Pith. sign in

REVIEW 5 minor 130 references

Randomization Tests in Randomized Saturation Designs

T0 review · 0 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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

Pith's one-line read The same cluster-level relabeling of saturation labels yields finite-sample tests for no-spillover and bounded-spillover nulls, asymptotic tests for average spillovers, and a finite-sample test for global monotonicity.

desk verdict Solid design-based toolkit for multi-saturation spillover tests; incremental but carefully proved and usable when K is small. read the letter →

arxiv 2607.04257 v1 pith:6YSKS63P submitted 2026-07-05 stat.ME econ.EM

Jizhou Liu , Azeem M. Shaikh , Liang Zhong This is my paper · ORCID
classification stat.MEecon.EM MSC 62G1062K99
keywords causalinferenceconditionalrandomizationtestinterferencerandomizedsaturationdesignspillovereffectsstudentizationpairwiseimputationmonotonenull
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 saturation designs assign whole clusters to different treatment intensities and then randomize who is treated inside each cluster. That setup is meant to measure spillover effects, but the scientifically interesting nulls—no spillover for untreated units, spillovers bounded by a chosen constant, zero average spillover, or monotone response as saturation rises—are only partially sharp, so ordinary reshuffling of the full assignment does not give a valid reference distribution. The paper shows that conditioning on untreated focal units in the two relevant saturation cells turns the problem into a simple cluster-level relabeling of those two labels. The same relabeling distribution, paired with null-specific statistics, delivers exact finite-sample tests for the unit-level equality and bounded nulls, and, after studentization, asymptotic tests for weak average-spillover nulls. For several ordered saturations it further constructs an unconditional pairwise-imputation test that controls size for the global monotone null. Simulations and a reanalysis of the Zomba cash-transfer experiment show how the procedures behave with few, unequally sized clusters.

What carries the argument

The focal-unit cluster-level relabeling distribution: after selecting a fixed number of untreated units from each cluster whose observed saturation is one of the two levels being compared, one reassigns those two saturation labels across the contrasted clusters while keeping the observed counts fixed. Under the partially sharp null the focal outcomes are invariant; under the bounded null the equality boundary supplies a least-favorable imputation; for average nulls the same relabelings calibrate a studentized statistic.

What would settle it

In a multi-saturation experiment whose clusters and margins match the paper’s design, apply the focal-set conditional test under a known unit-level equality or bounded null that holds by construction; if the resulting p-values are not stochastically larger than uniform, or if the studentized average-spillover test fails to control size as the number of clusters grows under the paper’s regularity conditions, the central validity claims fail.

Watch

Extended reading notes

Core claim

For any fixed pair of saturation levels, one common conditional device—select untreated focal units in the contrasted clusters and relabel those clusters’ saturation labels while preserving the observed margins—yields finite-sample valid randomization tests of the partially sharp no-spillover null and of the unit-level bounded-spillover null, and, when the test statistic is studentized at the cluster level, asymptotically valid tests of weighted average-spillover nulls. Separately, an unconditional pairwise-imputation statistic yields a finite-sample valid test of global monotonicity of untreated outcomes over an ordered set of saturations.

Load-bearing premise

Each unit’s outcome is allowed to depend on the assignment only through its own treatment status and the total number of treated units in its own cluster; there is no cross-cluster interference and no dependence on which particular peers are treated.

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

0 major / 5 minor

Summary. This paper develops randomization tests for spillover hypotheses in multi-saturation two-stage designs. For a fixed pair of saturations, it constructs a common conditional framework based on untreated focal units and cluster-level relabeling, delivering finite-sample conditional validity for a partially sharp no-spillover null and a bounded individual-spillover null (Theorem 3.1). For weak average-spillover nulls that do not impute missing outcomes, it pairs the same relabeling device with studentized Neyman statistics and proves unconditional asymptotic size control under stated regularity (Theorem 4.1). For ordered multi-saturation settings, it gives a finite-sample valid unconditional pairwise-imputation test of global monotonicity of untreated outcomes (Theorem 5.1). Calibrated simulations and a Zomba Cash Transfer application illustrate size, power, and implementation.

