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REVIEW 3 major objections 2 minor 300 references

Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis

T0 review · 3 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Cumulative natural direct and indirect effects decompose total effects while preserving skew-symmetry and additivity for continuous treatments.

desk verdict The paper defines CNDE and CNIE by decomposing local effects E[∂x Y_x] to fix skew-symmetry and additivity for continuous treatments, but the differentiability step sits outside the listed assumptions. read the letter →

arxiv 2606.22165 v1 pith:25ZYDKLR submitted 2026-06-20 stat.ME

Yuta Kawakami , Jin Tian This is my paper · ORCID
classification stat.ME
keywords causalmediationanalysisnaturaldirecteffectsindirectcontinuoustreatmentsskew-symmetryadditivityeffectdecomposition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

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.

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.

Watch

Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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)
    purpose: Direct effect measure satisfying skew-symmetry and additivity
    Newly introduced measure; no independent evidence outside the paper.
  • Cumulative natural indirect effect (CNIE)
    purpose: Indirect effect measure satisfying skew-symmetry and additivity
    Newly introduced measure; no independent evidence outside the paper.

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Cite this review

Pith. "Pith review of Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis." pith.science (2026). https://pith.science/paper/25ZYDKLR

@misc{pith2026260622165,
  author       = {Pith},
  title        = {Pith review of: Cumulative Natural Direct and Indirect Effects for Causal Mediation Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/25ZYDKLR}},
  note         = {Machine review of arXiv:2606.22165}
}
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.

Figures

Figures reproduced from arXiv: 2606.22165 by the authors.

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 functions F = {fV… view at source ↗
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 effect (L… view at source ↗
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 direct and indi… view at source ↗

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Works this paper leans on

300 extracted references · 17 canonical work pages

  1. [1]

    Wong, Wing Hung , title =

  2. [2]

    Engle , journal =

    Robert F. Engle , journal =. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation , urldate =

  3. [3]

    Randomization Inference for Treatment Effect Variation , publisher =

    Ding, Peng and Feller, Avi and Miratrix, Luke , keywords =. Randomization Inference for Treatment Effect Variation , publisher =. 2014 , copyright =

    2014

  4. [4]

    D. R. Cox , journal =. Interaction , urldate =

  5. [5]

    Gujarati , author=

    Basic econometrics/Damodar N. Gujarati , author=

  6. [6]

    International Journal of Mathematics Trends and Technology (IJMTT) , volume=

    On the calculus of Dirac delta function with some applications , author=. International Journal of Mathematics Trends and Technology (IJMTT) , volume=

  7. [7]

    , journal =

    Grimstad, Bjarne and Knudsen, Brage R. , journal =. Mathematical programming formulations for piecewise polynomial functions , volume =

  8. [8]

    and Pedregal, Diego J

    Young, Peter C. and Pedregal, Diego J. and Tych, Wlodek , journal =. Dynamic harmonic regression , volume =

Show all 300 references
  1. [9]

    Probability Theory and Statistical Inference: Econometric Modeling with Observational Data , year =

    Spanos, Aris , publisher =. Probability Theory and Statistical Inference: Econometric Modeling with Observational Data , year =

  2. [10]

    Inferring Cause and Effect in the Presence of Heteroscedastic Noise , volume =

    Xu, Sascha and Mian, Osman A and Marx, Alexander and Vreeken, Jilles , booktitle =. Inferring Cause and Effect in the Presence of Heteroscedastic Noise , volume =

  3. [11]

    T. S. Breusch and A. R. Pagan , journal =. A Simple Test for Heteroscedasticity and Random Coefficient Variation , urldate =

  4. [12]

    J. B. Diaz and F. T. Metcalf , journal =. On Iteration Procedures for Equations of the First Kind, Ax = y, and Picard's Criterion for the Existence of a Solution , urldate =

  5. [13]

    1910 , journal=

    Sur un th. 1910 , journal=

    1910

  6. [14]

    Integral equations of the first kind, inverse problems and regularization: a crash course , volume =

    C W Groetsch, , journal =. Integral equations of the first kind, inverse problems and regularization: a crash course , volume =

  7. [15]

    An Introduction to Partial Differential Equations , year =

    Pinchover, Yehuda and Rubinstein, Jacob , publisher =. An Introduction to Partial Differential Equations , year =

  8. [16]

    Calculus of variations / J

    Jost, J. Calculus of variations / J

  9. [17]

    Imbens , journal =

    Guido W. Imbens , journal =. Instrumental Variables: An Econometrician's Perspective , urldate =

  10. [18]

    1926 , publisher=

    An introduction to the study of integral equations , author=. 1926 , publisher=

    1926

  11. [19]

    Fuller , journal =

    Yasuo Amemiya and Wayne A. Fuller , journal =

  12. [20]

    Journal of Mathematical Analysis and Applications , volume=

    On compact operators , author=. Journal of Mathematical Analysis and Applications , volume=

  13. [21]

    McCormick , title =

    Chunxiao Li and Cynthia Rudin and Tyler H. McCormick , title =. Journal of Machine Learning Research , year =

  14. [22]

    Testing Additive Separability of Error Term in Nonparametric Structural Models , volume =

    Liangjun Su and Yundong Tu and Aman Ullah , journal =. Testing Additive Separability of Error Term in Nonparametric Structural Models , volume =

  15. [23]

    Baum and Mark E

    Christopher F. Baum and Mark E. Schaffer and Steven Stillman , journal =. Instrumental Variables and GMM: Estimation and Testing , volume =

  16. [24]

    Large Sample Properties of Generalized Method of Moments Estimators , urldate =

    Lars Peter Hansen , journal =. Large Sample Properties of Generalized Method of Moments Estimators , urldate =

  17. [25]

    Angrist and Guido W

    Joshua D. Angrist and Guido W. Imbens , journal =. Two-Stage Least Squares Estimation of Average Causal Effects in Models with Variable Treatment Intensity , volume =

  18. [26]

    The nonlinear two-stage least-squares estimator , volume =

    Takeshi Amemiya , journal =. The nonlinear two-stage least-squares estimator , volume =

  19. [27]

    Statistica Sinica , volume=

    AN EQUATION FOR THE IDENTIFICATION OF AVERAGE CAUSAL EFFECT IN NONLINEAR MODELS , author=. Statistica Sinica , volume=

  20. [28]

    2015 , publisher=

    Numerical analysis , author=. 2015 , publisher=

    2015

  21. [29]

    2009 , publisher=

    Measurement error models , author=. 2009 , publisher=

    2009

  22. [30]

    Errors-in-variables methods in system identification , volume =

    Torsten S. Errors-in-variables methods in system identification , volume =. Automatica , number =

  23. [31]

    Pattern Recognition and Machine Learning , year =

    Bishop, Christopher , month =. Pattern Recognition and Machine Learning , year =

  24. [32]

    Earnings, Schooling, and Ability Revisited

    Card, David. Earnings, Schooling, and Ability Revisited. 1994

    1994

  25. [33]

    and Newey, Whitney K

    Hausman, Jerry A. and Newey, Whitney K. and Woutersen, Tiemen and Chao, John C. and Swanson, Norman R. , journal =. Instrumental variable estimation with heteroskedasticity and many instruments , volume =

