Pith. sign in

REVIEW 4 major objections 3 minor 41 references

Causal Inference under Interference: Regression Adjustment and Optimality

T0 review · 4 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Under network interference, the regression-adjusted estimator is asymptotically normal, variance-optimal among linear adjustments, and matched by a √n-rate nonparametric counterpart.

desk verdict Solid, novel extension of regression adjustment to network interference, but the central CLT leans on an unverified extension of Li–Wager that needs to be checked before the results are load-bearing. read the letter →

arxiv 2502.06008 v2 pith:7YUWMIL7 submitted 2025-02-09 stat.ME

Xinyuan Fan , Chenlei Leng , Weichi Wu This is my paper · ORCID
classification stat.ME MSC 62F1262G2005C80
keywords averagetreatmenteffectnetworkinterferenceregressionadjustmentgraphonrandomgraphasymptoticoptimalityvarianceestimationnonparametrickernelsmoothingrandomizedcontrolledtrials
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

Randomized trials can still use regression adjustment when units interfere through a network: the paper proves that the classic trick of regressing outcomes on covariates within treatment and control groups, then comparing adjusted means, keeps its variance-reduction guarantee. Working with a graphon random-graph model (a latent-variable model for network edges) and assuming anonymous interference, meaning a unit's outcome depends only on its own treatment and the fraction of treated neighbors, the authors show the linear regression-adjusted estimator is $\sqrt{n}$-asymptotically normal and achieves the smallest asymptotic variance among all linear adjustments. They also derive an explicit variance formula, construct a consistent plug-in estimator for it, and give a kernel-and-trimming estimator that keeps the $\sqrt{n}$ rate for fixed covariate dimension while being optimal among nonlinear adjustments. The upshot for practice is that experimenters with networked data can report valid confidence intervals computed from the observed adjacency matrix alone, gaining the same efficiency that regression adjustment provides in classical trials.

What carries the argument

The carrying device is the graphon random-graph model of Definition 1, in which undirected edges are drawn independently as $\mathrm{Bernoulli}(\rho_n h(U_i,U_j))$ with $\sqrt{n}\rho_n \to \infty$, combined with the anonymous-interference assumption $Y_i = f_i(W_i, M_i/N_i)$, which makes the fraction of treated neighbors $M_i/N_i$ concentrate at $\pi$. That concentration lets a Taylor expansion of $f_i$ in its second argument produce the variance decomposition of Theorem 2, whose irreducible network component is governed by the constant $b = E[\,E(h(U_i,U_j)/E(h(U_i,U_j)\mid U_j)\mid U_i)^2\,]$. Optimality is carried by two comparison identities: the linear one $V(\alpha_1,\alpha_0) - V_{\mathrm{reg}} = \frac{1}{\pi(1-\pi)}u^\top\mathrm{cov}(z_1)u$, and the nonlinear one in Theorem 5, whose variance is minimized by the conditional-expectation choice $g_k = E(f_i(k,\pi)\mid z_i)$. For the nonparametric result the proof uses a U-statistics projection that deliberately excludes the network randomness and then controls the dependent remainder through the graphon structure, while variance estimation plugs an SVD-based low-rank approximation $\hat b = \frac1n\sum_i(\sum_j E_{ij}/\sum_k E_{jk})^2$ of the network constant into the formula.

What would settle it

Simulate the paper's design but break Assumption 1: let outcomes depend on a neighbor-indexed weighted fraction $\sum_j E_{ij} w_j W_j / \sum_j E_{ij} w_j$ with non-constant i.i.d. weights $w_j$, so the identity of treated neighbors matters; if $\sqrt{n}(\hat\tau - \tau)$ is no longer centered at zero with variance (4), for instance if coverage of the Theorem 4 interval drops far below nominal, the central claim would be shown to be confined to anonymous interference.

