Self-separated and self-connected models for mediator and outcome missingness in mediation analysis

Elizabeth A. Stuart; Fan Yang; Grace V. Ringlein; Razieh Nabi; Trang Quynh Nguyen

arxiv: 2411.07221 · v2 · submitted 2024-11-11 · 📊 stat.ME

Self-separated and self-connected models for mediator and outcome missingness in mediation analysis

Trang Quynh Nguyen , Razieh Nabi , Fan Yang , Grace V. Ringlein , Elizabeth A. Stuart This is my paper

Pith reviewed 2026-05-23 17:22 UTC · model grok-4.3

classification 📊 stat.ME

keywords mediation analysismissing dataidentificationshadow variablescausal inferencemissingness mechanismsconditional independence

0 comments

The pith

Identification of mediation effects remains possible when both mediator and outcome are missing under self-separated conditional independence or self-connected shadow variable models.

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

The paper develops two classes of missingness models for mediation analysis where the mediator and outcome may both be unobserved. Self-separated models achieve identification by imposing conditional independence assumptions that remove specific connections in the causal structure. Self-connected models instead draw on shadow variables, which can be built into the mediator, outcome, or covariates or added as auxiliaries at various positions, while also permitting dependencies from unobserved causes of missingness. The work reviews prior models, introduces new ones, and extends shadow variable theory to produce general identification templates. These templates are illustrated with two practical examples.

Core claim

Self-separated missingness models identify mediation quantities via conditional independence assumptions that eliminate certain edges, while self-connected models identify the same quantities by leveraging shadow variables placed at different points in the graph and by allowing unobserved common causes of the missingness indicators; together these classes yield identification templates and a substantial extension of shadow variable theory for the mediation setting.

What carries the argument

Self-separated missingness models (identification via conditional independence assumptions that remove connections) and self-connected missingness models (identification via shadow variables that may be built-in or auxiliary).

If this is right

Identification templates apply directly to mediation problems with missing mediator and outcome under the stated assumptions.
Models can incorporate dependencies from unobserved causes of missingness without losing identifiability where the shadow-variable conditions hold.
Shadow variables can be placed at multiple positions including mediator, outcome, covariates, or external auxiliaries.
The synthesis connects existing missingness models to new variants and supplies generally useful identification techniques beyond the mediation setting.

Where Pith is reading between the lines

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

The same self-separated and self-connected constructions could be tested for applicability to missingness in longitudinal or survival mediation settings.
Sensitivity analyses that vary the strength of the required independences or shadow-variable relations would quantify robustness of the identification results.
Empirical checks for the presence of usable shadow variables in observational datasets could guide whether self-connected models are feasible in a given study.

Load-bearing premise

The stated conditional independence assumptions hold or the required shadow variables exist with the necessary properties in the actual data-generating process.

What would settle it

A simulation or real dataset in which the conditional independences are violated or no suitable shadow variables are present would produce non-identifiable mediation effects and biased estimates under the proposed models.

Figures

Figures reproduced from arXiv: 2411.07221 by Elizabeth A. Stuart, Fan Yang, Grace V. Ringlein, Razieh Nabi, Trang Quynh Nguyen.

**Figure 3.** Figure 3: A “disjoint MAR” model A M Y R M R Y UA UY full form M Y RM RY reduced form 3 Self-separated missingness models In this section, we seek models with self-separated missingness that are submodels of the saturated models above where P(M | X, A) and P(Y | X, A, M) are identified based on certain conditional independence assumptions. In addition to identification, we pay close attention to model falsifiabilit… view at source ↗

read the original abstract

Missing data is a common challenge in studying treatment effects. In the context of mediation analysis, this paper addresses missingness in the mediator and outcome, focusing on identification. We first consider self-separated missingness models where identification is achieved by conditional independence assumptions. This model class is somewhat limited as it is constrained by the need to remove a certain number of connections from the model. We then turn to self-connected missingness models where identification relies on information from shadow variables. This model class turns out to contain substantial variation, allowing models with built-in shadow variables (mediator, outcome or covariates) and models with auxiliary shadow variables at different positions in the causal structure. To improve the practical value of the missingness mechanisms, we allow where possible for dependencies due to unobserved causes of the missingness, a feature often neglected. In this exploration, we review existing models, connect to new models, and develop theory where needed. This results in templates for identification in the mediation setting, generally useful identification techniques, and perhaps most importantly a synthesis and substantial extension of shadow variable theory. Two examples relate the models to practical considerations.