  26. [34]

    Newey and James L

    Whitney K. Newey and James L. Powell , journal =. Instrumental Variable Estimation of Nonparametric Models , urldate =

  27. [35]

    2017 , institution=

    Individual counterfactuals with multidimensional unobserved heterogeneity , author=. 2017 , institution=

    2017

  28. [36]

    Matzkin , journal =

    Rosa L. Matzkin , journal =. On independence conditions in nonseparable models: Observable and unobservable instruments , volume =

  29. [37]

    Nonlinear causal discovery with additive noise models , volume =

    Hoyer, Patrik and Janzing, Dominik and Mooij, Joris M and Peters, Jonas and Sch\". Nonlinear causal discovery with additive noise models , volume =. Advances in Neural Information Processing Systems , editor =

  30. [38]

    and Rubin, Donald B

    Imbens, Guido W. and Rubin, Donald B. , year=. Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction , publisher=

  31. [39]

    Health Service Research Methodology: A Focus on AIDS , publisher=

    James Robins , title =. Health Service Research Methodology: A Focus on AIDS , publisher=

  32. [40]

    2009 , edition =

    Judea Pearl , title =. 2009 , edition =

    2009

  33. [41]

    Methods of mathematical physics , volume=

    Lagrange’s interpolation formula , author=. Methods of mathematical physics , volume=. 1988 , publisher=

    1988

  34. [42]

    Morgan , journal =

    Luke Keele and Jason W. Morgan , journal =. HOW STRONG IS STRONG ENOUGH? STRENGTHENING INSTRUMENTS THROUGH MATCHING AND WEAK INSTRUMENT TESTS , urldate =

  35. [43]

    Trace Class Operators and Lidskii’s Theorem , author=

  36. [44]

    Unobserved Ability, Efficiency Wages, and Interindustry Wage Differentials , urldate =

    McKinley Blackburn and David Neumark , journal =. Unobserved Ability, Efficiency Wages, and Interindustry Wage Differentials , urldate =

  37. [45]

    , author=

    A method of successive approximations for Fredholm integral equations of the first kind. , author=. Uspeki mat Nauk , volume=

  38. [46]

    , publisher =

    Muscat, J. , publisher =. Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras , year =

  39. [47]

    Linear regression , pages=

    Multiple linear regression , author=. Linear regression , pages=. 2017 , publisher=

    2017

  40. [48]

    Journal of the American Statistical Association , year=

    Two-Stage Least Squares Estimation of Average Causal Effects in Models with Variable Treatment Intensity , author=. Journal of the American Statistical Association , year=

  41. [49]

    Stock , booktitle =

    J.H. Stock , booktitle =. Instrumental Variables in Statistics and Econometrics , year =

  42. [50]

    Kernel Instrumental Variable Regression , volume =

    Singh, Rahul and Sahani, Maneesh and Gretton, Arthur , booktitle =. Kernel Instrumental Variable Regression , volume =

  43. [51]

    Terza and Anirban Basu and Paul J

    Joseph V. Terza and Anirban Basu and Paul J. Rathouz , journal =. Two-stage residual inclusion estimation: Addressing endogeneity in health econometric modeling , volume =

  44. [52]

    and Tibshirani, R

    Hastie, T. and Tibshirani, R. and Friedman, J. , publisher =. The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition , year =

  45. [53]

    , journal =

    Thorndike, Robert L. , journal =. Who belongs in the family? , volume =

  46. [54]

    2011 , institution=

    Continuous Treatments , author=. 2011 , institution=

    2011

  47. [55]

    Wooldridge , title=

    Jeffrey M. Wooldridge , title=

  48. [56]

    Nguyen Viet, Cuong , title=

  49. [57]

    Bounds on Treatment Effects from Studies with Imperfect Compliance , volume =

    Alexander Balke and Judea Pearl , journal =. Bounds on Treatment Effects from Studies with Imperfect Compliance , volume =

  50. [58]

    arXiv: Methodology , year=

    Regression with genuinely functional errors-in-covariates , author=. arXiv: Methodology , year=

  51. [59]

    Functional Linear Regression: Dependence and Error Contamination , volume =

    Cheng Chen and Shaojun Guo and Xinghao Qiao , journal =. Functional Linear Regression: Dependence and Error Contamination , volume =

  52. [60]

    2010 , publisher=

    Econometric analysis of cross section and panel data , author=. 2010 , publisher=

    2010

  53. [61]

    Efficient Estimation of Conditional Variance Functions in Stochastic Regression , urldate =

    Jianqing Fan and Qiwei Yao , journal =. Efficient Estimation of Conditional Variance Functions in Stochastic Regression , urldate =

  54. [62]

    and Berger, R.L

    Casella, G. and Berger, R.L. , publisher =. Statistical Inference , year =

  55. [63]

    , publisher =

    Greene, W.H. , publisher =. Econometric Analysis , year =

  56. [64]

    An introduction to causal inference

    Pearl, Judea , journal =. An introduction to causal inference. , volume =

  57. [65]

    Asymptotic normality of support vector machine variants and other regularized kernel methods , volume =

    Robert Hable , journal =. Asymptotic normality of support vector machine variants and other regularized kernel methods , volume =

  58. [66]

    Asymptotic normality and confidence intervals for derivatives of 2-layers neural network in the random features model , volume =

    Shen, Yiwei and Bellec, Pierre C , booktitle =. Asymptotic normality and confidence intervals for derivatives of 2-layers neural network in the random features model , volume =

  59. [67]

    , publisher =

    DiBenedetto, E. , publisher =. Real Analysis , year =

  60. [68]

    Tchetgen Tchetgen and David E

    Mohammad Taha Bahadori and Eric J. Tchetgen Tchetgen and David E. Heckerman , title =. 2022 , booktitle =

    2022

  61. [69]

    Program Heterogeneity and Propensity Score Matching: An Application to the Evaluation of Active Labor Market Policies , urldate =

    Michael Lechner , journal =. Program Heterogeneity and Propensity Score Matching: An Application to the Evaluation of Active Labor Market Policies , urldate =

  62. [70]

    J. A. Hausman , journal =. Specification Tests in Econometrics , urldate =

  63. [71]

    Algebraic Geometry and Statistical Learning Theory , DOI=

    Watanabe, Sumio , year=. Algebraic Geometry and Statistical Learning Theory , DOI=

  64. [72]

    War and Wages , volume =

    Dylan S Small and Paul R Rosenbaum , journal =. War and Wages , volume =

  65. [73]

    Ridge: a computer program for calculating ridge regression estimates , author=

  66. [74]

    Identification and inference for econometric models: Essays in honor of Thomas Rothenberg , pages=

    Unobserved heterogeneity and estimation of average partial effects , author=. Identification and inference for econometric models: Essays in honor of Thomas Rothenberg , pages=

  67. [75]

    IRREGULAR

    Bryan S. Graham and James L. Powell , journal =. IDENTIFICATION AND ESTIMATION OF AVERAGE PARTIAL EFFECTS IN "IRREGULAR" CORRELATED RANDOM COEFFICIENT PANEL DATA MODELS , urldate =