Watch

Extended reading notes

Core claim

Under the anonymous-interference and graphon random-graph assumptions, the regression-adjusted estimator $\hat\tau = n^{-1}\mathbf{1}^\top X(\hat\beta_1 - \hat\beta_0)$ is consistent for $\tau = E(f_1(1,\pi) - f_1(0,\pi))$ and satisfies $\sqrt{n}(\hat\tau - \tau) \xrightarrow{d} N(0, V_{\mathrm{reg}})$, where $V_{\mathrm{reg}}$ has an explicit four-term form: a treated-group residual variance, a control-group residual variance, a covariate-projection term, and an irreducible network-interference term $b\pi(1-\pi)(E f_1^{(1)}(1,\pi) - E f_1^{(1)}(0,\pi))^2$. The central optimality result is that any other linear adjustment $\hat\tau(\alpha_1,\alpha_0)$ has asymptotic variance $V_{\mathrm{reg}} + \frac{1}{\pi(1-\pi)}u^\top \mathrm{cov}(z_1)u \geq V_{\mathrm{reg}}$ with $u = (1-\pi)(\alpha_1 - \beta_1^{(-1)}) + \pi(\alpha_0 - \beta_0^{(-1)})$, so the population regression coefficients attain the minimum. The same pattern recurs at the nonparametric level: among estimators $\hat\tau(g_1,g_0)$ with arbitrary bounded adjustment functions, the variance is minimized when $g_k(z) = E(f_k(k,\pi)\mid z)$, and a local-constant kernel estimator with trimming attains this bound while remaining $\sqrt{n}$-normal for any fixed covariate dimension $p$.

Load-bearing premise

Everything rests on the anonymous-interference assumption that a unit's outcome depends on its own treatment and the fraction of treated neighbors; if which particular neighbors are treated — their identities, tie strengths, or positions — matters, the estimator and the optimality claim are not guaranteed.

Editorial extensions

If this is right

  • Whenever covariates correlate with outcomes, regression adjustment strictly reduces the asymptotic variance under network interference, and the difference-in-means estimator is recovered as the special case $z_i=1$; adjusting with uninformative covariates does no harm.
  • The explicit variance formula yields valid confidence intervals from data alone: the plug-in estimator of Theorem 4 reaches nominal coverage in simulations, whereas the earlier conservative bound from Li and Wager (2022) can under-cover.
  • The nonparametric estimator retains the parametric $\sqrt{n}$ rate for any fixed covariate dimension, so the curse of dimensionality does not degrade the precision of the ATE estimate even at $p=10$, and its asymptotic variance is never larger than the linear estimator's.
  • Both results hold across dense and sparse graphs satisfying $\sqrt{n}\rho_n\to\infty$, in contrast to the sparse-only analyses of Leung (2022) and Gao and Ding (2023).
  • Omitting the network-interference term from the confidence interval lowers coverage markedly on real contact networks, so the irreducible $b$-term must be estimated, not ignored.

Reading between the lines

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

  • If the fraction-of-treated-neighbors restriction were relaxed to richer exposure mappings, such as which neighbors are treated or how strongly connected they are, the estimator would likely need exposure-propensity-type weights; nothing in the variance machinery suggests the current $\sqrt{n}$-optimality would survive unmodified, a boundary the paper itself flags as future work.
  • The plug-in estimator $\hat b$ illustrates a general principle for low-rank graphons: degree-scaled weighted neighbor sums are self-normalizing consistent estimators of conditional graphon integrals, a principle that likely transfers to other estimands under interference, such as spillover or indirect effects, whose variances would carry an analogous irreducible network term.
  • A practical stress test suggested by Theorem 6: with higher-order kernels chosen so that $2q-6 < p < 2q$, the $\sqrt{n}$ rate should persist for $p$ beyond 10, but the rule-of-thumb bandwidth of Remark 4 may need recalibration — a simulation the paper does not run.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper studies regression adjustment for estimating the average treatment effect under network interference, modeled by anonymous interference on a graphon random graph. It proposes a linear regression-adjusted estimator and proves its consistency, asymptotic normality, and variance optimality within a class of linear adjustments (Theorems 1-3); it then constructs a consistent estimator for the asymptotic variance using a low-rank graphon approximation (Theorem 4). It also proposes a nonparametric kernel-and-trimming estimator and claims sqrt(n)-asymptotic normality and variance optimality in a broader class (Theorems 5-6). The contributions are illustrated by simulations and a real contact-network application.