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

Paper introduces self-separated and self-connected missingness models for mediation analysis and extends shadow-variable theory, but identification still requires external justification of untestable assumptions.

read the letter

The core contribution is two new classes of missingness models tailored to mediator and outcome missingness. Self-separated models achieve identification via conditional independence assumptions that remove edges from the graph. Self-connected models instead rely on shadow variables, which can be built-in (mediator, outcome, covariates) or auxiliary, and the work allows unobserved common causes of missingness in several cases. The authors connect these to existing models, develop the needed identification theory, and supply templates plus two practical examples. That synthesis and extension of shadow-variable ideas to the mediation setting is the clearest advance over prior work on missing data in causal settings. It gives methodologists a broader menu of identification strategies when both mediator and outcome are incomplete. The main limitation is that both model classes rest on assumptions that cannot be verified from the observed data: specific conditional independences for the self-separated case and the existence plus independence properties of the shadow variables for the self-connected case. The abstract states these must be justified externally, so the results inherit the usual untestable foundation rather than removing it. No empirical checks or sensitivity analyses are described, which leaves open how robust the templates are when the assumptions hold only approximately. This is a specialized methods paper aimed at researchers already working on causal mediation and missing-data problems. A reader looking for identification templates in that niche could get value from the synthesis. It is coherent on its own terms and engages the relevant literature, so it deserves peer review even if the assumptions limit immediate applicability.

Referee Report

0 major / 3 minor

Summary. The manuscript develops identification results for mediation analysis when both the mediator and outcome are subject to missingness. It introduces self-separated missingness models that achieve identification via conditional independence assumptions removing certain edges from the causal graph, and self-connected models that rely on shadow variables (either built-in from the mediator/outcome/covariates or auxiliary at various positions). The work allows for dependencies induced by unobserved causes of missingness, reviews prior models, develops new theory as needed, and supplies identification templates along with two practical examples. The central contribution is framed as a synthesis and substantial extension of shadow-variable theory to this setting.

Significance. If the derivations are correct, the results supply concrete identification functionals and templates for a practically relevant missing-data problem in causal mediation analysis. The synthesis of shadow-variable approaches, combined with explicit allowance for unobserved missingness causes, could reduce reliance on complete-case or ad-hoc imputation methods and provide generally reusable identification techniques beyond the specific mediation setting.

minor comments (3)

The abstract notes that self-separated models are 'somewhat limited' by the need to remove connections; the main text should quantify this limitation (e.g., maximum number of edges removable while preserving identification) and delineate the precise graph structures for which the class remains useful.
The claim of a 'substantial extension of shadow variable theory' would benefit from an explicit comparison table or subsection contrasting the new model classes against the models reviewed in the literature section, highlighting which identification functionals are novel.
The two examples are described only at the abstract level; the main text should clarify whether they use real data, simulated data, or both, and report sensitivity checks when the external justification for the conditional independences or shadow-variable properties is varied.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive summary of our work on self-separated and self-connected missingness models in mediation analysis and for recommending minor revision. No specific major comments were provided in the report, so our response below addresses the overall assessment.

Circularity Check

0 steps flagged

Theoretical identification templates under external conditional independence and shadow-variable assumptions; derivation chain remains self-contained

full rationale

The paper presents identification results for mediator/outcome missingness by defining two model classes (self-separated via conditional independences; self-connected via shadow variables) and deriving templates from those assumptions. These assumptions are explicitly stated as requiring external justification rather than being derived from data or prior results within the paper. No equations reduce a claimed prediction or identification functional to a fitted parameter or self-defined quantity by construction. Existing models are reviewed and extended, but the central synthesis does not rely on load-bearing self-citations whose validity is presupposed without independent verification. This is standard theoretical model development with no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard causal graphical model assumptions plus new missingness mechanism assumptions (conditional independences for self-separated models and shadow variable properties for self-connected models). No free parameters or invented entities with independent evidence are described in the abstract; the new model classes themselves constitute the contribution rather than additional postulated entities.

axioms (2)

domain assumption Conditional independence assumptions required for identification in self-separated missingness models
Abstract states that identification in this class is achieved by these assumptions; they are load-bearing for the templates.
domain assumption Existence and positioning of shadow variables sufficient for identification in self-connected models
Abstract indicates that identification relies on information from shadow variables, which may be built-in or auxiliary.

pith-pipeline@v0.9.0 · 5740 in / 1378 out tokens · 31016 ms · 2026-05-23T17:22:59.362603+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