  68. [76]

    and Keueger, Alan B

    Angrist, Joshua D. and Keueger, Alan B. , journal =

  69. [77]

    Job Market Signaling , urldate =

    Michael Spence , journal =. Job Market Signaling , urldate =

  70. [78]

    Econometric evaluation of labour market policies , pages=

    Identification and estimation of causal effects of multiple treatments under the conditional independence assumption , author=. Econometric evaluation of labour market policies , pages=. 2001 , publisher=

    2001

  71. [79]

    The Case against Education: Why the Education System Is a Waste of Time and Money , urldate =

    Bryan Caplan , publisher =. The Case against Education: Why the Education System Is a Waste of Time and Money , urldate =

  72. [80]

    Estimation and Inference of Heterogeneous Treatment Effects using Random Forests , volume =

    Stefan Wager and Susan Athey , journal =. Estimation and Inference of Heterogeneous Treatment Effects using Random Forests , volume =

  73. [81]

    , journal =

    Athey, Susan and Imbens, Guido W. , journal =. Machine Learning Methods That Economists Should Know About , volume =

  74. [82]

    Metalearners for estimating heterogeneous treatment effects using machine learning , volume =

    S. Metalearners for estimating heterogeneous treatment effects using machine learning , volume =. Proceedings of the National Academy of Sciences , number =

  75. [83]

    Nie, X and Wager, S , journal =

  76. [84]

    Chernozhukov, Victor and Chetverikov, Denis and Demirer, Mert and Duflo, Esther and Hansen, Christian and Newey, Whitney and Robins, James , journal =

  77. [85]

    arXiv: Methodology , year=

    Asymptotic properties of a Nadaraya-Watson type estimator for regression functions of infinite order , author=. arXiv: Methodology , year=

  78. [86]

    Foster and Vasilis Syrgkanis , title=

    Dylan J. Foster and Vasilis Syrgkanis , title=. CoRR , volume=. 2019 , cdate=

    2019

  79. [87]

    Handbook of Econometrics , editor =

    Panel data , author =. Handbook of Econometrics , editor =. 1984 , chapter =

    1984

  80. [88]

    Use of the Singular Value Decomposition in Regression Analysis , volume =

    John Mandel , journal =. Use of the Singular Value Decomposition in Regression Analysis , volume =

  81. [89]

    and Newey, Whitney K

    Imbens, Guido W. and Newey, Whitney K. , journal =. Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity , volume =

  82. [90]

    Identification of Nonseparable Models Using Instruments With Small Support , volume =

    Torgovitsky, Alexander , journal =. Identification of Nonseparable Models Using Instruments With Small Support , volume =

  83. [91]

    , journal =

    Chen, Xiaohong and Chernozhukov, Victor and Lee, Sokbae and Newey, Whitney K. , journal =. Local Identification of Nonparametric and Semiparametric Models , volume =

  84. [92]

    1996 , journal =

    Degrees Matter: New Evidence on Sheepskin Effects in the Returns to Education , author =. 1996 , journal =

    1996

  85. [93]

    Bounded, efficient and multiply robust estimation of average treatment effects using instrumental variables

    Wang, Linbo and Tchetgen Tchetgen, Eric , journal =. Bounded, efficient and multiply robust estimation of average treatment effects using instrumental variables. , volume =

  86. [94]

    Structure and Interpretation of Computer Programs

    Harold Abelson and Gerald Jay Sussman and Julie Sussman. Structure and Interpretation of Computer Programs. 1985

    1985

  87. [95]

    Visual Information Extraction with Lixto

    Robert Baumgartner and Georg Gottlob and Sergio Flesca. Visual Information Extraction with Lixto. Proceedings of the 27th International Conference on Very Large Databases. 2001

    2001

  88. [96]

    Brachman and James G

    Ronald J. Brachman and James G. Schmolze. An overview of the KL-ONE knowledge representation system. Cognitive Science. 1985

    1985

  89. [97]

    Complexity results for nonmonotonic logics

    Georg Gottlob. Complexity results for nonmonotonic logics. Journal of Logic and Computation. 1992

    1992

  90. [98]

    Hypertree Decompositions and Tractable Queries

    Georg Gottlob and Nicola Leone and Francesco Scarcello. Hypertree Decompositions and Tractable Queries. Journal of Computer and System Sciences. 2002

    2002

  91. [99]

    Levesque

    Hector J. Levesque. Foundations of a functional approach to knowledge representation. Artificial Intelligence. 1984

    1984

  92. [100]

    Levesque

    Hector J. Levesque. A logic of implicit and explicit belief. Proceedings of the Fourth National Conference on Artificial Intelligence. 1984

    1984

  93. [101]

    On the compilability and expressive power of propositional planning formalisms

    Bernhard Nebel. On the compilability and expressive power of propositional planning formalisms. Journal of Artificial Intelligence Research. 2000

    2000

  94. [102]

    Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education

    Clancey, William J. Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education. Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI-83)

  95. [103]

    Classification Problem Solving

    Clancey, William J. Classification Problem Solving. Proceedings of the Fourth National Conference on Artificial Intelligence

  96. [104]

    , title =

    Robinson, Arthur L. , title =. 1980 , doi =. https://science.sciencemag.org/content/208/4447/1019.full.pdf , journal =

    1980

  97. [105]

    New Ways to Make Microcircuits Smaller---Duplicate Entry

    Robinson, Arthur L. New Ways to Make Microcircuits Smaller---Duplicate Entry. Science

  98. [106]

    Clancey and Glenn Rennels , abstract =

    Diane Warner Hasling and William J. Clancey and Glenn Rennels , abstract =. Strategic explanations for a diagnostic consultation system , journal =. 1984 , issn =. doi:https://doi.org/10.1016/S0020-7373(84)80003-6 , url =

    1984 doi

  99. [107]

    and Rennels, Glenn R

    Hasling, Diane Warner and Clancey, William J. and Rennels, Glenn R. and Test, Thomas. Strategic Explanations in Consultation---Duplicate. The International Journal of Man-Machine Studies

  100. [108]

    Poligon: A System for Parallel Problem Solving

    Rice, James. Poligon: A System for Parallel Problem Solving

  101. [109]

    Transfer of Rule-Based Expertise through a Tutorial Dialogue

    Clancey, William J. Transfer of Rule-Based Expertise through a Tutorial Dialogue

  102. [110]

    The Engineering of Qualitative Models

    Clancey, William J. The Engineering of Qualitative Models

  103. [111]

    2008 , eprint=

    Crime and punishment in scientific research , author=. 2008 , eprint=

    2008

  104. [112]

    Pluto: The 'Other' Red Planet

    NASA. Pluto: The 'Other' Red Planet

  105. [113]

    Imbens and Whitney K

    Victor Chernozhukov and Guido W. Imbens and Whitney K. Newey , journal =. Instrumental variable estimation of nonseparable models , volume =

  106. [114]

    The Propensity Score with Continuous Treatments

    Keisuke Hirano and Imbens, Guido W. The Propensity Score with Continuous Treatments. Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives. 2005. doi:10.1002/0470090456.ch7

