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The causal effects of modified treatment policies under network interference

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper shows that a new intervention class, the induced modified treatment policy, identifies causal effects of continuous exposures even when units interfere through a known network.

desk verdict The identification result for induced MTPs is real and the simulations are solid, but the efficiency theory is built on the wrong EIF template and needs substantial revision. read the letter →

arxiv 2412.02105 v3 pith:EF6FTEZ2 submitted 2024-12-03 stat.ME

Salvador V. Balkus , Scott W. Delaney , Nima S. Hejazi This is my paper · ORCID
classification stat.ME MSC 62D2062G0562G20
keywords modifiedtreatmentpoliciesnetworkinterferencesemiparametricefficiencytargetedmaximumlikelihoodcoareaformulacontinuousexposurescausalinferencezero-emissionvehicles
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

Modified treatment policies (MTPs) are a standard tool for estimating the causal effect of continuous exposures, but their identification assumes no interference: one unit's exposure does not affect another's outcome. This paper introduces the induced MTP, which composes the intervention rule d with the network summary function s, so that the intervention acts on the exposure summaries that actually drive outcomes under network interference. Under assumptions on positivity, no unmeasured confounding, piecewise smooth invertibility, and a summary coarea condition, the paper identifies the counterfactual mean of an induced MTP as a functional of observed data, and constructs semiparametric efficient one-step and targeted minimum loss estimators with cross-fitted machine learning. Simulations show that induced MTPs eliminate the identification bias that plagues classical estimators, and a California analysis of zero-emission vehicle uptake on NO2 finds larger and statistically stronger pollution reductions than analyses ignoring interference.

What carries the argument

The induced MTP is the composition $s \circ d$: first apply the investigator's modified treatment policy $d(a, l; \delta)$ to each unit's exposure, then apply the network summary function $s$ to obtain the counterfactual exposure summary $A^{s\circ d}_i = s_{F_i}(d(A, L; \delta), L)$. This composition converts the MTP question into a stochastic-intervention question on exposure summaries, for which semiparametric theory already exists. The coarea formula — the measure-theoretic change-of-variables identity for maps with non-square Jacobians — carries the identification step, and the efficient influence function (Equation 11) carries the estimation step, feeding a one-step bias-corrected estimator and a targeted maximum likelihood estimator that use cross-fitted super learning for the nuisance parameters $m$ and $r$.

What would settle it

Re-run the semi-synthetic commuting-network experiment with a deliberately misspecified summary function (for example, an unweighted sum instead of the commuter-weighted sum) and check whether the estimator's bias grows while the correctly specified version stays near zero; the gap isolates how much of the identification rests on the correct network mechanism.

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Extended reading notes

Core claim

The paper's central claim is that the causal effect of an induced MTP under network interference is identified by $\psi_n = \frac{1}{n}\sum_{i=1}^n E_P\big[ m(A^s_i, L^s_i) \cdot r(A^s_i, A^{s\circ d}_i, L^s_i) \cdot w(A, L, i) \big]$, where $m$ is the outcome regression on exposure and confounder summaries, $r$ is a density ratio between post- and pre-intervention exposure summaries, and $w$ is a deterministic Jacobian weight determined by the investigator's choice of $d$ and $s$. The identification argument applies the coarea formula, a change-of-variables for functions with non-square Jacobians, to justify re-expressing the counterfactual mean of the summary exposure in terms of observed data. Building on the semiparametric theory for network stochastic interventions, the paper derives the efficient influence function for this estimand and constructs one-step and TMLE estimators that are doubly robust and semiparametric efficient under the rate and regularity conditions of Ogburn et al. (2022).

Load-bearing premise

The load-bearing assumption is that the investigator knows the network $F$ and the summary function $s$ describing how interference propagates, and that $s$ is correctly specified; if the true mechanism differs, the induced MTP estimand will not correspond to the causal effect of the policy.

