This tutorial aims to an accessible introduction to \texttt{crumble} while minimizing technical complexity. We first review the mediation parameters implemented in \texttt{crumble} -- natural direct and indirect effects, randomized interventional effects, and recanting-twin effects. For each, we give the definition, interpretation, identification assumptions, and suitability in the presence or absence of intermediate confounding. Then, we demonstrate the usage of \texttt{crumble} by examining an example configuration. Next, we describe how \texttt{crumble} accommodates non-binary treatments through modified treatment policies. Finally, we illustrate the practical use of \texttt{crumble} through two case studies -- one with a binary treatment and one with a non-binary treatment -- based on the Job Search Intervention Study data.

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.

Desk Editor's Note private letter to a colleague

This is a clear tutorial on the crumble package that organizes existing mediation tools for applied users but introduces no new methods or results.

read the letter

The core of this paper is a practical walkthrough for the crumble R package on causal mediation. It reviews natural direct and indirect effects, randomized interventional effects, and recanting-twin effects, explains their identification assumptions including intermediate confounding, and shows how to fit them with code examples on the Job Search Intervention Study data. It also covers modified treatment policies for non-binary treatments. That structure is the main contribution: it pulls together definitions, assumptions, and usage in one accessible place without requiring readers to chase multiple theory papers first. The examples are straightforward and the handling of continuous or multi-dimensional mediators looks usable for real data problems in epidemiology or policy research. The paper does not claim new estimators or theorems, and it correctly positions itself as guidance rather than original theory. Its soundness rests on the prior semiparametric work behind crumble rather than any fresh derivations here. The case studies function as illustrations, not as tests that could falsify the approach or compare performance against alternatives. No circular reasoning appears, and the assumptions are stated plainly. This kind of tutorial is useful for applied researchers who already know basic causal inference but need a concrete entry point to these specific parameters. It is less relevant for methodologists looking for advances in identification or efficiency. I would bring it to a reading group focused on applied causal methods, but I would not cite it in my own methodological work. It deserves peer review because accessible, accurate documentation of working software improves how these tools get used in practice, even if the underlying package was developed elsewhere.

Referee Report

0 major / 4 minor

Summary. The manuscript is a tutorial for the crumble R package, which supports nonparametric estimation of causal mediation parameters including natural direct and indirect effects, randomized interventional effects, and recanting-twin effects. It reviews definitions, interpretations, and identification assumptions for each parameter (with and without intermediate confounding), explains accommodation of continuous or multi-dimensional mediators and non-binary treatments via modified treatment policies, and demonstrates usage through case studies on the Job Search Intervention Study data.

Significance. If the crumble implementation follows the referenced semiparametric results, the tutorial meaningfully lowers the barrier for applied researchers to conduct modern mediation analysis in settings with intermediate confounding or complex mediator structures. By providing accessible explanations and concrete examples rather than new theory, it addresses a practical gap and could increase appropriate use of these methods.

minor comments (4)

The tutorial sections reviewing identification assumptions for each effect could include brief practical guidance on assessing positivity and correct nuisance specification in the context of the Job Search Intervention Study example, to better support applied readers.
In the case study sections, clarify how the modified treatment policies are defined and implemented for the non-binary treatment example, including any specific parameter choices or sensitivity checks.
The manuscript would benefit from a short table comparing the mediation parameters (e.g., when each is identifiable, their interpretations, and handling of intermediate confounding) to aid quick reference.
Ensure that all code snippets are self-contained and that the package version and dependencies are explicitly stated for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review of the manuscript and for recommending acceptance. We are pleased that the tutorial is viewed as addressing a practical gap by providing accessible explanations and examples for modern causal mediation analysis using the crumble package.