    2005 doi

  107. [115]

    Kennedy and Zongming Ma and Matthew D

    Edward H. Kennedy and Zongming Ma and Matthew D. McHugh and Dylan S. Small , journal =. Non-parametric methods for doubly robust estimation of continuous treatment effects , urldate =

  108. [116]

    1995 , publisher=

    Numerical methods for the solution of ill-posed problems , author=. 1995 , publisher=

    1995

  109. [117]

    Imbens and Joshua D

    Guido W. Imbens and Joshua D. Angrist , journal =. Identification and Estimation of Local Average Treatment Effects , urldate =

  110. [118]

    Chapter 30 - The Causal Effect of Education on Earnings , volume =

    David Card , editor =. Chapter 30 - The Causal Effect of Education on Earnings , volume =

  111. [119]

    Jason Hartford and Greg Lewis and Kevin Leyton-Brown and Matt Taddy , booktitle =. Deep

  112. [120]

    Estimating the Returns to Schooling: Some Econometric Problems , urldate =

    Zvi Griliches , journal =. Estimating the Returns to Schooling: Some Econometric Problems , urldate =

  113. [121]

    Proceedings of the AAAI Workshop on Information Theoretic Causal Inference and Discovery , year=

    Causal Inference with Heteroscedastic Noise Models , author=. Proceedings of the AAAI Workshop on Information Theoretic Causal Inference and Discovery , year=

  114. [122]

    , publisher =

    Leoni, G. , publisher =. A First Course in Sobolev Spaces , year =

  115. [123]

    ERN: Cross-Sectional Models , year=

    The Economics of Exclusion Restrictions in IV Models , author=. ERN: Cross-Sectional Models , year=

  116. [124]

    , chapter =

    Hirano, Keisuke and Imbens, Guido W. , chapter =. The Propensity Score with Continuous Treatments , year =

  117. [125]

    Journal of Causal Inference , year=

    A Boosting Algorithm for Estimating Generalized Propensity Scores with Continuous Treatments , author=. Journal of Causal Inference , year=

  118. [126]

    Recursive partitioning for heterogeneous causal effects , volume =

    Susan Athey and Guido Imbens , journal =. Recursive partitioning for heterogeneous causal effects , volume =

  119. [127]

    Randomization inference for treatment effect variation , urldate =

    Peng Ding and Avi Feller and Luke Miratrix , journal =. Randomization inference for treatment effect variation , urldate =

  120. [128]

    Angrist , journal =

    Joshua D. Angrist , journal =. Treatment Effect Heterogeneity in Theory and Practice , urldate =

  121. [129]

    Journal of Causal Inference , number =

    Instruments with Heterogeneous Effects: Bias, Monotonicity, and Localness , author =. Journal of Causal Inference , number =

  122. [130]

    Bargagli-Stoffi and Kristof De Witte and Giorgio Gnecco , journal =

    Falco J. Bargagli-Stoffi and Kristof De Witte and Giorgio Gnecco , journal =

  123. [131]

    Machine Learning Estimation of Heterogeneous Treatment Effects with Instruments , volume =

    Syrgkanis, Vasilis and Lei, Victor and Oprescu, Miruna and Hei, Maggie and Battocchi, Keith and Lewis, Greg , booktitle =. Machine Learning Estimation of Heterogeneous Treatment Effects with Instruments , volume =

  124. [132]

    arXiv e-prints , pages=

    Towards R-learner of conditional average treatment effects with a continuous treatment: T-identification, estimation, and inference , author=. arXiv e-prints , pages=

  125. [133]

    Regression Shrinkage and Selection via the Lasso , urldate =

    Robert Tibshirani , journal =. Regression Shrinkage and Selection via the Lasso , urldate =

  126. [134]

    Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , urldate =

    Chunrong Ai and Xiaohong Chen , journal =. Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions , urldate =

  127. [135]

    Estimation in Partially Linear Models and Numerical Comparisons

    Liang, Hua , journal =. Estimation in Partially Linear Models and Numerical Comparisons. , volume =

  128. [136]

    Engle and C

    Robert F. Engle and C. W. J. Granger and John Rice and Andrew Weiss , journal =. Semiparametric Estimates of the Relation between Weather and Electricity Sales , volume =

  129. [137]

    Ronald Gallant and Douglas W

    A. Ronald Gallant and Douglas W. Nychka , journal =. Semi-Nonparametric Maximum Likelihood Estimation , urldate =

  130. [138]

    Newey , journal =

    Whitney K. Newey , journal =. Nonparametric Instrumental Variables Estimation , urldate =

  131. [139]

    Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves , urldate =

    Richard Blundell and Xiaohong Chen and Dennis Kristensen , journal =. Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves , urldate =

  132. [140]

    Handbook of Monte Carlo Methods , author=

  133. [141]

    P. M. Robinson , journal =. Root-N-Consistent Semiparametric Regression , urldate =

  134. [142]

    Sur un nouveau développement en série des fonctions , volume=

    Hermite, Charles , year=. Sur un nouveau développement en série des fonctions , volume=. doi:10.1017/CBO9780511702761.022 , booktitle=

    doi

  135. [143]

    Causal Inference With General Treatment Regimes , volume =

    Kosuke Imai and David A van Dyk , journal =. Causal Inference With General Treatment Regimes , volume =

  136. [144]

    Blei , journal =

    Yixin Wang and David M. Blei , journal =. The Blessings of Multiple Causes , volume =

  137. [145]

    Damelin and H.S

    S.B. Damelin and H.S. Jung and K.H. Kwon , keywords =. Convergence of Hermite and Hermite–Fejér Interpolation of Higher Order for Freud Weights , journal =. 2001 , issn =. doi:https://doi.org/10.1006/jath.2001.3619 , url =

    2001 doi

  138. [146]

    VanderWeele and Mirjam J

    Tyler J. VanderWeele and Mirjam J. Knol , pages =. A Tutorial on Interaction , title =. Epidemiologic Methods , doi =. 2014 , lastchecked =

    2014

  139. [147]

    Angrist and Alan B

    Joshua D. Angrist and Alan B. Krueger , title =. Journal of the American Statistical Association , volume =. 1992 , publisher =

    1992

  140. [148]

    Kernel methods for pattern analysis , author =

  141. [149]

    Altonji and Thomas A

    Joseph G. Altonji and Thomas A. Dunn , journal =. The Effects of Family Characteristics on the Return to Education , urldate =

  142. [150]

    , journal =

    Chen, Xiaohong and Christensen, Timothy M. , journal =. Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression , volume =

  143. [151]

    International Conference on Machine Learning , year=

    Ridge Regression Learning Algorithm in Dual Variables , author=. International Conference on Machine Learning , year=

  144. [152]

    The tariff on animal and vegetable oils , author=

  145. [153]

    Mostly Harmless Econometrics: An Empiricist's Companion , isbn =

    Angrist, Joshua and Pischke, Jörn-Steffen , year =. Mostly Harmless Econometrics: An Empiricist's Companion , isbn =

  146. [154]