Editorial extensions

If this is right

  • Ignoring network interference biases classical MTP estimates; the induced MTP removes this identification bias in simulations across Erdős–Rényi, scale-free, and Watts–Strogatz networks.
  • The one-step and network-TMLE estimators achieve the semiparametric efficiency bound while permitting cross-fitted machine learning for nuisance estimation, and the proposed variance estimator attains near-nominal coverage.
  • In the California ZEV analysis, the induced MTP effect estimate is over 1.3 times larger than the non-network MTP estimate, and confidence intervals shrink, providing statistically significant evidence where classical methods do not.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because identification hinges on a correctly specified summary function, a natural extension is a sensitivity analysis that perturbs $F$ and $s$ and reports how the estimated effect changes; the paper does not provide such a tool.
  • The induced MTP construction should transfer to longitudinal exposures with time-varying networks, but would require sequential regression procedures to handle summary measures of exposures under time-varying confounding.
  • The density ratio $r$ is the practical bottleneck: for nodes with very high degree the ratio can explode, so replacing density-ratio estimation with balancing weights such as Riesz regression is a promising testable modification.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces a new class of intervention, the induced modified treatment policy (MTP), to define and identify causal effects of continuous exposures under network interference. The target estimand is the counterfactual mean of Y under an MTP d applied to the exposure vector, summarized through a network function s. Under assumptions A1-A4, the authors identify this estimand by a functional (Eq. 8) involving an outcome regression m, a density ratio r, and a deterministic Jacobian weight w obtained via the coarea formula. They then propose one-step and TMLE estimators of this functional, claim semiparametric efficiency by adapting the stochastic-intervention influence function of Ogburn et al. (2022), and develop a network-adjusted variance estimator. The methodology is evaluated in simulations on synthetic and semi-synthetic networks and applied to estimate the effect of zero-emission vehicle uptake on NO2 in California.

Significance. If the identification result and estimators are correct, this is a useful contribution: it extends the MTP framework, which is popular for continuous exposures, to settings with network interference, and does so with a tractable estimand that avoids Monte Carlo nuisance estimation. The identification via the coarea formula is novel, and the simulations provide credible evidence that accounting for interference reduces bias. The authors also ship reproducible Julia code for the simulations and data analysis, which is a strength. However, the advertised semiparametric efficiency result is not supported by the provided derivation: the displayed influence function in Eq. (11) is the stochastic-intervention influence function with a data-dependent replacement density, not the efficient influence function of the induced MTP parameter. This affects the central efficiency claim and the variance estimator, and the paper's contribution is therefore contingent on repairing this theory.