Significance. The contribution is solid and useful for a design that is common in applied work but still thin on finite-sample inference for partially sharp spillover nulls. Extending Basse et al. (2019) from binary first-stage / single-treated-unit designs to multi-saturation designs with multiple treated units is a genuine methodological step. The shared focal-unit relabeling construction, the least-favorable shift for bounded nulls, the studentized weak-null asymptotics, and the monotone PIRT together form a coherent toolkit rather than isolated tricks. The appendix conditioning lemmas, super-uniformity argument, and three-term CLT decomposition make the validity claims checkable. The Zomba illustration and calibrated simulations are appropriately modest and show practical relevance when clusters per cell are few.

minor comments (5)
  1. [Section 6.1] In Section 6.1, the compound-label exposure mapping for Zomba is carefully stated, but a short explicit sentence that the reported tests are of schoolgirl-offer-label contrasts (not pure cash-transfer intensity) would further reduce misreading of Table 2.
  2. [Figures 1–2] Figures 1–2 would be easier to read if the panels listed the four outcomes or reported outcome-specific ranges rather than only averages over outcomes.
  3. [Remark 3.1] Remark 3.1 recommends kj = min{nj−mj(s), nj−mj(s′)}; a one-sentence note on sensitivity of p-values to smaller fixed kj would help applied users.
  4. [Section 2.2 / Conclusion] The paper correctly flags Assumption 2.1 as interpretational; a brief pointer in the conclusion to diagnostics or sensitivity for peer-identity dependence would be welcome without changing the theorems.
  5. [References] A few arXiv-style self-citations (Liu 2026; Liu & Zhong 2026) are listed as forthcoming; ensure final versions and DOIs are updated at production.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: design-based validity theorems are self-contained; self-citations supply building blocks without forcing the central claims.

full rationale

The paper’s load-bearing results are finite-sample and asymptotic validity theorems for randomization tests under a known assignment law (Theorems 3.1, 4.1, 5.1). Validity follows from standard conditional-randomization and pairwise-imputation arguments once focal units make the relevant exposures imputable (or least-favorable under the equality boundary), with full proofs in Appendices B–D. p-values are not fitted to outcomes; simulations and the Zomba application illustrate size/power and implementation rather than “predict” quantities forced by fitted constants. Self-citations (Zhong 2024 for PIRT; Liu 2026; Liu & Zhong 2026) acknowledge related frameworks, but the multi-saturation CRT, bounded-null shift, studentized weak-null result, and application are developed and proved here and are not equivalent to those citations by construction. Assumption 2.1 is an exposure-mapping interpretational premise, not a circular derivation step. Score 1 reflects only ordinary non-load-bearing self-citation, not circular reduction of the central claims.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims are design-based: they rest on the known two-stage randomization, partial-interference exposure mapping, and null-specific imputability or studentization. Researcher-chosen objects (δ, focal sizes, weights, SM, score functions) are free design choices, not data-fitted constants that force the result. No new physical entities are postulated.

free parameters (4)
  • bounded-null threshold δ
    Researcher-chosen constant defining Hs,s′δ,B; validity holds for any fixed δ, but substantive conclusions depend on this hand-chosen bound.
  • focal-set sizes kj
    Must be chosen before seeing labels and satisfy 1≤kj≤min{nj−mj(s),nj−mj(s′)}; default max untreated count is recommended but still a free design choice affecting power.
  • weak-null weights λjK
    Nonrandom weights defining the average target (equal-cluster vs unit-average); studentized validity is for the chosen weighted null, not a data-driven weight.
  • monotone saturation set SM and score transforms ψ,φ
    Ordered subset and monotone transforms in the PIRT statistic are part of the null/statistic choice and affect which units enter pairwise comparisons.
assumptions (6)
  • domain assumption Homogeneous partial interference: Yi depends only on own Di and cluster treated count (Assumption 2.1).
    Load-bearing exposure mapping; without it, tests target a misspecified exposure null.
  • domain assumption Two-stage complete randomization of saturations with fixed margins and within-cluster sampling of exactly mj(Aj) treated units.
    Known design law used for all reference distributions (Section 2.1).
  • domain assumption Finite-population fixed potential outcomes; randomness only from assignment and focal sampling.
    Standard Fisherian setup stated in Section 2.
  • standard math Super-uniformity of finite randomization p-values (Lemma B.1) and conditional uniformity of cluster relabelings (Lemma B.2).
    Standard probability facts used to prove Theorem 3.1.
  • domain assumption Assumption C.1: stable design fractions, transformed fourth-moment bounds, nondegenerate limiting variances, maximal negligibility for weak-null asymptotics.
    Regularity needed for Theorem 4.1; not free parameters but untestable asymptotic conditions.
  • ad hoc to paper Pairwise-imputable spillover-monotone statistic definition and pairwise ordering under H0,M (Definition 5.1, Proposition 5.1).
    Paper-specific statistic class that makes the unconditional PIRT work; builds on Zhong (2024) but specialized to saturation exposures.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Randomization Tests in Randomized Saturation Designs." pith.science (2026). https://pith.science/paper/6YSKS63P

@misc{pith2026260704257,
  author       = {Pith},
  title        = {Pith review of: Randomization Tests in Randomized Saturation Designs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YSKS63P}},
  note         = {Machine review of arXiv:2607.04257}
}
read the original abstract