Circularity Check

0 steps flagged

No significant circularity; tutorial references prior package and standard assumptions

full rationale

The paper is structured as an accessible tutorial on the existing crumble package rather than a new theoretical derivation. It reviews mediation parameters (natural direct/indirect effects, randomized interventional effects, recanting-twin effects), standard causal identification assumptions (no unmeasured confounding, consistency, positivity), and demonstrates usage on the Job Search Intervention Study data. No quantities are defined in terms of fitted parameters from the current analysis, no predictions reduce to inputs by construction, and no uniqueness theorems or ansatzes are smuggled via self-citation. The mention of prior crumble development is a normal reference to the package being tutorialized and does not bear the load of any derivation. The work is self-contained against external benchmarks of causal mediation literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The tutorial rests on standard causal inference assumptions rather than new free parameters or invented entities; it reviews identification conditions for each mediation parameter without introducing novel postulates.

axioms (1)

domain assumption No unmeasured confounding between treatment, mediator, and outcome; consistency; positivity; and correct specification of nuisance functions for identification of natural direct/indirect effects and related quantities.
Invoked throughout the sections defining and interpreting each mediation parameter and their suitability with or without intermediate confounding.

pith-pipeline@v0.9.0 · 5526 in / 1371 out tokens · 48470 ms · 2026-05-10T16:38:38.942708+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Inference for natural mediation effects under case-cohort sampling with applications in identifying COVID -19 vaccine correlates of protection

David Benkeser, Iv ´an D´ıaz, and Jialu Ran. Inference for natural mediation effects under case- cohort sampling with applications in identifying covid-19 vaccine correlates of protection.arXiv preprint arXiv:2103.02643,

work page arXiv
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Graphical Models for Processing Missing Data

doi: 10.1080/01621459.2021.1955691. URL https://doi.org/10.1080/01621459.2021.1955691. Daniel Falbel and Javier Luraschi.torch: Tensors and Neural Networks with ’GPU’ Acceleration,

work page doi:10.1080/01621459.2021.1955691 2021
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Introducing longi- tudinal modified treatment policies: a unified framework for studying complex exposures.arXiv preprint arXiv:2304.09460,

Katherine L Hoffman, Diego Salazar-Barreto, Kara E Rudolph, and Iv´an D´ıaz. Introducing longi- tudinal modified treatment policies: a unified framework for studying complex exposures.arXiv preprint arXiv:2304.09460,

work page arXiv
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Semiparametric doubly robust targeted double machine learning: a review

Edward H Kennedy. Semiparametric doubly robust targeted double machine learning: a review. arXiv preprint arXiv:2203.06469,

work page arXiv
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General targeted machine learning for modern causal mediation analysis.arXiv preprint arXiv:2408.14620,

Richard Liu, Nicholas T Williams, Kara E Rudolph, and Iv ´an D´ıaz. General targeted machine learning for modern causal mediation analysis.arXiv preprint arXiv:2408.14620,

work page arXiv
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Organic direct and indirect effects with post-treatment common causes of mediator and outcome.arXiv preprint arXiv:1510.02753,

Judith J Lok. Organic direct and indirect effects with post-treatment common causes of mediator and outcome.arXiv preprint arXiv:1510.02753,

work page arXiv
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Causal organic direct and indirect effects: closer to baron and kenny.arXiv preprint arXiv:1903.04697,

Judith J Lok. Causal organic direct and indirect effects: closer to baron and kenny.arXiv preprint arXiv:1903.04697,

work page arXiv 1903
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Assessing racial disparities in health- care expenditures via mediator distribution shifts.arXiv preprint arXiv:2504.21688,

Xiaxian Ou, Xinwei He, David Benkeser, and Razieh Nabi. Assessing racial disparities in health- care expenditures via mediator distribution shifts.arXiv preprint arXiv:2504.21688,

work page arXiv
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Amiram D Vinokur and Yaacov Schul

doi: 10.1097/EDE.0000000000000596. Amiram D Vinokur and Yaacov Schul. Mastery and inoculation against setbacks as active ingredi- ents in the jobs intervention for the unemployed.Journal of consulting and clinical psychology, 65(5):867,

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Recanting twins: addressing intermediate confounding in mediation analysis.arXiv preprint arXiv:2401.04450,

Tat-Thang V o, Nicholas Williams, Richard Liu, Kara E Rudolph, and Ivan Dıaz. Recanting twins: addressing intermediate confounding in mediation analysis.arXiv preprint arXiv:2401.04450,

work page arXiv
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R package version 0.1.2