[1]

The Moderator - Mediator Variable Distinction in Social Psychological Research : Conceptual , Strategic , and Statistical Considerations

Reuben M Baron and David A Kenny. The Moderator - Mediator Variable Distinction in Social Psychological Research : Conceptual , Strategic , and Statistical Considerations . Journal of Personality and Social Psychology, 51 0 (6): 0 1173--1182, 1986

work page 1986
[2]

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

David Benkeser, Iván Díaz, and Jialu Ran. Inference for natural mediation effects under case-cohort sampling with applications in identifying COVID -19 vaccine correlates of protection. arXiv:2103.02643 [math, stat], 2021

work page arXiv 2021
[3]

Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser, and James M. Robins. Identification In Missing Data Models Represented By Directed Acyclic Graphs . In Proceedings of The 35th Uncertainty in Artificial Intelligence Conference , pages 1149--1158. PMLR, 2020. ISSN: 2640-3498

work page 2020
[4]

Cole and Constantine E

Stephen R. Cole and Constantine E. Frangakis. The Consistency Statement in Causal Inference : A Definition or an Assumption ? Epidemiology, 20 0 (1), 2009. doi:10.1097/EDE.0b013e31818ef366

work page doi:10.1097/ede.0b013e31818ef366 2009
[5]

Ghazaleh Dashti, Katherine J

S. Ghazaleh Dashti, Katherine J. Lee, Julie A. Simpson, John B. Carlin, and Margarita Moreno-Betancur. Handling multivariable missing data in causal mediation analysis, 2024. arXiv:2403.17396 [stat]

work page arXiv 2024
[6]

Direct and Indirect Effects of Sequential Treatments

Vanessa Didelez, Philip Dawid, and Sara Geneletti. Direct and Indirect Effects of Sequential Treatments . In Proceedings of the Twenty - Second Conference on Uncertainty in Artificial Intelligence , 2006. doi:10.48550/arXiv.1206.6840

work page doi:10.48550/arxiv.1206.6840 2006
[7]

A new instrumental method for dealing with endogenous selection

Xavier d’Haultfoeuille. A new instrumental method for dealing with endogenous selection. Journal of Econometrics, 154 0 (1): 0 1--15, 2010. doi:10.1016/j.jeconom.2009.06.005

work page doi:10.1016/j.jeconom.2009.06.005 2010
[8]

Nonparametric causal mediation analysis for stochastic interventional (in)direct effects

Nima S Hejazi, Kara E Rudolph, Mark J Van Der Laan, and Iván Díaz. Nonparametric causal mediation analysis for stochastic interventional (in)direct effects. Biostatistics, 24 0 (3): 0 686--707, 2023. doi:10.1093/biostatistics/kxac002

work page doi:10.1093/biostatistics/kxac002 2023
[9]

A general approach to causal mediation analysis

Kosuke Imai, Luke Keele, and Dustin Tingley. A general approach to causal mediation analysis. Psychological Methods, 15 0 (4): 0 309--334, 2010 a . doi:10.1037/a0020761

work page doi:10.1037/a0020761 2010
[10]

Identification, Inference and Sensitivity Analysis for Causal Mediation Effects

Kosuke Imai, Luke Keele, and Teppei Yamamoto. Identification, Inference and Sensitivity Analysis for Causal Mediation Effects . Statistical Science, 25 0 (1), 2010 b . doi:10.1214/10-STS321

work page doi:10.1214/10-sts321 2010
[11]

Phillip S. Kott. Calibration Weighting When Model and Calibration Variables Can Differ . In Fulvia Mecatti, Pier Luigi Conti, and Maria Giovanna Ranalli, editors, Contributions to Sampling Statistics , pages 1--18. Springer International Publishing, Cham, 2014. ISBN 978-3-319-05320-2. doi:10.1007/978-3-319-05320-2_1

work page doi:10.1007/978-3-319-05320-2_1 2014
[12]

Nuisance mediators and missing data in mediation analyses of pain trials

Hopin Lee. Nuisance mediators and missing data in mediation analyses of pain trials. European Journal of Pain, 24 0 (9): 0 1651--1652, 2020. doi:10.1002/ejp.1637

work page doi:10.1002/ejp.1637 2020
[13]

E. L. Lehmann and Henry Scheffé. Completeness, Similar Regions , and Unbiased Estimation - Part I . Sankhyā: The Indian Journal of Statistics, 10 0 (4): 0 305--340, 1950. doi:10.1007/978-1-4614-1412-4_23

work page doi:10.1007/978-1-4614-1412-4_23 1950
[14]