Randomized saturation designs are widely used to study spillover effects in clustered populations. In these designs, clusters are first assigned to treatment saturation levels, and units are then randomized within clusters according to the assigned saturation. This paper develops randomization tests for such experiments under several null hypotheses that arise naturally in spillover analysis. For a fixed pair of saturation levels, we first study two individual-level hypotheses: a partially sharp null of no spillover effect for every untreated unit and a bounded null that restricts individual spillover effects by a prespecified constant. Both hypotheses can be tested using a common conditional randomization framework, with finite-sample validity obtained by combining the same focal-unit relabeling distribution with null-specific statistics. We then study weak average-spillover nulls and show that, although these nulls do not yield finite-sample exact conditional tests, studentized relabeling statistics deliver asymptotically valid randomization-based inference. Finally, for multiple ordered saturation levels, we develop a finite-sample valid unconditional pairwise-imputation test for global monotonicity of spillover effects. Simulations and an application to the Zomba Cash Transfer experiment illustrate the finite-sample behavior and practical implementation of the methods.

Figures

Figures reproduced from arXiv: 2607.04257 by the authors.

Figure 1
Figure 1. Calibrated power for pairwise bounded spillover nulls. The horizontal line marks [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. Calibrated power for monotone spillover nulls. The left panel uses the three-level [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

130 extracted references · 47 canonical work pages

  1. [1]

    2026 , eprint=

    Randomization Tests in Switchback Experiments , author=. 2026 , eprint=

    2026

  2. [2]

    2026 , eprint=

    Randomization Inference in Two-Sided Market Experiments , author=. 2026 , eprint=

    2026

  3. [4]

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

    Caughey, Devin and Dafoe, Allan and Li, Xinran and Miratrix, Luke , title =. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume =. 2023 , month =. doi:10.1093/jrsssb/qkad080 , url =

    doi:10.1093/jrsssb/qkad080 2023

  4. [5]

    Journal of the American Statistical Association , volume=

    Robust permutation tests for correlation and regression coefficients , author=. Journal of the American Statistical Association , volume=. 2017 , publisher=

    2017

  5. [6]

    DiCiccio and Thomas J

    Cyrus J. DiCiccio and Thomas J. DiCiccio and Joseph P. Romano , keywords =. Exact tests via multiple data splitting , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.spl.2020.108865 , url =

    work page doi:10.1016/j.spl.2020.108865 2020

  6. [7]

    2015 , publisher=

    Causal inference in statistics, social, and biomedical sciences , author=. 2015 , publisher=

    2015

  7. [8]

    Journal of the American Statistical Association , volume =

    Xinran Li and Peng Ding , title =. Journal of the American Statistical Association , volume =. 2017 , publisher =. doi:10.1080/01621459.2017.1295865 , URL =

    doi:10.1080/01621459.2017.1295865 2017

  8. [9]

    Journal of the American Statistical Association , volume =

    Guillaume Basse and Avi Feller , title =. Journal of the American Statistical Association , volume =. 2018 , publisher =. doi:10.1080/01621459.2017.1323641 , URL =

    doi:10.1080/01621459.2017.1323641 2018

Show all 130 references
  1. [10]

    Biometrika , pages=

    Asymptotic Validity and Finite-Sample Properties of Approximate Randomization Tests , author=. Biometrika , pages=. 2025 , publisher=

    2025

  2. [11]

    Biometrika , volume =

    Basse, G W and Feller, A and Toulis, P , title = ". Biometrika , volume =. 2019 , month =. doi:10.1093/biomet/asy072 , url =

    2019 doi

  3. [12]

    , title =

    Fisher, Ronald A. , title =. 1953 , publisher =

    1953

  4. [13]

    Lehmann and Joseph P

    Erich L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses (Third Ed.). 2005

    2005

  5. [14]

    , author=

    Planning of experiments. , author=. 1958 , publisher=

    1958

  6. [15]

    Journal of the American Statistical Association , year =

    Hudgens, Michael G and Halloran, M Elizabeth , title =. Journal of the American Statistical Association , year =

  7. [16]

    Aronow , title =

    Peter M. Aronow , title =. Sociological Methods & Research , volume =. 2012 , doi =. https://doi.org/10.1177/0049124112437535 , abstract =

    2012 doi

  8. [17]

    Imbens , title =

    Susan Athey and Dean Eckles and Guido W. Imbens , title =. Journal of the American Statistical Association , volume =. 2018 , publisher =. doi:10.1080/01621459.2016.1241178 , URL =

    2018 doi

  9. [18]

    The Annals of Applied Statistics , year=

    Estimating average causal effects under general interference, with application to a social network experiment , author=. The Annals of Applied Statistics , year=