URLhttps://CRAN.R-project.org/package=crumble. R package version 0.1.2. Nicholas T Williams, Oliver J Hines, and Kara E Rudolph. Riesz representers for the rest of us. arXiv preprint arXiv:2507.19413, 2025a. Nicholas T Williams, Anton Hung, and Kara E Rudolph. Re: don’t let your analysis go to seed: on the impact of random seed on machine learning-based c...

work page arXiv
[12]

lt15k " ~ 1 , 18income ==

31 A Incorporating MTPs into common mediation parameters This section briefly introduces the definition and identifications of the common mediation param- eters reviewed in §3 when MTPs are incorporated for non-binary treatment/exposures. Details on the estimation of these parameters are provided in §5 of (Liu et al., 2024). In what follows, we denoted 1 ...

work page 2024
[13]

Then, we explain howcrumbleaddresses these estimation challenges by examining its implementation details

C Technical Details for Estimation: Challenges and Solutions incrumble In this section, we briefly introduce the two estimation challenges that come up when eitherMor Zis continuous or high-dimensional. Then, we explain howcrumbleaddresses these estimation challenges by examining its implementation details. C.1 Challenges in Estimation One challenge comes...

work page 2021
[14]

However, not all identification formulas we reviewed in §3 are repeated conditional expectations ofY

or longitudinal causal effects (D ´ıaz et al., 2021), for example, can be applied to estimateψ( ˆP)without worrying about the data complexity. However, not all identification formulas we reviewed in §3 are repeated conditional expectations ofY. In fact, some of the identification formulas become hard to estimate when eitherMorZis continuous and/or high-di...

work page 2021
[15]

C.2 Implementation Details incrumble Thecrumble()function incrumbleencompasses all the functionalities for estimation

suggests that the conditions required for the Bayes reparameterization approach to achieve desirable statis- tical properties are more stringent than those needed for the approach implemented incrumble (see Remark 2 in (Liu et al., 2024)). C.2 Implementation Details incrumble Thecrumble()function incrumbleencompasses all the functionalities for estimation...

work page 2024
[16]

for more technical details about lines 4-7, and (Williams et al., 2025a) for another tutorial about Riesz regression, which was public after (Liu et al., 2024). We also note thatestimate phi n alpha()andestimate phi r alpha()in lines 4-7 rely on an input parameternn modulebecausecrumblesolves the unconstrained optimization prob- lems using deep learning o...

work page 2024

[1] [1]

Inference for natural mediation effects under case-cohort sampling with applications in identifying COVID -19 vaccine correlates of protection

David Benkeser, Iv ´an D´ıaz, and Jialu Ran. Inference for natural mediation effects under case- cohort sampling with applications in identifying covid-19 vaccine correlates of protection.arXiv preprint arXiv:2103.02643,

work page arXiv

[2] [2]

Graphical Models for Processing Missing Data

doi: 10.1080/01621459.2021.1955691. URL https://doi.org/10.1080/01621459.2021.1955691. Daniel Falbel and Javier Luraschi.torch: Tensors and Neural Networks with ’GPU’ Acceleration,

work page doi:10.1080/01621459.2021.1955691 2021

[3] [3]

Introducing longi- tudinal modified treatment policies: a unified framework for studying complex exposures.arXiv preprint arXiv:2304.09460,

Katherine L Hoffman, Diego Salazar-Barreto, Kara E Rudolph, and Iv´an D´ıaz. Introducing longi- tudinal modified treatment policies: a unified framework for studying complex exposures.arXiv preprint arXiv:2304.09460,

work page arXiv

[4] [4]

Semiparametric doubly robust targeted double machine learning: a review

Edward H Kennedy. Semiparametric doubly robust targeted double machine learning: a review. arXiv preprint arXiv:2203.06469,

work page arXiv

[5] [5]

General targeted machine learning for modern causal mediation analysis.arXiv preprint arXiv:2408.14620,

Richard Liu, Nicholas T Williams, Kara E Rudolph, and Iv ´an D´ıaz. General targeted machine learning for modern causal mediation analysis.arXiv preprint arXiv:2408.14620,

work page arXiv

[6] [6]