Significance. If the results are correct, the paper would extend the classical regression-adjustment optimality of Lin (2013) and Li and Ding (2017) to graphon network interference, provide a principled alternative to the conservative variance estimator of Li and Wager (2022), and give a nonparametric estimator that attains a root-n rate despite nonparametric smoothing. The paper contains useful explicit variance formulas, a clean decomposition in (6), a novel plug-in estimator of the graphon functional b, and a substantial set of supporting lemmas for the nonparametric analysis. The main weakness is that several load-bearing technical claims are delegated to unverified or omitted proofs, which prevents the paper from being fully convincing in its current form.

major comments (4)
  1. [Proof of Theorem 2, Step 1(a), after Eq. (32)] The central CLT step is asserted rather than proved. Immediately after Eq. (32), the paper states that Theorem 4 of Li and Wager (2022), particularly their Eq. (24), 'still holds' under the paper's weaker moment conditions on qi, with no derivation. This is the step that converts the bias bound into the asymptotic normality claim (33), and therefore underpins Theorem 2 and Theorem 3. Under Assumption 3, the outcome functions and their derivatives are only L2-bounded, whereas Li and Wager's proof is written under uniform boundedness assumptions. A genuine Lindeberg-type check is needed for the graphon-dependent terms, and the manuscript does not show that the variance of the leading term is finite or that the remainder vanishes. The same unverified extension is used later in the proof of Theorem 6, Step 4, where the non-i.i.d. term is handled by 'following the proof of Theorem 4 in Li and Wager (2022)'. This is a load-bearing gap and should be fixed by supplying a complete proof under Assumption 3, or by strengthening the assumptions to match Li and Wager.
  2. [Supplementary Material, Lemma 3] Lemma 3 asserts consistency of the estimators for Ef^{(1)}_1(1,pi) and Ef^{(1)}_1(0,pi) by claiming that Theorem 6 of Li and Wager (2022) 'still holds' under the paper's slightly weaker assumptions. No proof is provided, and no check is given that the weaker moment conditions suffice for the variance-estimation argument. Since Theorem 4, and hence the paper's second advertised contribution, relies on Lemma 3, this is a load-bearing technical gap. The authors should either prove the claimed extension or impose conditions under which the Li-Wager theorem applies directly.
  3. [Proof of Theorem 5] The proof of Theorem 5 is omitted with the comment 'The proof can be obtained by following the steps in proof of Theorem 3. We omit it as there is no further difficulty.' Theorem 5 is not a minor corollary: it establishes the asymptotic variance formula (15) for the entire class of nonlinear adjustments and is the basis for the claim that the oracle conditional-mean choices are variance optimal. The omission is not acceptable for a load-bearing theorem in this paper. A full proof, or at least a detailed proof of the variance formula (15) and its minimization, should be included.
  4. [Section 2, Assumption 3 and Eq. (6)] Assumption 3 states E(xx^T) ⪰ 0, but the definitions of beta_1, beta_0 and the variance formulas use Mxx^{-1} and the inverse of E(z1 z1^T). If Mxx is only positive semidefinite, these objects are not well-defined. The assumption should require positive definiteness, or the paper should explicitly handle the singular case. This is a technical point, but it affects the statement of all main theorems.
minor comments (3)
  1. [Supplementary Material, Lemma 13] The statement of Lemma 13 contains a typo: it reads 'Eδ12 = 0, Eδ12 = o(n^{-1/2})' in the second assertion; from the proof it should be Eδ22 = o(n^{-1/2}).
  2. [Section 2, Remark after Theorem 2] The paper says Vreg is 'the first such result under the random graph asymptotic setup'. Given the earlier literature on regression adjustment under interference, a more precise comparison with Aronow and Samii (2017) and Gao and Ding (2023) would help the reader calibrate the novelty claim.
  3. [Section 5, Table 5] In the empirical application the network is fixed and is not generated by a graphon, yet the variance estimator uses a graphon-based approximation with r = 10. The paper should briefly acknowledge that this is an approximation for the real network and discuss the sensitivity to the choice of r.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimality results are derived from explicit assumptions and external benchmarks, with no load-bearing self-citation or fit-as-prediction reduction.

full rationale

The paper's central claims are self-contained against the assumptions it states. Theorem 2 derives the asymptotic variance Vreg by a Cramer-Wold decomposition of the regression-adjusted estimator; the population regression coefficients beta1 and beta0 arise as probability limits of the least-squares fits, not as parameters fitted to force the theorem. Theorem 3's optimality statement is an algebraic comparison, tilde V(alpha1, alpha0) - Vreg = (1/(pi(1-pi))) u^T cov(z1) u >= 0, which is a direct calculation from the model, not a restatement of the estimand. The network term b is a graphon functional defined in equation (5) and is estimated by the plug-in estimator in equation (10) with consistency proved via Lemma 4; it is not an input recycled as a prediction. The proofs invoke Theorem 4 of Li and Wager (2022) and Theorem 6 of Li and Wager (2022) as external benchmarks, and while the extension to weaker moment conditions is asserted without a full verification in the text, reliance on an external theorem is not circularity because the cited result is not authored by the present paper and does not reduce the present derivation to its own inputs. No self-citation chain of the kind described in the circularity patterns appears: the citations to Lin (2013), Li and Ding (2017), and Li and Wager (2022) are conventional external prior work. The only caveat is a possible proof gap concerning whether Li and Wager's Theorem 4 indeed extends under Assumption 3, but that is a correctness and rigor concern, not a circularity concern.