Identifiability and estimation of causal mediation effects with missing data

Wei Li and Xiao-Hua Zhou. Identifiability and estimation of causal mediation effects with missing data. Statistics in Medicine, 36 0 (25): 0 3948--3965, 2017. doi:10.1002/sim.7413

work page doi:10.1002/sim.7413 2017
[15]

Tchetgen Tchetgen

Yilin Li, Wang Miao, Ilya Shpitser, and Eric J. Tchetgen Tchetgen. A self-censoring model for multivariate nonignorable nonmonotone missing data. Biometrics, 79 0 (4): 0 3203--3214, 2023. doi:10.1111/biom.13916

work page doi:10.1111/biom.13916 2023
[16]

Estimating Mixed Memberships With Sharp Eigenvector Deviations

Daniel Malinsky, Ilya Shpitser, and Eric J. Tchetgen Tchetgen. Semiparametric Inference for Nonmonotone Missing - Not -at- Random Data : The No Self - Censoring Model . Journal of the American Statistical Association, 117 0 (539): 0 1415--1423, 2022. doi:10.1080/01621459.2020.1862669

work page doi:10.1080/01621459.2020.1862669 2022
[17]

Tchetgen Tchetgen

Wang Miao and Eric J. Tchetgen Tchetgen. On varieties of doubly robust estimators under missingness not at random with a shadow variable. Biometrika, 103 0 (2): 0 475--482, 2016. doi:10.1093/biomet/asw016

work page doi:10.1093/biomet/asw016 2016
[18]

T., and Geng, Z

Wang Miao, Lan Liu, Eric Tchetgen Tchetgen, and Zhi Geng. Identification, Doubly Robust Estimation , and Semiparametric Efficiency Theory of Nonignorable Missing Data With a Shadow Variable , 2015. arXiv:1509.02556v2

work page arXiv 2015
[19]

Tchetgen Tchetgen, and Zhi Geng

Wang Miao, Lan Liu, Yilin Li, Eric J. Tchetgen Tchetgen, and Zhi Geng. Identification and Semiparametric Efficiency Theory of Nonignorable Missing Data with a Shadow Variable . ACM / IMS Journal of Data Science, 1 0 (2), 2024. doi:10.1145/3592389

work page doi:10.1145/3592389 2024
[20]

On Handling Self -masking and Other Hard Missing Data Problems

Karthika Mohan. On Handling Self -masking and Other Hard Missing Data Problems . In AAAI Spring Symposium 2019 , 2019. URL https://why19.causalai.net/papers/mohan-why19.pdf

work page 2019
[21]

Graphical Models for Processing Missing Data

Karthika Mohan and Judea Pearl. Graphical Models for Processing Missing Data . Journal of the American Statistical Association, 116 0 (534): 0 1023--1037, 2021. doi:10.1080/01621459.2021.1874961

work page doi:10.1080/01621459.2021.1874961 2021
[22]

Graphical Models for Inference with Missing Data

Karthika Mohan, Judea Pearl, and Jin Tian. Graphical Models for Inference with Missing Data . In Advances in Neural Information Processing Systems , volume 26. Curran Associates, Inc., 2013

work page 2013
[23]

On Testability and Goodness of Fit Tests in Missing Data Models

Razieh Nabi and Rohit Bhattacharya. On Testability and Goodness of Fit Tests in Missing Data Models . In Proceedings of the Thirty - Ninth Conference on Uncertainty in Artificial Intelligence , pages 1467--1477. PMLR, 2023. ISSN: 2640-3498

work page 2023
[24]

Full Law Identification in Graphical Models of Missing Data : Completeness Results

Razieh Nabi, Rohit Bhattacharya, and Ilya Shpitser. Full Law Identification in Graphical Models of Missing Data : Completeness Results . In Proceedings of the 37th International Conference on Machine Learning , pages 7153--7163. PMLR, 2020. URL https://proceedings.mlr.press/v119/nabi20a.html

work page 2020
[25]

Trang Quynh Nguyen, Ian Schmid, and Elizabeth A. Stuart. Clarifying causal mediation analysis for the applied researcher: Defining effects based on what we want to learn. Psychological Methods, 26 0 (2): 0 255--271, 2021. doi:10.1037/met0000299

work page doi:10.1037/met0000299 2021
[26]

Direct and Indirect Effects

Judea Pearl. Direct and Indirect Effects . In Proceedings of the seventeenth conference on uncertainty and artificial intelligence, pages 411--420. 2001

work page 2001
[27]