major comments (4)
  1. [Section 3.3, Eq. (11)] The displayed influence function is not the efficient influence function for the induced MTP parameter. In the no-interference limit (s the identity, w=1), Eq. (11) reduces to r(A,L)(Y-m(A,L)) + E[m(d(A,L),L)|L] - psi, whereas the classical MTP EIF (Haneuse & Rotnitzky, 2013; Diaz & van der Laan, 2012) is r(A,L)(Y-m(A,L)) + m(d(A,L),L) - psi. The difference U = m(d(A,L),L) - E[m(d(A,L),L)|L] is a mean-zero function of A given L and lies in the treatment-mechanism tangent space; an influence function must satisfy E[phi S_a] = dpsi/depsilon for treatment scores S_a, and with the conditional-expectation plug-in term E[(E[m|L]-psi)S_a] = 0, so the pathwise derivative condition fails. The derivation in Supplement S3, which plugs the MTP replacement density into the stochastic-intervention EIF of Ogburn et al. (2022), therefore does not yield an EIF for the induced MTP, and the claimed semiparametric efficiency of the one-step estimator (Eq. 14) and TMLE is not established.
  2. [Section 3.3, Eq. (10) and Supplement S3] The representation of an induced MTP as a stochastic intervention is not valid for the purpose of applying the stochastic-intervention EIF. The replacement density \bar p^*(as_i|ls_i) defined in Eq. (10) depends on the observed exposure through as_i^{d-1} = s_{F_i}(d^{-1}(A,L),L), so the intervention is not a fixed stochastic intervention with a replacement distribution independent of the observed data. The stochastic-intervention EIF of Ogburn et al. (2022) requires a fixed replacement distribution; otherwise the pathwise derivative must include the additional contribution from the dependence of the intervention value on the observed exposure. The change-of-variables step in S3 (Eqs. S6-S8) therefore does not justify Eq. (11).
  3. [Section 3.5 and Supplement S5 (Lemma S3, Theorem S3)] The variance estimator proof is invalid because it conflates the estimating function used in Eq. (12), which contains the conditional expectation E[m(As^d,Ls)|L], with the actual influence function of the one-step estimator, which uses m(As^d,Ls) as in Eq. (14). Lemma S3 states Var(phi(O_i) - hat psi(|F_i|)) = Var(hat psi_OS); this equality requires phi to be the influence function of hat psi_OS, but the difference between the two plug-in terms is a non-negligible mean-zero function of A given L that contributes to the variance of the estimator. Consequently, the consistency of hat sigma^2 for the variance of hat psi_OS and hat psi_TMLE is not established.
  4. [Supplement S5, Lemma S2] The proof of Lemma S2 asserts an unstated assumption, namely |N(|F_i|)| proportional to |N(|F_j|)| for all i,j, claiming it follows from the positivity assumption; this proportionality is not implied by Assumption A1 and is not stated in the main text. The subsequent consistency and variance proofs rely on this degree-strata proportionality, which may fail for the types of network structures (e.g., scale-free) considered in the simulations. Either the assumption should be stated in the main text and its plausibility discussed, or the variance estimator should be shown consistent without it.
minor comments (4)
  1. [Section 3.5, Eq. (16)] The text does not specify how hat psi(|F_i|) is computed in practice (for example, whether cross-fitting is used for the within-stratum one-step estimators); please clarify the estimation procedure and its data-splitting scheme.
  2. [Supplement S6, Theorem S4] The bound on the covariance sum is given as o(K_max^2/n) = o(1/C_n), but K_max^2/n is not necessarily o(1/C_n) since C_n <= n/K_max^2; the proof needs an additional argument (for example, uniform boundedness and decay of covariances) to establish the required rate.
  3. [Section 3.2, Eq. (8)] The notation r(As_i, As_i^{d-1}, Ls_i) in the main text is inconsistent with the definition r(as, as^{d-1}, ls) in the identifiability display; please make the argument order and superscripts consistent.
  4. [Section 5.1] The statement that the induced MTP effect strengthens prior evidence is based on a single observational dataset with a known network summary and assumed summary function; please soften the causal interpretation or add a sensitivity analysis for the choice of network and summary function.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: identification and efficiency claims are derived from stated assumptions and external prior results, not from fitted inputs or self-citation chains.

full rationale

The central derivation chain is self-contained relative to its assumptions. The induced MTP estimand (Eq. 5) is defined as E[(1/n) sum_i Y(sFi(d(A,L),L))]; Eq. 8 is obtained by iterated expectation under Assumptions A1-A4, the coarea formula (Lemma S1, Theorem S1), and change of variables, with w defined to make the transformation valid. This is a derivation, not a definitional identity: w and r are not fitted to the target; m and r are nuisance parameters estimated from data, and the efficiency claim rests on the external CLT of Ogburn et al. (2022) and EIFs of van der Laan (2014) and Sofrygin & van der Laan (2017), none authored by the present authors. The only self-citations are to the authors' software packages (CausalTables.jl, ModifiedTreatment.jl), which are implementation tools, not load-bearing evidence for the statistical claims. The simulations and data analysis evaluate the proposed estimators against known or externally observed quantities; no fitted parameter is relabeled as a prediction. The skeptic's concern about the form of Eq. 11 is a potential correctness issue in the EIF derivation (the replacement density depends on observed A, which may require an additional term), not a circularity: the displayed EIF is not equal to the target by construction, and the one-step/TMLE estimators use m(As^d,Ls) directly. Per the scoring rules, such correctness risks do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 1 invented entities

The central claim rests on the SCM, the four identification assumptions A1-A4, and the CLT of Ogburn et al. The most fragile element is the hidden degree-strata proportionality assumption used for variance estimation; it is not listed among the main assumptions and may fail in skewed networks.