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

The central claims rest on five stated modeling assumptions plus standard U-statistic projection theory and Li-Wager (2022) lemmas. No free parameters are fitted; bandwidth and trimming are tuning choices that do not enter the theoretical optimality claim. No new entities (forces, particles, dimensions) are introduced.

assumptions (7)
  • domain assumption Anonymous interference: Yi = fi(Wi, Mi/Ni)
    Section 2 Assumption 1. The entire estimand and estimator are defined under this; if outcomes depend on the identity of treated neighbors, all theorems fail.
  • domain assumption Graphon model with sqrt(n) rho_n -> infinity, bounded graphon and positive minimum degree
    Section 2 Assumption 2. Needed for the network to be dense enough that Mi/Ni concentrates around pi and the CLT holds.
  • domain assumption Moment conditions on covariates and outcome derivatives
    Section 2 Assumption 3. Imposes E|x|^4 < infinity and finite L2 norms of x f^(k); weaker than Li-Wager but still required for the Taylor expansion and CLT.
  • domain assumption Low-rank graphon with bounded eigenfunctions and rho_n -> 0
    Section 3 Assumption 4. Needed for the plug-in estimator b and the PC balancing estimators of E f^(1); affects Theorem 4 variance estimator consistency.
  • domain assumption Kernel, density, and trimming regularity conditions (Assumption 5)
    Section 4 Assumption 5. Includes q-th order kernel, smooth marginal density, bandwidth/trimming constraints, and tail condition on S1; supports Theorem 6.
  • domain assumption External lemmas from Li and Wager (2022), e.g., Lemma 15 on degree concentration and moment bounds
    Invoked in proofs of Lemma 1, 2, 4, 9 and Theorem 6 to control Mi/Ni and neighbor sums; an unproved background result from the prior literature.
  • standard math U-statistic projection theory (Powell et al., 1986; Korolyuk, 2013)
    Used in Lemmas 8-11 to approximate kernel-type statistics by i.i.d. sums.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Causal Inference under Interference: Regression Adjustment and Optimality." pith.science (2026). https://pith.science/paper/7YUWMIL7

@misc{pith2026250206008,
  author       = {Pith},
  title        = {Pith review of: Causal Inference under Interference: Regression Adjustment and Optimality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YUWMIL7}},
  note         = {Machine review of arXiv:2502.06008}
}
read the original abstract

In randomized controlled trials without interference, regression adjustment is widely used to enhance the efficiency of treatment effect estimation. This paper extends this efficiency principle to settings with network interference, where a unit's response may depend on the treatments assigned to its neighbors in a network. We make three key contributions: (1) we establish a central limit theorem for a linear regression-adjusted estimator and prove its optimality in achieving the smallest asymptotic variance within a class of linear adjustments; (2) we develop a novel, consistent estimator for the asymptotic variance of this linear estimator; and (3) we propose a nonparametric estimator that integrates kernel smoothing and trimming techniques, demonstrating its asymptotic normality and its optimality in minimizing asymptotic variance within a broader class of nonlinear adjustments. Extensive simulations validate the superior performance of our estimators, and a real-world data application illustrates their practical utility. Our findings underscore the power of regression-based methods and reveal the potential of kernel-and-trimming-based approaches for further enhancing efficiency under network interference.

Figures

Figures reproduced from arXiv: 2502.06008 by the authors.