Can one assess whether missing data are missing at random in medical studies? Statistical Methods in Medical Research, 15 0 (3): 0 213--234, 2006

Richard F Potthoff, Gail E Tudor, Karen S Pieper, and Vic Hasselblad. Can one assess whether missing data are missing at random in medical studies? Statistical Methods in Medical Research, 15 0 (3): 0 213--234, 2006. doi:10.1191/0962280206sm448oa

work page doi:10.1191/0962280206sm448oa 2006
[28]

Robins and Sander Greenland

James M. Robins and Sander Greenland. Identifiability and Exchangeability for Direct and Indirect Effects . Epidemiology, 3 0 (2): 0 143--155, 1992. doi:10.1097/00001648-199203000-00013

work page doi:10.1097/00001648-199203000-00013 1992
[29]

Donald B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66 0 (5): 0 688--701, 1974. doi:10.1037/h0037350

work page doi:10.1037/h0037350 1974
[30]

Welch, Toby J

Jerzy Splawa-Neyman. On the application of probability theory to agricultural experiments. Essay on principles. Section 9 [ Translated and edited by DM Dabrowska and TP Speed ; republished in Statistical Science, 1990, 5(4):465-472 ]. Annals of Agricultural Sciences, 10: 0 1--51, 1923. doi:10.1214/ss/1177012031

work page doi:10.1214/ss/1177012031 1990
[31]

Ogburn, and Ilya Shpitser

Ranjani Srinivasan, Rohit Bhattacharya, Razieh Nabi, Elizabeth L. Ogburn, and Ilya Shpitser. Graphical Models of Entangled Missingness , April 2023. URL http://arxiv.org/abs/2304.01953. arXiv:2304.01953

work page arXiv 2023
[32]

Semiparametric estimation with data missing not at random using an instrumental variable

BaoLuo Sun, Lan Liu, Wang Miao, Kathleen Wirth, James Robins, and Eric J Tchetgen Tchetgen. Semiparametric estimation with data missing not at random using an instrumental variable. Statistica Sinica, 28 0 (4): 0 1965, 2018

work page 1965
[33]

VanderWeele, Stijn Vansteelandt, and James M

Tyler J. VanderWeele, Stijn Vansteelandt, and James M. Robins. Effect Decomposition in the Presence of an Exposure - Induced Mediator - Outcome Confounder :. Epidemiology, 25 0 (2): 0 300--306, 2014. doi:10.1097/EDE.0000000000000034

work page doi:10.1097/ede.0000000000000034 2014
[34]

An Instrumental Variable Approach for Identification and Estimation with Nonignorable Nonresponse

Sheng Wang, Jun Shao, and Jae Kwang Kim. An Instrumental Variable Approach for Identification and Estimation with Nonignorable Nonresponse . Statistica Sinica, 24 0 (3): 0 1097--1116, 2014

work page 2014
[35]

Causal inference with confounders missing not at random

S Yang, L Wang, and P Ding. Causal inference with confounders missing not at random. Biometrika, 106 0 (4): 0 875--888, 2019. doi:10.1093/biomet/asz048

work page doi:10.1093/biomet/asz048 2019
[36]

Methods for mediation analysis with missing data

Zhiyong Zhang and Lijuan Wang. Methods for mediation analysis with missing data. Psychometrika, 78 0 (1): 0 154--184, 2013. doi:10.1007/s11336-012-9301-5

work page doi:10.1007/s11336-012-9301-5 2013
[37]

Semiparametric Pseudo - Likelihoods in Generalized Linear Models With Nonignorable Missing Data

Jiwei Zhao and Jun Shao. Semiparametric Pseudo - Likelihoods in Generalized Linear Models With Nonignorable Missing Data . Journal of the American Statistical Association, 110 0 (512): 0 1577--1590, 2015. doi:10.1080/01621459.2014.983234

work page doi:10.1080/01621459.2014.983234 2015
[38]

Population-level balance in signed networks

Shuozhi Zuo, Debashis Ghosh, Peng Ding, and Fan Yang. Mediation analysis with the mediator and outcome missing not at random. Journal of the American Statistical Association, 2024. doi:10.1080/01621459.2024.2359132

work page doi:10.1080/01621459.2024.2359132 2024

[1] [1]

The Moderator - Mediator Variable Distinction in Social Psychological Research : Conceptual , Strategic , and Statistical Considerations