assumptions (7)
  • domain assumption SCM in Equation (4): L_i = f_L(eps_Li); A_i = f_A(Ls_i, eps_Ai); Y_i = f_Y(As_i, Ls_i, eps_Yi) with errors independent except within distance-2 neighborhoods.
    Defines the data-generating process with network interference; used throughout Section 3.
  • domain assumption A1 Summary positivity: if (sFi(a,l), sFi(l)) in support, then (sFi(ad,l), sFi(l)) in support.
    Required for the density ratio to be well-defined in the identification formula (Section 3.2).
  • domain assumption A2 No unmeasured confounding: Y(sFi(A,L)) independent of sFi(A,L) given L.
    Standard causal assumption adapted to summary exposures; used in the identification proof (Section S2).
  • domain assumption A3 Piecewise smooth invertibility of the MTP d(a,l;delta).
    Standard MTP condition ensuring existence of the efficient influence function (Haneuse and Rotnitzky 2013).
  • ad hoc to paper A4 Summary coarea: sqrt(det(Ja sFi(a,l) Ja sFi(a,l)^T)) > 0 almost everywhere.
    Introduced to extend the change-of-variables argument to non-square summary functions; necessary for identification per Theorem S1.
  • ad hoc to paper Proportional degree-strata sizes: |N(|Fi|)| proportional to |N(|Fj|)| for all i,j.
    Hidden assumption in the variance estimator consistency proof (Lemma S2); not stated among the main assumptions.
  • standard math CLT and bounded estimating function conditions from Ogburn et al. (2022, Theorem 1).
    Borrowed to establish asymptotic normality of the proposed estimators.
invented entities (1)
  • Induced modified treatment policy (MTP) independent evidence
    purpose: Defines a counterfactual intervention on a continuous exposure under network interference by composing an MTP with a network summary function.
    It is a mathematical construct whose consequences are tested in simulations and the California application; its causal interpretation depends on the investigator's choice of summary and network, so it has no external physical evidence but is falsifiable through applications.

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Cite this review

Pith. "Pith review of The causal effects of modified treatment policies under network interference." pith.science (2026). https://pith.science/paper/EF6FTEZ2

@misc{pith2026241202105,
  author       = {Pith},
  title        = {Pith review of: The causal effects of modified treatment policies under network interference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EF6FTEZ2}},
  note         = {Machine review of arXiv:2412.02105}
}
read the original abstract

Modified treatment policies are a widely applicable class of interventions useful for studying the causal effects of continuous exposures. Approaches to evaluating their causal effects assume no interference, meaning that such effects cannot be learned from data in settings where the exposure of one unit affects the outcomes of others, as is common in spatial or network data. We introduce a new class of intervention, induced modified treatment policies, which we show identify such causal effects in the presence of network interference. Building on recent developments for causal inference in networks, we provide flexible, semi-parametric efficient estimators of the statistical estimand. Numerical experiments demonstrate that an induced modified treatment policy can eliminate the causal, or identification, bias that results from network interference. We use the methodology developed to evaluate the effect of zero-emission vehicle uptake on air pollution in California, strengthening prior evidence.

Figures

Figures reproduced from arXiv: 2412.02105 by the authors.

Figure 1
Figure 1. How an induced MTP arises as the composition of MTP and summary functions [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Asymptotic performance of network-TMLE in simulation on multiple network structures. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Estimated effect sizes measuring the expected difference in NO [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Estimated effect sizes measuring the expected difference in NO [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

85 extracted references · 71 canonical work pages · cited by 2 Pith papers

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Aronow, P. M. & Samii, C. (2017). Estimating average causal effects under general interference, with application to a social network experiment. The Annals of Applied Statistics 11, 1912--1947

    2017

  4. [4]

    , Eckles, D

    Athey, S. , Eckles, D. & Imbens, G. W. (2017). Exact p-values for network interference. Journal of the American Statistical Association 113, 230–240

    2017

  5. [5]

    & Hejazi, N

    Balkus, S. & Hejazi, N. (2024). ModifiedTreatment.jl . https://github.com/salbalkus/ModifiedTreatment.jl

    2024

  6. [6]

    Balkus, S. V. & Hejazi, N. S. (2025). Causaltables.jl: Simulating and storing data for statistical causal inference in julia. Journal of Open Source Software 10, 7580

    work page 2025

  7. [7]

    , Edelman, A

    Bezanson, J. , Edelman, A. , Karpinski, S. & Shah, V. B. (2017). Julia: A fresh approach to numerical computing. SIAM Review 59, 65–98

    work page 2017

  8. [8]