Figure 1
Figure 1. shows the distributions of ˆτnp for p = 1 and p = 5. We observe that the sample distributions closely match the theoretical distribution derived in Theorem 6 (shown in red). Notably, our ˆτnp exhibits good performance regardless of whether network interference is present or absent. Moreover, the sample distribution for p = 5 fits well with the limiting distribution, indicating that our estimator, ˆτnp, can mitigate … view at source ↗
Figure 2
Figure 2. A comparison of MSEs between ˆτnp (the estimator after nonparametric regression adjustment) and ˆτ (the estimator after linear regression adjustment). To demonstrate the performance gain of ˆτnp over ˆτ , we plot in [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Aggregated contact networks for students during the morning (left) and midday (right) periods. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 27 canonical work pages

  1. [1]

    Aronow, P. M. and Samii, C. (2017). Estimating average causal effects under general interference, with application to a social network experiment. Annals of Applied Statistics , 11(4):1912--1947

    work page 2017

  2. [2]

    Athey, S., Eckles, D., and Imbens, G. W. (2018). Exact p-values for network interference. Journal of the American Statistical Association , 113(521):230--240

    2018

  3. [3]

    Banerjee, A. (2007). A method of estimating the average derivative. Journal of Econometrics , 136(1):65--88

    work page 2007

  4. [4]

    Beemer, J., Spoon, K., He, L., Fan, J., and Levine, R. A. (2018). Ensemble learning for estimating individualized treatment effects in student success studies. International Journal of Artificial Intelligence in Education , 28:315--335

    work page 2018

  5. [5]

    M., Tavares, M., Resende, J., et al

    Bragan c a, A., Newton, P., Cohn, A., Assun c \ a o, J., Camboim, C., de Faveri, D., Farinelli, B., Perego, V. M., Tavares, M., Resende, J., et al. (2022). Extension services can promote pasture restoration: Evidence from brazil’s low carbon agriculture plan. Proceedings of the National Academy of Sciences , 119(12):e2114913119

    work page 2022

  6. [6]

    D., and Sadoulet, E

    Cai, J., Janvry, A. D., and Sadoulet, E. (2015). Social networks and the decision to insure. American Economic Journal: Applied Economics , 7(2):81--108

    work page 2015

  7. [7]

    Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., and Newey, W. (2017). Double/debiased/neyman machine learning of treatment effects. American Economic Review , 107(5):261--265

    work page 2017

  8. [8]

    Chin, A. (2018). Central limit theorems via stein's method for randomized experiments under interference. arXiv preprint arXiv:1804.03105

    work page Pith review arXiv 2018

Show all 41 references
  1. [9]

    Donnat, C., Klopp, O., and Verzelen, N. (2024). One-bit total variation denoising over networks with applications to partially observed epidemics. arXiv preprint arXiv:2405.00619

    2024 arXiv

  2. [10]

    Duflo, E., Greenstone, M., Pande, R., and Ryan, N. (2013). Truth-telling by third-party auditors and the response of polluting firms: Experimental evidence from india. The Quarterly Journal of Economics , 128(4):1499--1545

    2013

  3. [11]

    P., and Zhang, Y

    Fan, Q., Hsu, Y.-C., Lieli, R. P., and Zhang, Y. (2022). Estimation of conditional average treatment effects with high-dimensional data. Journal of Business & Economic Statistics , 40(1):313--327

    2022

  4. [12]

    and Ding, P

    Gao, M. and Ding, P. (2023). Causal inference in network experiments: regression-based analysis and design-based properties. arXiv preprint arXiv:2309.07476

    2023 arXiv

  5. [13]

    Gemmetto, V., Barrat, A., and Cattuto, C. (2014). Mitigation of infectious disease at school: targeted class closure vs school closure. BMC Infectious Diseases , 14:1--10

    2014

  6. [14]

    Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory , 24(3):726--748

    2008

  7. [15]

    and Stoker, T

    H \"a rdle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association , 84(408):986--995

    1989

  8. [16]

    Hudgens, M. G. and Halloran, M. E. (2008). Toward causal inference with interference. Journal of the American Statistical Association , 103(482):832--842

    2008

  9. [17]

    Imbens, G. W. and Rubin, D. B. (2015). Causal inference for statistics, social, and biomedical sciences: An introduction. Cambridge University Press

    2015

  10. [18]

    Korolyuk, V. S. (2013). Theory of U-statistics , volume 273. Springer Science & Business Media

    2013

  11. [19]

    Lavergne, P. (2008). A cauchy-schwarz inequality for expectation of matrices. Discussion papers, Department of Economics, Simon Fraser University

    2008

  12. [20]

    Leung, M. P. (2022). Causal inference under approximate neighborhood interference. Econometrica , 90(1):267--293

    2022

  13. [21]

    and Wager, S

    Li, S. and Wager, S. (2022). Random graph asymptotics for treatment effect estimation under network interference. The Annals of Statistics , 50(4):2334--2358