    Developing a protocol for observational comparative effectiveness research: A user's guide , year=

    Estimation and reporting of heterogeneity of treatment effects , author=. Developing a protocol for observational comparative effectiveness research: A user's guide , year=

  147. [155]

    Stuart Geman and Chii-Ruey Hwang , journal =

  148. [156]

    Convex Optimization , DOI=

    Boyd, Stephen and Vandenberghe, Lieven , year=. Convex Optimization , DOI=

  149. [157]

    arXiv preprint arXiv:1908.09237 , year=

    The Ridge Path Estimator for Linear Instrumental Variables , author=. arXiv preprint arXiv:1908.09237 , year=

    1908

  150. [158]

    COLT/EuroCOLT , year=

    A Generalized Representer Theorem , author=. COLT/EuroCOLT , year=

  151. [159]

    Discrete Multivariate Analysis: Theory and Practice , author=

  152. [160]

    Estimation with Many Instrumental Variables , urldate =

    Christian Hansen and Jerry Hausman and Whitney Newey , journal =. Estimation with Many Instrumental Variables , urldate =

  153. [161]

    Envelope Theorems for Arbitrary Choice Sets , volume =

    Milgrom, Paul and Segal, Ilya , journal =. Envelope Theorems for Arbitrary Choice Sets , volume =

  154. [162]

    International Conference on Learning Representations , year=

    Learning Causal Models from Conditional Moment Restrictions by Importance Weighting , author=. International Conference on Learning Representations , year=

  155. [163]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , month =

    D. Journal of the Royal Statistical Society Series B: Statistical Methodology , month =

  156. [164]

    Stock , journal =

    James H. Stock , journal =. Nonparametric Policy Analysis , volume =

  157. [165]

    Assessing the Causal Effect of Policies: An Example Using Stochastic Interventions , volume =

    D. Assessing the Causal Effect of Policies: An Example Using Stochastic Interventions , volume =

  158. [166]

    Population Intervention Causal Effects Based on Stochastic Interventions , urldate =

    Iván Díaz Muñoz and Mark van der Laan , journal =. Population Intervention Causal Effects Based on Stochastic Interventions , urldate =

  159. [167]

    A Calculus for Stochastic Interventions:Causal Effect Identification and Surrogate Experiments , volume =

    Correa, Juan and Bareinboim, Elias , journal =. A Calculus for Stochastic Interventions:Causal Effect Identification and Surrogate Experiments , volume =

  160. [168]

    Deep learning , volume =

    LeCun, Yann and Bengio, Yoshua and Hinton, Geoffrey , journal =. Deep learning , volume =

  161. [169]

    Principal stratification--a goal or a tool? , volume =

    Pearl, Judea , journal =. Principal stratification--a goal or a tool? , volume =

  162. [170]

    Principal stratification--uses and limitations

    Vanderweele, Tyler J , journal =. Principal stratification--uses and limitations. , volume =

  163. [171]

    Neurocomputing , year=

    Stochastic Intervention for Causal Inference via Reinforcement Learning , author=. Neurocomputing , year=

  164. [172]

    Provably Efficient Causal Reinforcement Learning with Confounded Observational Data , volume =

    Wang, Lingxiao and Yang, Zhuoran and Wang, Zhaoran , booktitle =. Provably Efficient Causal Reinforcement Learning with Confounded Observational Data , volume =

  165. [173]

    Stochastic intervention for causal inference via reinforcement learning , volume =

    Tri Dung Duong and Qian Li and Guandong Xu , journal =. Stochastic intervention for causal inference via reinforcement learning , volume =

  166. [174]

    Tutorial in biostatistics

    Wong, W K and Lachenbruch, P A , journal =. Tutorial in biostatistics. Designing studies for dose response. , volume =

  167. [175]

    Adaptive dose finding based on t-statistic for dose-response trials

    Ivanova, Anastasia and Bolognese, James A and Perevozskaya, Inna , journal =. Adaptive dose finding based on t-statistic for dose-response trials. , volume =

  168. [176]

    Statistics in medicine , volume=

    Statistical principles for clinical trials (ICH E9): an introductory note on an international guideline , author=. Statistics in medicine , volume=. 1999 , publisher=

    1999

  169. [177]

    Principal stratification in causal inference

    Frangakis, Constantine E and Rubin, Donald B , journal =. Principal stratification in causal inference. , volume =

  170. [178]

    Angrist and Guido W

    Joshua D. Angrist and Guido W. Imbens and Donald B. Rubin , journal =. Identification of Causal Effects Using Instrumental Variables , urldate =

  171. [179]

    Online Algorithms and Stochastic Approximations , booktitle =

    Bottou, L\'. Online Algorithms and Stochastic Approximations , booktitle =. 1998 , url =

    1998

  172. [180]

    The dose-response relationship in Phase I clinical trials and beyond: use, meaning, and assessment , volume =

    G. The dose-response relationship in Phase I clinical trials and beyond: use, meaning, and assessment , volume =. Pharmacology & Therapeutics , number =

  173. [181]

    Deep Generalized Method of Moments for Instrumental Variable Analysis , volume =

    Bennett, Andrew and Kallus, Nathan and Schnabel, Tobias , booktitle =. Deep Generalized Method of Moments for Instrumental Variable Analysis , volume =

  174. [182]

    A definition of causal effect for epidemiological research , volume =

    M A Hern. A definition of causal effect for epidemiological research , volume =. Journal of Epidemiology and Community Health , month =

  175. [183]

    First Conference on Causal Learning and Reasoning , year=

    Identifying Principal Stratum Causal Effects Conditional on a Post-treatment Intermediate Response , author=. First Conference on Causal Learning and Reasoning , year=

  176. [184]

    Advances in Neural Information Processing Systems , editor=

    Double Machine Learning Density Estimation for Local Treatment Effects with Instruments , author=. Advances in Neural Information Processing Systems , editor=. 2021 , url=

    2021

  177. [185]

    2022 , eprint=

    Instrumental Variables in Causal Inference and Machine Learning: A Survey , author=. 2022 , eprint=

    2022

  178. [186]

    Heckman and Richard Robb , journal =

    James J. Heckman and Richard Robb , journal =. Alternative methods for evaluating the impact of interventions: An overview , volume =

  179. [187]

    Telser , journal =

    Lester G. Telser , journal =. Iterative Estimation of a Set of Linear Regression Equations , urldate =

  180. [188]

    Wooldridge , journal =

    Jeffrey M. Wooldridge , journal =. Control Function Methods in Applied Econometrics , urldate =

  181. [189]

    Kennedy , journal =

    Edward H. Kennedy , journal =. Nonparametric Causal Effects Based on Incremental Propensity Score Interventions , volume =

  182. [190]

    Dual Instrumental Variable Regression , volume =

    Muandet, Krikamol and Mehrjou, Arash and Lee, Si Kai and Raj, Anant , booktitle =. Dual Instrumental Variable Regression , volume =

  183. [191]

    Efficient Contextual Bandits with Continuous Actions , volume =

    Majzoubi, Maryam and Zhang, Chicheng and Chari, Rajan and Krishnamurthy, Akshay and Langford, John and Slivkins, Aleksandrs , booktitle =. Efficient Contextual Bandits with Continuous Actions , volume =