    Bickel, P. J. , Klaassen, C. A. , Bickel, P. J. , Ritov, Y. , Klaassen, J. , Wellner, J. A. & Ritov, Y. (1993). Efficient and adaptive estimation for semiparametric models, vol. 4. Springer

    work page 1993

Show all 85 references
  1. [9]

    , Fogarty, C

    Bong, H. , Fogarty, C. B. , Levina, L. & Zhu, J. (2024). Unraveling heterogeneous treatment effects in networks: A non-parametric approach based on node connectivity. arXiv:2410.11797

    2024 arXiv

  2. [10]

    Light-duty vehicle population in California

    California Energy Commission (2024). Light-duty vehicle population in California

    2024

  3. [11]

    & Chu, C

    Cheng, K. & Chu, C. (2004). Semiparametric density estimation under a two-sample density ratio model. Bernoulli 10

    2004

  4. [12]

    , Chetverikov, D

    Chernozhukov, V. , Chetverikov, D. , Demirer, M. , Duflo, E. , Hansen, C. , Newey, W. & Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Journal 21, C1–C68

    2018

  5. [13]

    , Newey, W

    Chernozhukov, V. , Newey, W. K. , Quintas-Martinez, V. & Syrgkanis, V. (2021). Automatic debiased machine learning via Riesz regression. arXiv preprint arXiv:2104.14737

    2021 arXiv

  6. [14]

    Clark, D. A. & Handcock, M. S. (2024). Causal inference over stochastic networks. Journal of the Royal Statistical Society Series A: Statistics in Society 187, 772–795

    2024

  7. [15]

    (2022 a )

    Cooper, M. (2022 a ). Satellite-derived ground level NO2 concentrations, 2005-2019

    2022

  8. [16]

    (2022 b )

    Cooper, M. (2022 b ). Tropomi-derived ground level NO2 concentrations (2019 annual mean)

    2022

  9. [17]

    Cox, D. R. (1958). Planning of experiments

    1958

  10. [18]

    Davies, M. M. & van der Laan, M. J. (2016). Optimal spatial prediction using ensemble machine learning. The International Journal of Biostatistics 12, 179--201

    2016

  11. [19]

    , Anenberg, S

    de Souza, P. , Anenberg, S. , Makarewicz, C. , Shirgaokar, M. , Duarte, F. , Ratti, C. , Durant, J. L. , Kinney, P. L. & Niemeier, D. (2023). Quantifying disparities in air pollution exposures across the United States using home and work addresses. Environmental Science & Tech...

    2023

  12. [20]

    & Hejazi, N

    D \' az, I. & Hejazi, N. S. (2020). Causal mediation analysis for stochastic interventions. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 82, 661--683

    2020

  13. [21]

    , Hejazi, N

    D \' az, I. , Hejazi, N. S. , Rudolph, K. E. & van der Laan , M. J. (2021). Non-parametric efficient causal mediation with intermediate confounders. Biometrika 108, 627--641

    2021

  14. [22]

    , Hoffman, K

    D \' az, I. , Hoffman, K. L. & Hejazi, N. S. (2024). Causal survival analysis under competing risks using longitudinal modified treatment policies. Lifetime Data Analysis 30, 213--236

    2024

  15. [23]

    & van der Laan , M

    D \'i az, I. & van der Laan , M. (2012). Population intervention causal effects based on stochastic interventions. Biometrics 68, 541--549

    2012

  16. [24]

    & van der Laan , M

    D\' i az, I. & van der Laan , M. J. (2018). Stochastic treatment regimes. In Targeted Learning in Data Science: Causal Inference for Complex Longitudinal Studies. Springer Science & Business Media, pp. 167--180

    2018

  17. [25]

    , Williams, N

    D \'i az, I. , Williams, N. , Hoffman, K. L. & Schenck, E. J. (2023). Nonparametric causal effects based on longitudinal modified treatment policies. Journal of the American Statistical Association 118, 846--857

    2023

  18. [26]

    & Wartenberg, D

    Elliott, P. & Wartenberg, D. (2004). Spatial epidemiology: Current approaches and future challenges. Environmental Health Perspectives 112, 998--1006