Reuben M Baron and David A Kenny. The Moderator - Mediator Variable Distinction in Social Psychological Research : Conceptual , Strategic , and Statistical Considerations . Journal of Personality and Social Psychology, 51 0 (6): 0 1173--1182, 1986

work page 1986

[2] [2]

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

David Benkeser, Iván Díaz, and Jialu Ran. Inference for natural mediation effects under case-cohort sampling with applications in identifying COVID -19 vaccine correlates of protection. arXiv:2103.02643 [math, stat], 2021

work page arXiv 2021

[3] [3]

Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser, and James M. Robins. Identification In Missing Data Models Represented By Directed Acyclic Graphs . In Proceedings of The 35th Uncertainty in Artificial Intelligence Conference , pages 1149--1158. PMLR, 2020. ISSN: 2640-3498

work page 2020

[4] [4]

Cole and Constantine E

Stephen R. Cole and Constantine E. Frangakis. The Consistency Statement in Causal Inference : A Definition or an Assumption ? Epidemiology, 20 0 (1), 2009. doi:10.1097/EDE.0b013e31818ef366

work page doi:10.1097/ede.0b013e31818ef366 2009

[5] [5]

Ghazaleh Dashti, Katherine J

S. Ghazaleh Dashti, Katherine J. Lee, Julie A. Simpson, John B. Carlin, and Margarita Moreno-Betancur. Handling multivariable missing data in causal mediation analysis, 2024. arXiv:2403.17396 [stat]

work page arXiv 2024

[6] [6]

Direct and Indirect Effects of Sequential Treatments

Vanessa Didelez, Philip Dawid, and Sara Geneletti. Direct and Indirect Effects of Sequential Treatments . In Proceedings of the Twenty - Second Conference on Uncertainty in Artificial Intelligence , 2006. doi:10.48550/arXiv.1206.6840

work page doi:10.48550/arxiv.1206.6840 2006

[7] [7]

A new instrumental method for dealing with endogenous selection

Xavier d’Haultfoeuille. A new instrumental method for dealing with endogenous selection. Journal of Econometrics, 154 0 (1): 0 1--15, 2010. doi:10.1016/j.jeconom.2009.06.005

work page doi:10.1016/j.jeconom.2009.06.005 2010

[8] [8]

Nonparametric causal mediation analysis for stochastic interventional (in)direct effects

Nima S Hejazi, Kara E Rudolph, Mark J Van Der Laan, and Iván Díaz. Nonparametric causal mediation analysis for stochastic interventional (in)direct effects. Biostatistics, 24 0 (3): 0 686--707, 2023. doi:10.1093/biostatistics/kxac002

work page doi:10.1093/biostatistics/kxac002 2023

[9] [9]

A general approach to causal mediation analysis

Kosuke Imai, Luke Keele, and Dustin Tingley. A general approach to causal mediation analysis. Psychological Methods, 15 0 (4): 0 309--334, 2010 a . doi:10.1037/a0020761

work page doi:10.1037/a0020761 2010

[10] [10]

Identification, Inference and Sensitivity Analysis for Causal Mediation Effects

Kosuke Imai, Luke Keele, and Teppei Yamamoto. Identification, Inference and Sensitivity Analysis for Causal Mediation Effects . Statistical Science, 25 0 (1), 2010 b . doi:10.1214/10-STS321

work page doi:10.1214/10-sts321 2010

[11] [11]

Phillip S. Kott. Calibration Weighting When Model and Calibration Variables Can Differ . In Fulvia Mecatti, Pier Luigi Conti, and Maria Giovanna Ranalli, editors, Contributions to Sampling Statistics , pages 1--18. Springer International Publishing, Cham, 2014. ISBN 978-3-319-05320-2. doi:10.1007/978-3-319-05320-2_1

work page doi:10.1007/978-3-319-05320-2_1 2014

[12] [12]

Nuisance mediators and missing data in mediation analyses of pain trials

Hopin Lee. Nuisance mediators and missing data in mediation analyses of pain trials. European Journal of Pain, 24 0 (9): 0 1651--1652, 2020. doi:10.1002/ejp.1637

work page doi:10.1002/ejp.1637 2020

[13] [13]

E. L. Lehmann and Henry Scheffé. Completeness, Similar Regions , and Unbiased Estimation - Part I . Sankhyā: The Indian Journal of Statistics, 10 0 (4): 0 305--340, 1950. doi:10.1007/978-1-4614-1412-4_23

work page doi:10.1007/978-1-4614-1412-4_23 1950

[14] [14]