  10. [19]

    Romano , title =

    EunYi Chung and Joseph P. Romano , title =. The Annals of Statistics , number =. 2013 , doi =

    2013

  11. [20]

    ETS Research Bulletin Series , volume =

    Rubin, Donald , title =. ETS Research Bulletin Series , volume =. doi:https://doi.org/10.1002/j.2333-8504.1972.tb00631.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.2333-8504.1972.tb00631.x , abstract =

    1972 doi

  12. [21]

    Rubin , title =

    Donald B. Rubin , title =. Journal of the American Statistical Association , volume =. 1980 , publisher =. doi:10.1080/01621459.1980.10477517 , URL =

    1980 doi

  13. [22]

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

    Masoero, Lorenzo and Vijaykumar, Suhas and Richardson, Thomas S and McQueen, James and Rosen, Ido and Burdick, Brian and Bajari, Pat and Imbens, Guido , title =. Journal of the Royal Statistical Society Series B: Statistical Methodology , pages =. 2026 , month =. doi:10.1093/j...

    2026 doi

  14. [23]

    Advances in Neural Information Processing Systems , volume=

    Long-term causal effects via behavioral game theory , author=. Advances in Neural Information Processing Systems , volume=

  15. [24]

    , title =

    Johari, Ramesh and Li, Hannah and Liskovich, Inessa and Weintraub, Gabriel Y. , title =. Management Science , volume =. 2022 , doi =. https://doi.org/10.1287/mnsc.2021.4247 , abstract =

    2022 doi

  16. [25]

    Double randomized online experiments , author=

  17. [26]

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

    Puelz, David and Basse, Guillaume and Feller, Avi and Toulis, Panos , title = ". Journal of the Royal Statistical Society Series B: Statistical Methodology , volume =. 2021 , month =. doi:10.1111/rssb.12478 , url =

    2021 doi

  18. [27]

    Canay and Joseph P

    Ivan A. Canay and Joseph P. Romano and Azeem M. Shaikh , journal =. RANDOMIZATION TESTS UNDER AN APPROXIMATE SYMMETRY ASSUMPTION , volume =

  19. [28]

    DiCiccio and Joseph P

    Cyrus J. DiCiccio and Joseph P. Romano , title =. Journal of the American Statistical Association , volume =. 2017 , publisher =. doi:10.1080/01621459.2016.1202117 , URL =

    2017 doi

  20. [29]

    Covariate-adjusted Fisher randomization tests for the average treatment effect , journal =

    Anqi Zhao and Peng Ding , keywords =. Covariate-adjusted Fisher randomization tests for the average treatment effect , journal =. 2021 , note =. doi:https://doi.org/10.1016/j.jeconom.2021.04.007 , url =

    2021 doi

  21. [30]

    Journal of the American Statistical Association , volume =

    Jason Wu and Peng Ding , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2020.1750415 , URL =

    2021 doi

  22. [31]

    Biometrika , volume =

    Ding, Peng and Dasgupta, Tirthankar , title = ". Biometrika , volume =. 2017 , month =. doi:10.1093/biomet/asx059 , url =

    2017 doi

  23. [32]

    2024 , eprint=

    Multiple Randomization Designs: Estimation and Inference with Interference , author=. 2024 , eprint=

    2024

  24. [33]

    Econometrica , volume =

    Basse, Guillaume and Ding, Peng and Feller, Avi and Toulis, Panos , title =. Econometrica , volume =. doi:https://doi.org/10.3982/ECTA20134 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA20134 , abstract =

    doi

  25. [34]

    Imbens and Lorenzo Masoero and James McQueen and Thomas S

    Patrick Bajari and Brian Burdick and Guido W. Imbens and Lorenzo Masoero and James McQueen and Thomas S. Richardson and Ido M. Rosen , title =. Statistical Science , number =. 2023 , doi =

    2023

  26. [35]

    Lehmann, E. L. and Romano, Joseph P. , biburl =

  27. [36]

    Aronow and Cyrus Samii , title =

    Peter M. Aronow and Cyrus Samii , title =. The Annals of Applied Statistics , number =. 2017 , doi =

    2017

  28. [37]

    Journal of Economic Perspectives , Volume =

    Athey, Susan and Luca, Michael , Title =. Journal of Economic Perspectives , Volume =. 2019 , Month =. doi:10.1257/jep.33.1.209 , URL =

    2019 doi

  29. [38]

    Trustworthy Online Controlled Experiments: A Practical Guide to A/B Testing , publisher=

    Kohavi, Ron and Tang, Diane and Xu, Ya , year=. Trustworthy Online Controlled Experiments: A Practical Guide to A/B Testing , publisher=

  30. [39]