    2004

  19. [27]

    , Spohn, M.-L

    Emmenegger, C. , Spohn, M.-L. , Elmer, T. & B \"u hlmann, P. (2023). Treatment effect estimation with observational network data using machine learning. arXiv:2206.14591

    2023 arXiv

  20. [28]

    , Rapp, R

    Faustini, A. , Rapp, R. & Forastiere, F. (2014). Nitrogen dioxide and mortality: review and meta-analysis of long-term studies. European Respiratory Journal 44, 744–753

    2014

  21. [29]

    & Papies, D

    Fuhr, J. & Papies, D. (2024). Double machine learning meets panel data -- promises, pitfalls, and potential solutions. arXiv:2409.01266

    2024 arXiv

  22. [30]

    , Johnston, J

    Garcia, E. , Johnston, J. , McConnell, R. , Palinkas, L. & Eckel, S. P. (2023). California's early transition to electric vehicles: Observed health and air quality co-benefits. Science of The Total Environment , 161761

    2023

  23. [31]

    , Hoffman, K

    Gilbert, B. , Hoffman, K. L. , Williams, N. , Rudolph, K. E. , Schenck, E. J. & D \' az, I. (2024). Identification and estimation of mediational effects of longitudinal modified treatment policies. arXiv:2403.09928

    2024

  24. [32]

    , Pierse, N

    Gillespie-Bennett, J. , Pierse, N. , Wickens, K. , Crane, J. & Howden-Chapman, P. (2010). The respiratory health effects of nitrogen dioxide in children with asthma. European Respiratory Journal 38, 303–309

    2010

  25. [33]

    & van der Laan, M

    Gruber, S. & van der Laan, M. J. (2010). A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome. The International Journal of Biostatistics 6

    2010

  26. [34]

    Halloran, M. E. & Hudgens, M. G. (2016). Dependent happenings: A recent methodological review. Current Epidemiology Reports 3, 297--305

    2016

  27. [35]

    & Rotnitzky, A

    Haneuse, S. & Rotnitzky, A. (2013). Estimation of the effect of interventions that modify the received treatment. Statistics in Medicine 32, 5260--5277

    2013

  28. [36]

    Hejazi, N. S. , Rudolph, K. E. , van der Laan , M. J. & D \' az, I. (2023). Nonparametric causal mediation analysis for stochastic interventional (in)direct effects. Biostatistics 24, 686--707

    2023

  29. [37]

    Hesterberg, T. W. , Bunn, W. B. , McClellan, R. O. , Hamade, A. K. , Long, C. M. & Valberg, P. A. (2009). Critical review of the human data on short-term nitrogen dioxide ( NO2 ) exposures: Evidence for NO2 no-effect levels. Critical Reviews in Toxicology 39, 743–781

    2009

  30. [38]

    Hoffman, K. L. , Salazar-Barreto, D. , Williams, N. T. , Rudolph, K. E. & D \' az, I. (2024). Studying continuous, time-varying, and/or complex exposures using longitudinal modified treatment policies. Epidemiology 35, 667--675

    2024

  31. [39]

    & Yanagi, T

    Hoshino, T. & Yanagi, T. (2023). Causal inference with noncompliance and unknown interference. Journal of the American Statistical Association , 1--12

    2023

  32. [40]

    Hubbard, A. E. , Kherad-Pajouh, S. & van der Laan, M. J. (2016). Statistical inference for data adaptive target parameters. The International Journal of Biostatistics 12, 3–19

    2016

  33. [41]

    Hudgens, M. G. & Halloran, M. E. (2008). Toward causal inference with interference. Journal of the American Statistical Association 103, 832--842

    2008

  34. [42]

    Kennedy, E. H. (2022). Semiparametric doubly robust targeted double machine learning: A review. arXiv:2203.06469

    2022 arXiv

  35. [43]

    Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally asymptotically linear estimators. The Annals of Statistics , 1548--1562

    1987

  36. [44]

    Koshevnik, Y. A. & Levit, B. Y. (1977). On a non-parametric analogue of the information matrix. Theory of Probability & Its Applications 21, 738--753

    1977

  37. [45]