  184. [192]

    Proceedings of the Thirty-Second Conference on Learning Theory , pages =

    Contextual bandits with continuous actions: Smoothing, zooming, and adapting , author =. Proceedings of the Thirty-Second Conference on Learning Theory , pages =. 2019 , editor =

    2019

  185. [193]

    Conference on Uncertainty in Artificial Intelligence , year=

    Identifying Dynamic Sequential Plans , author=. Conference on Uncertainty in Artificial Intelligence , year=

  186. [194]

    International Conference on Artificial Intelligence and Statistics , year=

    Policy Evaluation and Optimization with Continuous Treatments , author=. International Conference on Artificial Intelligence and Statistics , year=

  187. [195]

    Foundations of Machine Learning , urldate =

    Mehryar Mohri and Afshin Rostamizadeh and Ameet Talwalkar , publisher =. Foundations of Machine Learning , urldate =

  188. [196]

    Yau and Roderick J

    Linda H.Y. Yau and Roderick J. Little , journal =. Inference for the Complier-Average Causal Effect from Longitudinal Data Subject to Noncompliance and Missing Data, with Application to a Job Training Assessment for the Unemployed , urldate =

  189. [197]

    Guo, Lulu and Qian, Yi and Xie, Hui , journal =. Assessing complier average causal effects from longitudinal trials with multiple endpoints and treatment noncompliance: An application to a study of Arthritis Health Journal , volume =

  190. [198]

    Causal Inference With Interference and Noncompliance in Two-Stage Randomized Experiments , volume =

    Kosuke Imai and Zhichao Jiang and Anup Malani , journal =. Causal Inference With Interference and Noncompliance in Two-Stage Randomized Experiments , volume =

  191. [199]

    International Conference on Learning Representations , year=

    The role of over-parametrization in generalization of neural networks , author=. International Conference on Learning Representations , year=

  192. [200]

    Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation , volume =

    Wei, Colin and Ma, Tengyu , booktitle =. Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation , volume =

  193. [201]

    and Krueger, Alan B

    Angrist, Joshua D. and Krueger, Alan B. , Title =. Journal of Economic Perspectives , Volume =. 2001 , Month =. doi:10.1257/jep.15.4.69 , URL =

    2001 doi

  194. [202]

    Natural Experiments

    Mark R. Rosenzweig and Kenneth I. Wolpin , journal =. Natural "Natural Experiments" in Economics , urldate =

  195. [203]

    Natural Experiments and Quasi-Natural Experiments , author=

  196. [204]

    Proceedings of the 40th International Conference on Machine Learning , pages =

    Instrumental Variable Estimation of Average Partial Causal Effects , author =. Proceedings of the 40th International Conference on Machine Learning , pages =. 2023 , volume =

    2023

  197. [205]

    and Kennedy, Edward H

    Mauro, Jacqueline A. and Kennedy, Edward H. and Nagin, Daniel , journal =

  198. [206]

    , Title =

    Imbens, Guido W. , Title =. Journal of Economic Literature , Volume =. 2010 , Month =. doi:10.1257/jel.48.2.399 , URL =

    2010 doi

  199. [207]

    Profiling Compliers and Noncompliers for Instrumental-Variable Analysis , volume=

    Marbach, Moritz and Hangartner, Dominik , year=. Profiling Compliers and Noncompliers for Instrumental-Variable Analysis , volume=. Political Analysis , publisher=. doi:10.1017/pan.2019.48 , number=

    2019 doi

  200. [208]

    An IV Model of Quantile Treatment Effects , urldate =

    Victor Chernozhukov and Christian Hansen , journal =. An IV Model of Quantile Treatment Effects , urldate =

  201. [209]

    Aronow and Allison Carnegie , journal =

    Peter M. Aronow and Allison Carnegie , journal =. Beyond LATE: Estimation of the Average Treatment Effect with an Instrumental Variable , urldate =

  202. [210]

    Suppressed for Anonymity , author=

  203. [211]

    Financial and Economic Review , year=2022, volume=

    Zoltan Hermann and Hedvig Horvath and Attila Lindner , title=. Financial and Economic Review , year=2022, volume=. doi:, url=

    2022

  204. [212]

    Angrist and Kathryn Graddy and Guido W

    Joshua D. Angrist and Kathryn Graddy and Guido W. Imbens , journal =. The Interpretation of Instrumental Variables Estimators in Simultaneous Equations Models with an Application to the Demand for Fish , urldate =

  205. [213]

    Criteria for Surrogate End Points Based on Causal Distributions , urldate =

    Chuan Ju and Zhi Geng , journal =. Criteria for Surrogate End Points Based on Causal Distributions , urldate =

  206. [214]

    Surrogate measures and consistent surrogates

    Vanderweele, Tyler J , journal =. Surrogate measures and consistent surrogates. , volume =

  207. [215]

    , title =

    Prentice, Ross L. , title =. Statistics in Medicine , volume =. doi:https://doi.org/10.1002/sim.4780080407 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/sim.4780080407 , year =

    doi

  208. [216]

    Sharp bounds on causal effects using a surrogate endpoint

    Kuroki, Manabu , journal =. Sharp bounds on causal effects using a surrogate endpoint. , volume =

  209. [217]

    Lauritzen and Odd O

    Steffen L. Lauritzen and Odd O. Aalen and Donald B. Rubin and Elja Arjas , journal =. Discussion on Causality [with Reply] , urldate =

  210. [218]

    Stochastic Dominance and Expected Utility: Survey and Analysis , urldate =

    Haim Levy , journal =. Stochastic Dominance and Expected Utility: Survey and Analysis , urldate =

  211. [219]

    Barrett and Stephen G

    Garry F. Barrett and Stephen G. Donald , journal =. Consistent Tests for Stochastic Dominance , urldate =

  212. [220]

    The annals of mathematical statistics , pages=

    On a test of whether one of two random variables is stochastically larger than the other , author=. The annals of mathematical statistics , pages=. 1947 , publisher=

    1947

  213. [221]

    E. L. Lehmann , journal =. Ordered Families of Distributions , urldate =

  214. [222]

    Criteria for Surrogate End Points , urldate =

    Hua Chen and Zhi Geng and Jinzhu Jia , journal =. Criteria for Surrogate End Points , urldate =

  215. [223]

    Related causal frameworks for surrogate outcomes

    Joffe, Marshall M and Greene, Tom , journal =. Related causal frameworks for surrogate outcomes. , volume =

  216. [224]

    2014 , publisher=

    Gdpnow: A model for gdp'nowcasting' , author=. 2014 , publisher=

    2014

  217. [225]

    Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence , pages =

    Pearl, Judea , title =. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence , pages =. 2001 , isbn =

    2001

  218. [226]

    Ordered Hypotheses for Multiple Treatments: A Significance Test for Linear Ranks , urldate =

    Ellis Batten Page , journal =. Ordered Hypotheses for Multiple Treatments: A Significance Test for Linear Ranks , urldate =

  219. [227]