    Management Science , volume =

    Wager, Stefan and Xu, Kuang , title =. Management Science , volume =. 2021 , doi =. https://doi.org/10.1287/mnsc.2020.3844 , abstract =

    2021 doi

  31. [40]

    and Wooldridge, Jeffrey M

    Abadie, Alberto and Athey, Susan and Imbens, Guido W. and Wooldridge, Jeffrey M. , title =. Econometrica , volume =. doi:https://doi.org/10.3982/ECTA12675 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA12675 , abstract =

    doi

  32. [41]

    2003 , publisher=

    Experimentation Matters: Unlocking the Potential of New Technologies for Innovation , author=. 2003 , publisher=

    2003

  33. [42]

    Proceedings of the Fifteenth ACM Conference on Economics and Computation , pages =

    Blake, Thomas and Coey, Dominic , title =. Proceedings of the Fifteenth ACM Conference on Economics and Computation , pages =. 2014 , isbn =. doi:10.1145/2600057.2602837 , abstract =

    2014 doi

  34. [43]

    The Review of Economics and Statistics , volume =

    Comola, Margherita and Prina, Silvia , title =. The Review of Economics and Statistics , volume =. 2021 , month =. doi:10.1162/rest_a_00908 , url =

    2021 doi

  35. [44]

    Banking the poor via savings accounts: Evidence from a field experiment , journal =

    Silvia Prina , keywords =. Banking the poor via savings accounts: Evidence from a field experiment , journal =. 2015 , issn =. doi:https://doi.org/10.1016/j.jdeveco.2015.01.004 , url =

    2015 doi

  36. [45]

    63Challenges in Banking the Rural Poor: Evidence from Kenya’s Western Province , booktitle =

    Dupas, Pascaline and Green, Sarah and Keats, Anthony and Robinson, Jonathan , isbn =. 63Challenges in Banking the Rural Poor: Evidence from Kenya’s Western Province , booktitle =. 2016 , month =. doi:10.7208/chicago/9780226315867.003.0002 , url =

    2016 doi

  37. [47]

    Inference for two-stage experiments under covariate-adaptive randomization , journal =

    Jizhou Liu , keywords =. Inference for two-stage experiments under covariate-adaptive randomization , journal =. 2026 , issn =. doi:https://doi.org/10.1016/j.jeconom.2026.106189 , url =

    2026 doi

  38. [48]

    Hudgens , title =

    Lan Liu and Michael G. Hudgens , title =. Journal of the American Statistical Association , volume =. 2014 , publisher =. doi:10.1080/01621459.2013.844698 , note =

    2014 doi

  39. [49]

    Exact Confidence Intervals in the Presence of Interference

    Rigdon, Joseph and Hudgens, Michael G. Exact Confidence Intervals in the Presence of Interference. Stat Probab Lett

  40. [50]

    International conference on machine learning , pages=

    Estimation of causal peer influence effects , author=. International conference on machine learning , pages=. 2013 , organization=

    2013

  41. [51]

    The Quarterly journal of economics , volume=

    Cash or condition? Evidence from a cash transfer experiment , author=. The Quarterly journal of economics , volume=. 2011 , publisher=

    2011

  42. [52]

    Review of Economics and Statistics , volume=

    Optimal design of experiments in the presence of interference , author=. Review of Economics and Statistics , volume=. 2018 , publisher=

    2018

  43. [53]

    Average Direct and Indirect Causal Effects under Interference , publisher =

    Hu, Yuchen and Li, Shuangning and Wager, Stefan , keywords =. Average Direct and Indirect Causal Effects under Interference , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2104.03802 , url =

    2021 doi arXiv

  44. [54]

    Airoldi and Fabrizia Mealli , title =

    Laura Forastiere and Edoardo M. Airoldi and Fabrizia Mealli , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2020.1768100 , URL =

    2021 doi

  45. [55]

    Journal of the American Statistical Association , volume =

    Kosuke Imai and Zhichao Jiang and Anup Malani , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2020.1775612 , URL =

    2021 doi

  46. [56]

    Bugni and Ivan A

    Federico A. Bugni and Ivan A. Canay and Azeem M. Shaikh , title =. Journal of the American Statistical Association , volume =. 2018 , publisher =. doi:10.1080/01621459.2017.1375934 , note =

    2018 doi

  47. [57]

    and Shaikh, Azeem M

    Bugni, Federico and Canay, Ivan A. and Shaikh, Azeem M. and Tabord-Meehan, Max , title =. Journal of Political Economy Microeconomics , volume =. 2025 , doi =. https://doi.org/10.1086/732836 , abstract =

    2025 doi

  48. [58]