    , Zhang, D

    Liu, J. , Zhang, D. & Tchetgen Tchetgen , E. J. (2025). Auto-doubly robust estimation of causal effects on a network. arXiv

    2025

  38. [46]

    Morrison, C. N. , Mair, C. F. , Bates, L. , Duncan, D. T. , Branas, C. C. , Bushover, B. R. , Mehranbod, C. A. , Gobaud, A. N. , Uong, S. , Forrest, S. , Roberts, L. & Rundle, A. G. (2024). Defining spatial epidemiology: A systematic review and re-orientation. Epidemiology

    2024

  39. [47]

    Negro, L. (2022). Sample distribution theory using coarea formula. Communications in Statistics - Theory and Methods 53, 1864–1889

    2022

  40. [48]

    Ogburn, E. L. , Sofrygin, O. , D \'i az, I. & van der Laan , M. J. (2022). Causal inference for social network data. Journal of the American Statistical Association , 1--15

    2022

  41. [49]

    , Karmakar, B

    Ohnishi, Y. , Karmakar, B. & Sabbaghi, A. (2022). Degree of interference: A general framework for causal inference under interference. arXiv:2210.17516

    2022 arXiv

  42. [50]

    Pearl, J. (2000). Causality: Models, reasoning and inference. Cambridge University Press 19, 3

    2000

  43. [51]

    Pearl, J. (2010). On the consistency rule in causal inference: Axiom, definition, assumption, or theorem? Epidemiology 21, 872–875

    2010

  44. [52]

    Pebesma, E. (2018). Simple Features for R: Standardized Support for Spatial Vector Data . The R Journal 10, 439--446

    2018

  45. [53]

    & Wefelmeyer, W

    Pfanzagl, J. & Wefelmeyer, W. (1985). Contributions to a general asymptotic statistical theory. Statistics & Risk Modeling 3, 379--388

    1985

  46. [54]

    Phillips, R. V. , van der Laan , M. J. , Lee, H. & Gruber, S. (2023). Practical considerations for specifying a super learner. International Journal of Epidemiology 52, 1276--1285

    2023

  47. [55]

    areal: An R package for areal weighted interpolation

    Prener , Christopher , Revord & Charles (2019). areal: An R package for areal weighted interpolation . Journal of Open Source Software 4

    2019

  48. [56]

    Qin, J. (1998). Inferences for case-control and semiparametric two-sample density ratio models. Biometrika 85, 619--630

    1998

  49. [57]

    R: A Language and Environment for Statistical Computing

    R Core Team (2025). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria

    2025

  50. [58]

    Reich, B. J. , Yang, S. , Guan, Y. , Giffin, A. B. , Miller, M. J. & Rappold, A. (2021). A review of spatial causal inference methods for environmental and epidemiological applications. International Statistical Review 89, 605--634

    2021

  51. [59]

    Robins, J. M. , Hern \'a n, M. & Siebert, U. (2004). Effects of multiple interventions. In Comparative Quantification of Health Risks: Global and Regional Burden of Disease Attributable to Selected Major Risk Factors, M. Ezzati, A. D. Lopez, A. Rodgers & M. C. J. L, eds. World...

    2004

  52. [60]

    Rubin, D. B. (1980). Randomization analysis of experimental data: The fisher randomization test comment. Journal of the American Statistical Association 75, 591--593

    1980

  53. [61]

    S\" a vje, F. (2023). Causal inference with misspecified exposure mappings: separating definitions and assumptions. Biometrika 111, 1–15

    2023

  54. [62]

    , Braun, D

    Shin, H. , Braun, D. , Irene, K. , Audirac, M. & Antonelli, J. (2023). A spatial interference approach to account for mobility in air pollution studies with multivariate continuous treatments. arXiv:2305.14194

    2023 arXiv

  55. [63]

    & van der Laan , M

    Sofrygin, O. & van der Laan , M. J. (2017). Semi-parametric estimation and inference for the mean outcome of the single time-point intervention in a causally connected population. Journal of Causal Inference 5, 20160003

    2017

  56. [64]

    , Suzuki, T

    Sugiyama, M. , Suzuki, T. & Kanamori, T. (2012). Density Ratio Estimation in Machine Learning. Cambridge University Press