    D. J. Bartholomew , journal =. A Test Of Homogeneity for Ordered Alternatives , urldate =

  220. [228]

    Finite-time Analysis of the Multiarmed Bandit Problem , volume =

    Auer, Peter and Cesa-Bianchi, Nicol. Finite-time Analysis of the Multiarmed Bandit Problem , volume =. Machine Learning , number =

  221. [229]

    Mathematics of Operations Research , volume=

    The multi-armed bandit problem: decomposition and computation , author=. Mathematics of Operations Research , volume=. 1987 , publisher=

    1987

  222. [230]

    A tutorial on propensity score estimation for multiple treatments using generalized boosted models

    McCaffrey, Daniel F and Griffin, Beth Ann and Almirall, Daniel and Slaughter, Mary Ellen and Ramchand, Rajeev and Burgette, Lane F , journal =. A tutorial on propensity score estimation for multiple treatments using generalized boosted models. , volume =

  223. [231]

    Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze , volume=

    Teoria statistica delle classi e calcolo delle probabilita , author=. Pubblicazioni del R Istituto Superiore di Scienze Economiche e Commericiali di Firenze , volume=

  224. [232]

    A general approach to causal mediation analysis

    Imai, Kosuke and Keele, Luke and Tingley, Dustin , journal =. A general approach to causal mediation analysis. , volume =

  225. [233]

    Statistical Science , number =

    Kosuke Imai and Luke Keele and Teppei Yamamoto , title =. Statistical Science , number =. 2010 , doi =

    2010

  226. [234]

    Identifiability and exchangeability for direct and indirect effects

    Robins, J M and Greenland, S , journal =. Identifiability and exchangeability for direct and indirect effects. , volume =

  227. [235]

    Mediation analysis with principal stratification

    Gallop, Robert and Small, Dylan S and Lin, Julia Y and Elliott, Michael R and Joffe, Marshall and Ten Have, Thomas R , journal =. Mediation analysis with principal stratification. , volume =

  228. [236]

    Conference on Uncertainty in Artificial Intelligence , year=

    Non-Parametric Path Analysis in Structural Causal Models , author=. Conference on Uncertainty in Artificial Intelligence , year=

  229. [237]

    Path Coefficients and Path Regressions: Alternative or Complementary Concepts? , urldate =

    Sewall Wright , journal =. Path Coefficients and Path Regressions: Alternative or Complementary Concepts? , urldate =

  230. [238]

    Journal of Causal Inference , year=2013, volume=

    Pearl, Judea , title=. Journal of Causal Inference , year=2013, volume=. doi:10.1515/jci-2013-0003 , url=

    2013 doi

  231. [239]

    , author=

    The moderator--mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. , author=. Journal of personality and social psychology , volume=. 1986 , publisher=

    1986

  232. [240]

    The annals of mathematical statistics , volume=

    The method of path coefficients , author=. The annals of mathematical statistics , volume=. 1934 , publisher=

    1934

  233. [241]

    Proceedings of the 19th International Joint Conference on Artificial Intelligence , pages =

    Avin, Chen and Shpitser, Ilya and Pearl, Judea , title =. Proceedings of the 19th International Joint Conference on Artificial Intelligence , pages =. 2005 , publisher =

    2005

  234. [242]

    Peter Hall and Rodney C. L. Wolff and Qiwei Yao , journal =. Methods for Estimating a Conditional Distribution Function , urldate =

  235. [243]

    2004 , publisher=

    Nonparametric Estimation of Smooth Conditional Distributions , author=. 2004 , publisher=

    2004

  236. [244]

    Pathway Analysis, Causal Mediation, and the Identification of Causal Mechanisms

    R \" o th, Leonce. Pathway Analysis, Causal Mediation, and the Identification of Causal Mechanisms. Causality in Policy Studies: a Pluralist Toolbox. 2023. doi:10.1007/978-3-031-12982-7_6

    2023 doi

  237. [245]

    , journal =

    Hafeman, Danella M. , journal =

  238. [246]

    , author=

    Path Analysis-a primer. , author=. 1975 , publisher=

    1975

  239. [247]

    Racine , journal =

    Qi Li and Jeffrey S. Racine , journal =. Nonparametric Estimation of Conditional CDF and Quantile Functions with Mixed Categorical and Continuous Data , urldate =

  240. [248]

    Journal of Statistical Software , author=

    Nonparametric Econometrics: The np Package , volume=. Journal of Statistical Software , author=. 2008 , pages=. doi:10.18637/jss.v027.i05 , number=

    2008 doi

  241. [249]

    Assessing natural direct and indirect effects through multiple pathways

    Lange, Theis and Rasmussen, Mette and Thygesen, Lau Caspar , journal =. Assessing natural direct and indirect effects through multiple pathways. , volume =

  242. [250]

    Mediation Analysis with Multiple Mediators

    VanderWeele, T J and Vansteelandt, S , journal =. Mediation Analysis with Multiple Mediators. , volume =

  243. [251]

    Multiple mediation analysis with survival outcomes: With an application to explore racial disparity in breast cancer survival

    Yu, Qingzhao and Wu, Xiaocheng and Li, Bin and Scribner, Richard A , journal =. Multiple mediation analysis with survival outcomes: With an application to explore racial disparity in breast cancer survival. , volume =

  244. [252]

    Steen, Johan and Loeys, Tom and Moerkerke, Beatrijs and Vansteelandt, Stijn , journal =

  245. [253]

    De-confounding Causal Inference Using Latent Multiple-Mediator Pathways , volume =

    Yubai Yuan and Annie Qu , journal =. De-confounding Causal Inference Using Latent Multiple-Mediator Pathways , volume =

  246. [254]

    Causal mediation analysis with multiple mediators

    Daniel, R M and De Stavola, B L and Cousens, S N and Vansteelandt, S , journal =. Causal mediation analysis with multiple mediators. , volume =

  247. [255]

    Generalized causal mediation analysis

    Albert, Jeffrey M and Nelson, Suchitra , journal =. Generalized causal mediation analysis. , volume =

  248. [256]

    Biometrika , month =

    D. Biometrika , month =

  249. [257]

    Statistics and its Interface , volume=

    Conceptual issues concerning mediation, interventions and composition , author=. Statistics and its Interface , volume=. 2009 , publisher=

    2009

  250. [258]

    2022 , publisher=

    Introduction to algorithms , author=. 2022 , publisher=

    2022

  251. [259]

    Causal Diagrams for Empirical Research , urldate =

    Judea Pearl , journal =. Causal Diagrams for Empirical Research , urldate =

  252. [260]

    2018 , publisher=

    The book of why: the new science of cause and effect , author=. 2018 , publisher=

    2018

  253. [261]

    Therapist-guided online metacognitive intervention for excessive worry: a randomized controlled trial with mediation analysis

    Wahlund, Tove and Hesser, Hugo and Perrin, Sean and Johansson, Sanna and Huhn, Vilgot and S. Therapist-guided online metacognitive intervention for excessive worry: a randomized controlled trial with mediation analysis. , volume =. Cogn Behav Ther , month =

  254. [262]