    Optimality of Matched-Pair Designs in Randomized Controlled Trials , publisher =

    Bai, Yuehao , keywords =. Optimality of Matched-Pair Designs in Randomized Controlled Trials , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2206.07845 , url =

    2022 doi arXiv

  49. [59]

    van der Vaart, A. W. , year =. Asymptotic statistics , volume =

  50. [60]

    Romano and Azeem M

    Yuehao Bai and Joseph P. Romano and Azeem M. Shaikh , title =. Journal of the American Statistical Association , volume =. 2021 , publisher =. doi:10.1080/01621459.2021.1883437 , URL =

    2021 doi

  51. [61]

    Inference for Matched Tuples and Fully Blocked Factorial Designs , publisher =

    Bai, Yuehao and Liu, Jizhou and Tabord-Meehan, Max , keywords =. Inference for Matched Tuples and Fully Blocked Factorial Designs , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2206.04157 , url =

    2022 doi arXiv

  52. [62]

    2010 , publisher=

    Large sample techniques for statistics , author=. 2010 , publisher=

    2010

  53. [63]

    and Tabord-Meehan, Max , keywords =

    Bai, Yuehao and Liu, Jizhou and Shaikh, Azeem M. and Tabord-Meehan, Max , keywords =. Inference in Cluster Randomized Trials with Matched Pairs , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2211.14903 , url =

    2022 doi arXiv

  54. [64]

    , title =

    Foos, Florian and de Rooij, Eline A. , title =. American Journal of Political Science , volume =. doi:https://doi.org/10.1111/ajps.12270 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/ajps.12270 , abstract =

    doi

  55. [65]

    The Review of Economics and Statistics , pages =

    Cruces, Guillermo and Tortarolo, Dario and Vazquez-Bare, Gonzalo , title =. The Review of Economics and Statistics , pages =. 2025 , month =. doi:10.1162/rest_a_01552 , url =

    2025 doi

  56. [66]

    Identification and estimation of spillover effects in randomized experiments , journal =

    Gonzalo Vazquez-Bare , keywords =. Identification and estimation of spillover effects in randomized experiments , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.jeconom.2021.10.014 , url =

    2022 doi

  57. [67]

    American Economic Review , Volume =

    Dhar, Diva and Jain, Tarun and Jayachandran, Seema , Title =. American Economic Review , Volume =. 2022 , Month =. doi:10.1257/aer.20201112 , URL =

    2022 doi

  58. [68]

    The Quarterly Journal of Economics , volume =

    Duflo, Esther and Saez, Emmanuel , title = ". The Quarterly Journal of Economics , volume =. 2003 , month =. doi:10.1162/00335530360698432 , url =

    2003 doi

  59. [69]

    and Cristia, Julian and Cueto, Santiago and Malamud, Ofer and Cruz-Aguayo, Yyannu , Title =

    Beuermann, Diether W. and Cristia, Julian and Cueto, Santiago and Malamud, Ofer and Cruz-Aguayo, Yyannu , Title =. American Economic Journal: Applied Economics , Volume =. 2015 , Month =. doi:10.1257/app.20130267 , URL =

    2015 doi

  60. [70]

    Deterring or Displacing Electoral Irregularities? Spillover Effects of Observers in a Randomized Field Experiment in Ghana , urldate =

    Nahomi Ichino and Matthias Schündeln , journal =. Deterring or Displacing Electoral Irregularities? Spillover Effects of Observers in a Randomized Field Experiment in Ghana , urldate =

  61. [71]

    Direct and spillover effects of agricultural technology adoption programs: Experimental evidence from the Dominican Republic , type =

    Juli\'. Direct and spillover effects of agricultural technology adoption programs: Experimental evidence from the Dominican Republic , type =. doi:10.18235/0001742 , keywords =

    doi

  62. [72]

    http://www.nber.org/papers/w25627

    Haushofer, Johannes and Ringdal, Charlotte and Shapiro, Jeremy P and Wang, Xiao Yu. Income Changes and Intimate Partner Violence: Evidence from Unconditional Cash Transfers in Kenya. 2019. doi:10.3386/w25627 , URL = "http://www.nber.org/papers/w25627", abstract =

    2019 doi

  63. [73]

    The Quarterly Journal of Economics , volume =

    Haushofer, Johannes and Shapiro, Jeremy , title = ". The Quarterly Journal of Economics , volume =. 2016 , month =. doi:10.1093/qje/qjw025 , url =

    2016 doi

  64. [74]

    American Economic Journal: Applied Economics , Volume =

    Hidrobo, Melissa and Peterman, Amber and Heise, Lori , Title =. American Economic Journal: Applied Economics , Volume =. 2016 , Month =. doi:10.1257/app.20150048 , URL =

    2016 doi

  65. [75]

    http://www.nber.org/papers/w29576

    Malani, Anup and Holtzman, Phoebe and Imai, Kosuke and Kinnan, Cynthia and Miller, Morgen and Swaminathan, Shailender and Voena, Alessandra and Woda, Bartosz and Conti, Gabriella. Effect of Health Insurance in India: A Randomized Controlled Trial. 2021. doi:10.3386/w29576 , UR...