    2012

  57. [65]

    Tchetgen Tchetgen , E. J. , Fulcher, I. R. & Shpitser, I. (2021). Auto-G-Computation of causal effects on a network. Journal of the American Statistical Association 116, 833--844

    2021

  58. [66]

    Tchetgen Tchetgen , E. J. & VanderWeele, T. J. (2012). On causal inference in the presence of interference. Statistical Methods in Medical Research 21, 55--75

    2012

  59. [67]

    , Josey, K

    Tec, M. , Josey, K. , Mudele, O. & Dominici, F. (2024). Causal estimation of exposure shifts with neural networks and an application to inform air quality standards in the us. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, KDD '24. New...

    2024

  60. [68]

    National ambient air quality standards ( NAAQS ) for nitrogen dioxide

    United States EPA (2018). National ambient air quality standards ( NAAQS ) for nitrogen dioxide

    2018

  61. [69]

    Census Bureau (2024)

    U.S. Census Bureau (2024). LEHD origin-destination employment statistics data (2002-2021). https://lehd.ces.census.gov/data/#lodes. Accessed: 2024-06-11

    2024

  62. [70]

    Environmental Protection Agency (2024)

    U.S. Environmental Protection Agency (2024). Smart location mapping. https://www.epa.gov/smartgrowth/smart-location-mapping. Accessed: 2024-06-10

    2024

  63. [71]

    van der Laan , M. J. (2014). Causal inference for a population of causally connected units. Journal of Causal Inference 2, 13--74

    2014

  64. [72]

    van der Laan, M. J. , Dudoit, S. & Keles, S. (2004). Asymptotic optimality of likelihood-based cross-validation. Statistical Applications in Genetics and Molecular Biology 3, 1–23

    2004

  65. [73]

    van der Laan, M. J. , Polley, E. C. & Hubbard, A. E. (2007). Super learner. Statistical Applications in Genetics and Molecular Biology 6

    2007

  66. [74]

    van der Laan , M. J. & Robins, J. M. (2003). Unified Methods for Censored Longitudinal Data and Causality. Springer

    2003

  67. [75]

    van der Laan , M. J. & Rose, S. (2011). Targeted Learning: Causal Inference for Observational and Experimental Data, vol. 4. Springer

    2011

  68. [76]

    van der Laan , M. J. & Rubin, D. (2006). Targeted maximum likelihood learning. The International Journal of Biostatistics 2

    2006

  69. [77]

    van der Vaart, A. W. , Dudoit, S. & van der Laan, M. J. (2006). Oracle inequalities for multi-fold cross validation. Statistics & Decisions 24, 351--371

    2006

  70. [78]

    & Herman, M

    Walker, K. & Herman, M. (2024). tidycensus: Load US Census Boundary and Attribute Data as 'tidyverse' and 'sf'-Ready Data Frames. R package version 1.6.3

    2024

  71. [79]

    Young, J. G. , Hern \'a n, M. A. & Robins, J. M. (2014). Identification, estimation and approximation of risk under interventions that depend on the natural value of treatment using observational data. Epidemiologic Methods 3, 1--19

    2014

  72. [80]

    & van der Laan , M

    Zheng, W. & van der Laan , M. J. (2010). Asymptotic theory for cross-validated targeted maximum likelihood estimation. bepress

    2010

  73. [81]

    , Liu, V

    Zigler, C. , Liu, V. , Mealli, F. & Forastiere, L. (2023). Bipartite interference and air pollution transport: Estimating health effects of power plant interventions. arXiv:2012.04831

    2023 arXiv

  74. [82]

    Zivich, P. N. , Hudgens, M. G. , Brookhart, M. A. , Moody, J. , Weber, D. J. & Aiello, A. E. (2022). Targeted maximum likelihood estimation of causal effects with interference: A simulation study. Statistics in Medicine 41, 4554--4577

    2022

  75. [83]

    Billingsley, P. (2012). Probability and Measure. Wiley

    2012

  76. [84]

    Federer, H. (1959). Curvature measures. Transactions of the American Mathematical Society 93, 418–491

    1959

  77. [85]

    Federer, H. (1969). Geometric measure theory. Springer

    1969

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.