    Chapter 18 - Vestibular roll tilt thresholds partially mediate age-related effects on balance , editor =

    Sinem Balta Beylergil and Faisal Karmali and Wei Wang and Maria Carolina. Chapter 18 - Vestibular roll tilt thresholds partially mediate age-related effects on balance , editor =. 2019 , booktitle =. doi:https://doi.org/10.1016/bs.pbr.2019.04.019 , url =

    2019 doi

  255. [263]

    The Role of Breastfeeding on Respiratory Outcomes Later in Childhood

    Di Filippo, Paola and Lizzi, Mauro and Raso, Massimiliano and Di Pillo, Sabrina and Chiarelli, Francesco and Attanasi, Marina , journal =. The Role of Breastfeeding on Respiratory Outcomes Later in Childhood. , volume =

  256. [264]

    Tchetgen Tchetgen and Ilya Shpitser , title =

    Eric J. Tchetgen Tchetgen and Ilya Shpitser , title =. The Annals of Statistics , number =. 2012 , doi =

    2012

  257. [265]

    Policy-relevant proportions for direct effects

    VanderWeele, Tyler J , journal =. Policy-relevant proportions for direct effects. , volume =

  258. [266]

    Holland , journal =

    Paul W. Holland , journal =. Statistics and Causal Inference , urldate =

  259. [267]

    1994 , publisher=

    Encyclopaedia of Mathematics (set) , author=. 1994 , publisher=

    1994

  260. [268]

    and Knafl, Kathleen A

    Knafl, George J. and Knafl, Kathleen A. and Grey, Margaret and Dixon, Jane and Deatrick, Janet A. and Gallo, Agatha M. , journal =. Incorporating nonlinearity into mediation analyses , volume =

  261. [269]

    Highly structured stochastic systems , pages=

    Semantics of causal DAG models and the identification of direct and indirect effects , author=. Highly structured stochastic systems , pages=. 2003 , publisher=

    2003

  262. [270]

    A three-way decomposition of a total effect into direct, indirect, and interactive effects

    VanderWeele, Tyler J , journal =. A three-way decomposition of a total effect into direct, indirect, and interactive effects. , volume =

  263. [271]

    A unification of mediation and interaction: a 4-way decomposition

    VanderWeele, Tyler J , journal =. A unification of mediation and interaction: a 4-way decomposition. , volume =

  264. [272]

    2015 , publisher=

    Explanation in causal inference: methods for mediation and interaction , author=. 2015 , publisher=

    2015

  265. [273]

    Wright, Sewall , journal =

  266. [274]

    and Vanderweele, T.J

    Ding, P. and Vanderweele, T.J. and Robins, J. M. , journal =

  267. [275]

    Journal of Educational and Behavioral Statistics , volume=

    Treatment effects on ordinal outcomes: Causal estimands and sharp bounds , author=. Journal of Educational and Behavioral Statistics , volume=. 2018 , publisher=

    2018

  268. [276]

    Annual review of statistics and its application , volume=

    Evaluation of causal effects and local structure learning of causal networks , author=. Annual review of statistics and its application , volume=. 2019 , publisher=

    2019

  269. [277]

    Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding

    Ding, Peng and Vanderweele, Tyler J , journal =. Sharp sensitivity bounds for mediation under unmeasured mediator-outcome confounding. , volume =

  270. [278]

    and Vansteelandt, Stijn , journal =

    VanderWeele, Tyler J. and Vansteelandt, Stijn , journal =

  271. [279]

    Causal Inference: What If , year =

    Hern\'. Causal Inference: What If , year =

  272. [280]

    Xia, Fan and Chan, Kwun Chuen Gary , journal =

  273. [281]

    Miles, C H and Shpitser, I and Kanki, P and Meloni, S and Tchetgen Tchetgen, E J , journal =

  274. [282]

    1995 , publisher=

    The merriam-webster dictionary , author=. 1995 , publisher=

    1995

  275. [283]

    2012 , publisher=

    Manifolds, tensor analysis, and applications , author=. 2012 , publisher=

    2012

  276. [284]

    Tracing Causal Paths from Experimental and Observational Data , volume =

    Zhou, Xiang and Yamamoto, Teppei , journal =. Tracing Causal Paths from Experimental and Observational Data , volume =

  277. [285]

    Counterfactual Graphical Models for Longitudinal Mediation Analysis With Unobserved Confounding , volume =

    Shpitser, Ilya , journal =. Counterfactual Graphical Models for Longitudinal Mediation Analysis With Unobserved Confounding , volume =

  278. [286]

    Shpitser, Ilya and Pearl, Judea , title =. J. Mach. Learn. Res. , month =. 2008 , issue_date =

    2008

  279. [287]

    The 22nd International Conference on Artificial Intelligence and Statistics , pages=

    A potential outcomes calculus for identifying conditional path-specific effects , author=. The 22nd International Conference on Artificial Intelligence and Statistics , pages=. 2019 , organization=

    2019

  280. [288]

    Annual review of public health , volume=

    Mediation analysis: a practitioner's guide , author=. Annual review of public health , volume=. 2016 , publisher=

    2016

  281. [289]

    Efron , journal =

    B. Efron , journal =

  282. [290]

    2007 , publisher=

    Nonparametric econometrics: theory and practice , author=. 2007 , publisher=

    2007

  283. [291]

    Plos one , volume=

    Four-way decomposition of effect of cigarette smoking and body mass index on serum lipid profiles , author=. Plos one , volume=. 2022 , publisher=

    2022

  284. [292]

    Journal of statistical theory and practice , volume=

    Assessing natural direct and indirect effects for a continuous exposure and a dichotomous outcome , author=. Journal of statistical theory and practice , volume=. 2016 , publisher=

    2016

  285. [293]

    2018 , publisher=

    Density estimation for statistics and data analysis , author=. 2018 , publisher=

    2018

  286. [294]

    arXiv preprint arXiv:2105.09254 , year=

    Multiply robust causal mediation analysis with continuous treatments , author=. arXiv preprint arXiv:2105.09254 , year=

  287. [295]

    arXiv preprint arXiv:2501.06969 , year=

    Doubly robust inference on causal derivative effects for continuous treatments , author=. arXiv preprint arXiv:2501.06969 , year=

  288. [296]

    Journal of Applied Econometrics , volume=

    Direct and indirect effects of continuous treatments based on generalized propensity score weighting , author=. Journal of Applied Econometrics , volume=. 2020 , publisher=

    2020

  289. [297]

    Econometric Reviews , volume=

    Nonparametric estimation of mediation effects with a general treatment , author=. Econometric Reviews , volume=. 2024 , publisher=

    2024

  290. [298]

    Biometrika , volume=

    Proximal mediation analysis , author=. Biometrika , volume=. 2023 , publisher=

    2023

  291. [299]

    arXiv preprint arXiv:2112.13898 , year=

    Causal mediation with instrumental variables , author=. arXiv preprint arXiv:2112.13898 , year=

  292. [300]

    arXiv preprint arXiv:2602.23877 , year=

    Difference-in-differences for mediation analysis using double machine learning , author=. arXiv preprint arXiv:2602.23877 , year=

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.