    2022

  14. [22]

    and Ding, P

    Li, X. and Ding, P. (2017). General forms of finite population central limit theorems with applications to causal inference. Journal of the American Statistical Association , 112(520):1759--1769

    2017

  15. [23]

    Lin, W. (2013). Agnostic notes on regression adjustments to experimental data: Reexamining Freedman’s critique . The Annals of Applied Statistics , 7(1):295 -- 318

    2013

  16. [24]

    Liu, H., Ren, J., and Yang, Y. (2024). Randomization-based joint central limit theorem and efficient covariate adjustment in randomized block 2^k factorial experiments. Journal of the American Statistical Association , 119(545):136--150

    2024

  17. [25]

    and Yang, Y

    Liu, H. and Yang, Y. (2020). Regression-adjusted average treatment effect estimates in stratified randomized experiments. Biometrika , 107(4):935--948

    2020

  18. [26]

    Lov \'a sz, L. (2012). Large networks and graph limits , volume 60. American Mathematical Soc

    2012

  19. [27]

    Lu, X., Wang, Y., and Zhang, Z. (2024). Adjusting auxiliary variables under approximate neighborhood interference. arXiv preprint arXiv:2411.19789

    2024 arXiv

  20. [28]

    Marmor, Y., Abbey, A., Shahar, Y., and Mokryn, O. (2023). Assessing individual risk and the latent transmission of covid-19 in a population with an interaction-driven temporal model. Scientific Reports , 13(1):12955

    2023

  21. [29]

    I., Mishler, A

    Naimi, A. I., Mishler, A. E., and Kennedy, E. H. (2023). Challenges in obtaining valid causal effect estimates with machine learning algorithms. American Journal of Epidemiology , 192(9):1536--1544

    2023

  22. [30]

    L., Shepherd, H., and Aronow, P

    Paluck, E. L., Shepherd, H., and Aronow, P. M. (2016). Changing climates of conflict: A social network experiment in 56 schools. Proceedings of the National Academy of Sciences , 113(3):566--571

    2016

  23. [31]

    E., Beck, C

    Par \'e , P. E., Beck, C. L., and Ba s ar, T. (2020). Modeling, estimation, and analysis of epidemics over networks: An overview. Annual Reviews in Control , 50:345--360

    2020

  24. [32]

    E., Liu, J., Beck, C

    Par \'e , P. E., Liu, J., Beck, C. L., Kirwan, B. E., and Ba s ar, T. (2018). Analysis, estimation, and validation of discrete-time epidemic processes. IEEE Transactions on Control Systems Technology , 28(1):79--93

    2018

  25. [33]

    Park, C., Chen, G., Yu, M., and Kang, H. (2024). Minimum resource threshold policy under partial interference. Journal of the American Statistical Association , 119(548):2881--2894

    2024

  26. [34]

    L., Stock, J

    Powell, J. L., Stock, J. H., Stoker, T. M., et al. (1986). Semiparametric estimation of weighted average derivatives

    1986

  27. [35]

    and Takasaki, Y

    Sato, R. and Takasaki, Y. (2019). Peer effects on vaccination behavior: experimental evidence from rural nigeria. Economic Development and Cultural Change , 68(1):93--129

    2019

  28. [36]

    S \"a vje, F., Aronow, P., and Hudgens, M. (2021). Average treatment effects in the presence of unknown interference. Annals of Statistics , 49(2):673

    2021

  29. [37]

    Shi, C., Blei, D., and Veitch, V. (2019). Adapting neural networks for the estimation of treatment effects. Advances in Neural Information Processing Systems , 32

    2019

  30. [38]

    Stehl \'e , J., Voirin, N., Barrat, A., Cattuto, C., Isella, L., Pinton, J.-F., Quaggiotto, M., Van den Broeck, W., R \'e gis, C., Lina, B., et al. (2011). High-resolution measurements of face-to-face contact patterns in a primary school. PloS one , 6(8):e23176

    2011

  31. [39]

    and Ding, P

    Su, F. and Ding, P. (2021). Model-assisted analyses of cluster-randomized experiments. Journal of the Royal Statistical Society Series B: Statistical Methodology , 83(5):994--1015

    2021

  32. [40]

    Viviano, D. (2024). Policy targeting under network interference. Review of Economic Studies , page rdae041

    2024

  33. [41]

    and Ding, P

    Zhao, A. and Ding, P. (2022). Reconciling design-based and model-based causal inferences for split-plot experiments. The Annals of Statistics , 50(2):1170--1192

    2022

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.