Organic direct and indirect effects with post-treatment common causes of mediator and outcome.arXiv preprint arXiv:1510.02753,

Judith J Lok. Organic direct and indirect effects with post-treatment common causes of mediator and outcome.arXiv preprint arXiv:1510.02753,

work page arXiv

[7] [7]

Causal organic direct and indirect effects: closer to baron and kenny.arXiv preprint arXiv:1903.04697,

Judith J Lok. Causal organic direct and indirect effects: closer to baron and kenny.arXiv preprint arXiv:1903.04697,

work page arXiv 1903

[8] [8]

Assessing racial disparities in health- care expenditures via mediator distribution shifts.arXiv preprint arXiv:2504.21688,

Xiaxian Ou, Xinwei He, David Benkeser, and Razieh Nabi. Assessing racial disparities in health- care expenditures via mediator distribution shifts.arXiv preprint arXiv:2504.21688,

work page arXiv

[9] [9]

Amiram D Vinokur and Yaacov Schul

doi: 10.1097/EDE.0000000000000596. Amiram D Vinokur and Yaacov Schul. Mastery and inoculation against setbacks as active ingredi- ents in the jobs intervention for the unemployed.Journal of consulting and clinical psychology, 65(5):867,

work page doi:10.1097/ede.0000000000000596

[10] [10]

Recanting twins: addressing intermediate confounding in mediation analysis.arXiv preprint arXiv:2401.04450,

Tat-Thang V o, Nicholas Williams, Richard Liu, Kara E Rudolph, and Ivan Dıaz. Recanting twins: addressing intermediate confounding in mediation analysis.arXiv preprint arXiv:2401.04450,

work page arXiv

[11] [11]

R package version 0.1.2

URLhttps://CRAN.R-project.org/package=crumble. R package version 0.1.2. Nicholas T Williams, Oliver J Hines, and Kara E Rudolph. Riesz representers for the rest of us. arXiv preprint arXiv:2507.19413, 2025a. Nicholas T Williams, Anton Hung, and Kara E Rudolph. Re: don’t let your analysis go to seed: on the impact of random seed on machine learning-based c...

work page arXiv

[12] [12]

lt15k " ~ 1 , 18income ==

31 A Incorporating MTPs into common mediation parameters This section briefly introduces the definition and identifications of the common mediation param- eters reviewed in §3 when MTPs are incorporated for non-binary treatment/exposures. Details on the estimation of these parameters are provided in §5 of (Liu et al., 2024). In what follows, we denoted 1 ...

work page 2024

[13] [13]

Then, we explain howcrumbleaddresses these estimation challenges by examining its implementation details

C Technical Details for Estimation: Challenges and Solutions incrumble In this section, we briefly introduce the two estimation challenges that come up when eitherMor Zis continuous or high-dimensional. Then, we explain howcrumbleaddresses these estimation challenges by examining its implementation details. C.1 Challenges in Estimation One challenge comes...

work page 2021

[14] [14]

However, not all identification formulas we reviewed in §3 are repeated conditional expectations ofY

or longitudinal causal effects (D ´ıaz et al., 2021), for example, can be applied to estimateψ( ˆP)without worrying about the data complexity. However, not all identification formulas we reviewed in §3 are repeated conditional expectations ofY. In fact, some of the identification formulas become hard to estimate when eitherMorZis continuous and/or high-di...

work page 2021

[15] [15]

C.2 Implementation Details incrumble Thecrumble()function incrumbleencompasses all the functionalities for estimation

suggests that the conditions required for the Bayes reparameterization approach to achieve desirable statis- tical properties are more stringent than those needed for the approach implemented incrumble (see Remark 2 in (Liu et al., 2024)). C.2 Implementation Details incrumble Thecrumble()function incrumbleencompasses all the functionalities for estimation...

work page 2024

[16] [16]

for more technical details about lines 4-7, and (Williams et al., 2025a) for another tutorial about Riesz regression, which was public after (Liu et al., 2024). We also note thatestimate phi n alpha()andestimate phi r alpha()in lines 4-7 rely on an input parameternn modulebecausecrumblesolves the unconstrained optimization prob- lems using deep learning o...

work page 2024