    2021 doi

  66. [76]

    The Quarterly Journal of Economics , volume =

    Muralidharan, Karthik and Sundararaman, Venkatesh , title = ". The Quarterly Journal of Economics , volume =. 2015 , month =. doi:10.1093/qje/qjv013 , url =

    2015 doi

  67. [77]

    American Economic Journal: Economic Policy , Volume =

    Banerjee, Abhijit and Chattopadhyay, Raghabendra and Duflo, Esther and Keniston, Daniel and Singh, Nina , Title =. American Economic Journal: Economic Policy , Volume =. 2021 , Month =. doi:10.1257/pol.20190664 , URL =

    2021 doi

  68. [78]

    Biometrics , volume =

    Jiang, Zhichao and Imai, Kosuke and Malani, Anup , title =. Biometrics , volume =. 2022 , keywords =. doi:https://doi.org/10.1111/biom.13782 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/biom.13782 , abstract =

    2022 doi

  69. [79]

    Kinnan, Cynthia and Malani, Anup and Voena, Alessandra and Conti, Gabriella and Imai, Kosuke , title =

  70. [80]

    Nature Human Behaviour , year=

    Rogers, Todd and Feller, Avi , title=. Nature Human Behaviour , year=. doi:10.1038/s41562-018-0328-1 , url=

    doi

  71. [81]

    Journal of Public Economics , author =

    Experimental evidence on scaling up education reforms in. Journal of Public Economics , author =. 2018 , keywords =. doi:10.1016/j.jpubeco.2018.08.007 , abstract =

    2018 doi

  72. [82]

    Inducing positive sorting through performance pay:

    Brown, Christina and Andrabi, Tahir , year =. Inducing positive sorting through performance pay:

  73. [83]

    Microenterprise growth and the flypaper effect: Evidence from a randomized experiment in Ghana , journal =

    Marcel Fafchamps and David McKenzie and Simon Quinn and Christopher Woodruff , keywords =. Microenterprise growth and the flypaper effect: Evidence from a randomized experiment in Ghana , journal =. 2014 , issn =. doi:https://doi.org/10.1016/j.jdeveco.2013.09.010 , url =

    2014 doi

  74. [84]

    American Economic Journal: Applied Economics , Volume =

    de Mel, Suresh and McKenzie, David and Woodruff, Christopher , Title =. American Economic Journal: Applied Economics , Volume =. 2013 , Month =. doi:10.1257/app.5.2.122 , URL =

    2013 doi

  75. [85]

    American Economic Journal: Applied Economics , Volume =

    McKenzie, David and Puerto, Susana , Title =. American Economic Journal: Applied Economics , Volume =. 2021 , Month =. doi:10.1257/app.20180340 , URL =

    2021 doi

  76. [86]

    Designing Representative and Balanced Experiments by Local Randomization , publisher =

    Cytrynbaum, Max , keywords =. Designing Representative and Balanced Experiments by Local Randomization , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2111.08157 , url =

    2021 doi arXiv

  77. [87]

    2023 , eprint=

    Optimal Stratification of Survey Experiments , author=. 2023 , eprint=

    2023

  78. [88]

    Jiang, Liang and Phillips, Peter C. B. and Tao, Yubo and Zhang, Yichong , keywords =. Regression-Adjusted Estimation of Quantile Treatment Effects under Covariate-Adaptive Randomizations , publisher =. 2021 , copyright =. doi:10.48550/ARXIV.2105.14752 , url =

    2021 doi arXiv

  79. [89]

    and Tang, Haihan and Zhang, Yichong , keywords =

    Jiang, Liang and Linton, Oliver B. and Tang, Haihan and Zhang, Yichong , keywords =. Improving Estimation Efficiency via Regression-Adjustment in Covariate-Adaptive Randomizations with Imperfect Compliance , publisher =. 2022 , copyright =. doi:10.48550/ARXIV.2201.13004 , url =

    2022 doi arXiv

  80. [90]

    and Shaikh, Azeem M

    Bai, Yuehao and Jiang, Liang and Romano, Joseph P. and Shaikh, Azeem M. and Zhang, Yichong , keywords =. Covariate Adjustment in Experiments with Matched Pairs , publisher =. 2023 , copyright =. doi:10.48550/ARXIV.2302.04380 , url =

    2023 doi arXiv

  81. [91]