Identifiability and estimation of causal mediation effects with missing data

Wei Li and Xiao-Hua Zhou. Identifiability and estimation of causal mediation effects with missing data. Statistics in Medicine, 36 0 (25): 0 3948--3965, 2017. doi:10.1002/sim.7413

work page doi:10.1002/sim.7413 2017

[15] [15]

Tchetgen Tchetgen

Yilin Li, Wang Miao, Ilya Shpitser, and Eric J. Tchetgen Tchetgen. A self-censoring model for multivariate nonignorable nonmonotone missing data. Biometrics, 79 0 (4): 0 3203--3214, 2023. doi:10.1111/biom.13916

work page doi:10.1111/biom.13916 2023

[16] [16]

Estimating Mixed Memberships With Sharp Eigenvector Deviations

Daniel Malinsky, Ilya Shpitser, and Eric J. Tchetgen Tchetgen. Semiparametric Inference for Nonmonotone Missing - Not -at- Random Data : The No Self - Censoring Model . Journal of the American Statistical Association, 117 0 (539): 0 1415--1423, 2022. doi:10.1080/01621459.2020.1862669

work page doi:10.1080/01621459.2020.1862669 2022

[17] [17]

Tchetgen Tchetgen

Wang Miao and Eric J. Tchetgen Tchetgen. On varieties of doubly robust estimators under missingness not at random with a shadow variable. Biometrika, 103 0 (2): 0 475--482, 2016. doi:10.1093/biomet/asw016

work page doi:10.1093/biomet/asw016 2016

[18] [18]

T., and Geng, Z

Wang Miao, Lan Liu, Eric Tchetgen Tchetgen, and Zhi Geng. Identification, Doubly Robust Estimation , and Semiparametric Efficiency Theory of Nonignorable Missing Data With a Shadow Variable , 2015. arXiv:1509.02556v2

work page arXiv 2015

[19] [19]

Tchetgen Tchetgen, and Zhi Geng

Wang Miao, Lan Liu, Yilin Li, Eric J. Tchetgen Tchetgen, and Zhi Geng. Identification and Semiparametric Efficiency Theory of Nonignorable Missing Data with a Shadow Variable . ACM / IMS Journal of Data Science, 1 0 (2), 2024. doi:10.1145/3592389

work page doi:10.1145/3592389 2024

[20] [20]

On Handling Self -masking and Other Hard Missing Data Problems

Karthika Mohan. On Handling Self -masking and Other Hard Missing Data Problems . In AAAI Spring Symposium 2019 , 2019. URL https://why19.causalai.net/papers/mohan-why19.pdf

work page 2019

[21] [21]

Graphical Models for Processing Missing Data

Karthika Mohan and Judea Pearl. Graphical Models for Processing Missing Data . Journal of the American Statistical Association, 116 0 (534): 0 1023--1037, 2021. doi:10.1080/01621459.2021.1874961

work page doi:10.1080/01621459.2021.1874961 2021

[22] [22]

Graphical Models for Inference with Missing Data

Karthika Mohan, Judea Pearl, and Jin Tian. Graphical Models for Inference with Missing Data . In Advances in Neural Information Processing Systems , volume 26. Curran Associates, Inc., 2013

work page 2013

[23] [23]

On Testability and Goodness of Fit Tests in Missing Data Models

Razieh Nabi and Rohit Bhattacharya. On Testability and Goodness of Fit Tests in Missing Data Models . In Proceedings of the Thirty - Ninth Conference on Uncertainty in Artificial Intelligence , pages 1467--1477. PMLR, 2023. ISSN: 2640-3498

work page 2023

[24] [24]

Full Law Identification in Graphical Models of Missing Data : Completeness Results

Razieh Nabi, Rohit Bhattacharya, and Ilya Shpitser. Full Law Identification in Graphical Models of Missing Data : Completeness Results . In Proceedings of the 37th International Conference on Machine Learning , pages 7153--7163. PMLR, 2020. URL https://proceedings.mlr.press/v119/nabi20a.html

work page 2020

[25] [25]

Trang Quynh Nguyen, Ian Schmid, and Elizabeth A. Stuart. Clarifying causal mediation analysis for the applied researcher: Defining effects based on what we want to learn. Psychological Methods, 26 0 (2): 0 255--271, 2021. doi:10.1037/met0000299

work page doi:10.1037/met0000299 2021

[26] [26]

Direct and Indirect Effects

Judea Pearl. Direct and Indirect Effects . In Proceedings of the seventeenth conference on uncertainty and artificial intelligence, pages 411--420. 2001

work page 2001

[27] [27]

Can one assess whether missing data are missing at random in medical studies? Statistical Methods in Medical Research, 15 0 (3): 0 213--234, 2006

Richard F Potthoff, Gail E Tudor, Karen S Pieper, and Vic Hasselblad. Can one assess whether missing data are missing at random in medical studies? Statistical Methods in Medical Research, 15 0 (3): 0 213--234, 2006. doi:10.1191/0962280206sm448oa

work page doi:10.1191/0962280206sm448oa 2006

[28] [28]

Robins and Sander Greenland

James M. Robins and Sander Greenland. Identifiability and Exchangeability for Direct and Indirect Effects . Epidemiology, 3 0 (2): 0 143--155, 1992. doi:10.1097/00001648-199203000-00013

work page doi:10.1097/00001648-199203000-00013 1992

[29] [29]

Donald B. Rubin. Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66 0 (5): 0 688--701, 1974. doi:10.1037/h0037350

work page doi:10.1037/h0037350 1974

[30] [30]

Welch, Toby J

Jerzy Splawa-Neyman. On the application of probability theory to agricultural experiments. Essay on principles. Section 9 [ Translated and edited by DM Dabrowska and TP Speed ; republished in Statistical Science, 1990, 5(4):465-472 ]. Annals of Agricultural Sciences, 10: 0 1--51, 1923. doi:10.1214/ss/1177012031

work page doi:10.1214/ss/1177012031 1990

[31] [31]

Ogburn, and Ilya Shpitser

Ranjani Srinivasan, Rohit Bhattacharya, Razieh Nabi, Elizabeth L. Ogburn, and Ilya Shpitser. Graphical Models of Entangled Missingness , April 2023. URL http://arxiv.org/abs/2304.01953. arXiv:2304.01953

work page arXiv 2023

[32] [32]

Semiparametric estimation with data missing not at random using an instrumental variable

BaoLuo Sun, Lan Liu, Wang Miao, Kathleen Wirth, James Robins, and Eric J Tchetgen Tchetgen. Semiparametric estimation with data missing not at random using an instrumental variable. Statistica Sinica, 28 0 (4): 0 1965, 2018

work page 1965

[33] [33]

VanderWeele, Stijn Vansteelandt, and James M

Tyler J. VanderWeele, Stijn Vansteelandt, and James M. Robins. Effect Decomposition in the Presence of an Exposure - Induced Mediator - Outcome Confounder :. Epidemiology, 25 0 (2): 0 300--306, 2014. doi:10.1097/EDE.0000000000000034

work page doi:10.1097/ede.0000000000000034 2014

[34] [34]

An Instrumental Variable Approach for Identification and Estimation with Nonignorable Nonresponse

Sheng Wang, Jun Shao, and Jae Kwang Kim. An Instrumental Variable Approach for Identification and Estimation with Nonignorable Nonresponse . Statistica Sinica, 24 0 (3): 0 1097--1116, 2014

work page 2014

[35] [35]

Causal inference with confounders missing not at random

S Yang, L Wang, and P Ding. Causal inference with confounders missing not at random. Biometrika, 106 0 (4): 0 875--888, 2019. doi:10.1093/biomet/asz048

work page doi:10.1093/biomet/asz048 2019

[36] [36]

Methods for mediation analysis with missing data

Zhiyong Zhang and Lijuan Wang. Methods for mediation analysis with missing data. Psychometrika, 78 0 (1): 0 154--184, 2013. doi:10.1007/s11336-012-9301-5

work page doi:10.1007/s11336-012-9301-5 2013

[37] [37]

Semiparametric Pseudo - Likelihoods in Generalized Linear Models With Nonignorable Missing Data

Jiwei Zhao and Jun Shao. Semiparametric Pseudo - Likelihoods in Generalized Linear Models With Nonignorable Missing Data . Journal of the American Statistical Association, 110 0 (512): 0 1577--1590, 2015. doi:10.1080/01621459.2014.983234

work page doi:10.1080/01621459.2014.983234 2015

[38] [38]

Population-level balance in signed networks

Shuozhi Zuo, Debashis Ghosh, Peng Ding, and Fan Yang. Mediation analysis with the mediator and outcome missing not at random. Journal of the American Statistical Association, 2024. doi:10.1080/01621459.2024.2359132

work page doi:10.1080/01621459.2024.2359132 2024