    American Economic Journal: Applied Economics , Volume =

    Bruhn, Miriam and McKenzie, David , Title =. American Economic Journal: Applied Economics , Volume =. 2009 , Month =. doi:10.1257/app.1.4.200 , URL =

    2009 doi

  82. [92]

    Handbook of economic field experiments , volume=

    The econometrics of randomized experiments , author=. Handbook of economic field experiments , volume=. 2017 , publisher=

    2017

  83. [93]

    2022 , eprint=

    At What Level Should One Cluster Standard Errors in Paired and Small-Strata Experiments? , author=. 2022 , eprint=

    2022

  84. [94]

    and Canay, Ivan A

    Bugni, Federico A. and Canay, Ivan A. and Shaikh, Azeem M. , title =. Quantitative Economics , volume =. doi:https://doi.org/10.3982/QE1150 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/QE1150 , abstract =

    doi

  85. [95]

    Journal of the American Statistical Association , volume =

    Michael E Sobel , title =. Journal of the American Statistical Association , volume =. 2006 , publisher =. doi:10.1198/016214506000000636 , URL =

    2006 doi

  86. [96]

    Econometrica , volume =

    Miguel, Edward and Kremer, Michael , title =. Econometrica , volume =. doi:https://doi.org/10.1111/j.1468-0262.2004.00481.x , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1468-0262.2004.00481.x , abstract =

    2004 doi

  87. [97]

    Rubin , title =

    Kari Lock Morgan and Donald B. Rubin , title =. The Annals of Statistics , number =. 2012 , doi =

    2012

  88. [98]

    Quantitative Economics , volume =

    Cytrynbaum, Max , title =. Quantitative Economics , volume =. doi:https://doi.org/10.3982/QE2475 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.3982/QE2475 , year =

    doi

  89. [99]

    2024 , eprint=

    On the Efficiency of Finely Stratified Experiments , author=. 2024 , eprint=

    2024

  90. [100]

    Journal of Applied Econometrics , volume =

    Bai, Yuehao and Hsieh, Meng Hsuan and Liu, Jizhou and Tabord-Meehan, Max , title =. Journal of Applied Econometrics , volume =. doi:https://doi.org/10.1002/jae.3025 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/jae.3025 , abstract =

    doi

  91. [101]

    2022 , eprint=

    Rerandomization and covariate adjustment in split-plot designs , author=. 2022 , eprint=

    2022

  92. [102]

    The Annals of Statistics , number =

    Anqi Zhao and Peng Ding , title =. The Annals of Statistics , number =. 2022 , doi =

    2022

  93. [103]

    Aislinn and McIntosh, Craig and Özler, Berk , title = "

    Baird, Sarah and Bohren, J. Aislinn and McIntosh, Craig and Özler, Berk , title = ". The Review of Economics and Statistics , volume =. 2018 , month =. doi:10.1162/rest_a_00716 , url =

    2018 doi

  94. [104]

    Design of randomized experiments to measure social interaction effects , journal =

    Keisuke Hirano and Jinyong Hahn , keywords =. Design of randomized experiments to measure social interaction effects , journal =. 2010 , issn =. doi:https://doi.org/10.1016/j.econlet.2009.09.023 , url =

    2010 doi

  95. [105]

    Annales d'Economie et de Statistique , pages=

    Estimation of the conditional variance in paired experiments , author=. Annales d'Economie et de Statistique , pages=

  96. [106]

    2015 , publisher=

    Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction , author=. 2015 , publisher=

    2015

  97. [107]

    Aronow, P. M. (2012). A general method for detecting interference between units in randomized experiments. Sociological Methods & Research, 41 3--16

    2012

  98. [108]

    , Eckles, D

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

    2018

  99. [109]

    , Bohren, J

    Baird, S. , Bohren, J. A. , McIntosh, C. and \"O zler, B. (2018). Optimal design of experiments in the presence of interference. Review of Economics and Statistics, 100 844--860

    2018

  100. [110]

    , McIntosh, C

    Baird, S. , McIntosh, C. and \"O zler, B. (2011). Cash or condition? evidence from a cash transfer experiment. The Quarterly journal of economics, 126 1709--1753

    2011

  101. [111]

    , Ding, P

    Basse, G. , Ding, P. , Feller, A. and Toulis, P. (2024). Randomization tests for peer effects in group formation experiments. Econometrica, 92 567--590

    2024

  102. [112]

    and Feller, A

    Basse, G. and Feller, A. (2018). Analyzing two-stage experiments in the presence of interference. Journal of the American Statistical Association, 113 41--55

    2018

  103. [113]

    Basse, G. W. , Feller, A. and Toulis, P. (2019). Randomization tests of causal effects under interference . Biometrika, 106 487--494

    2019

  104. [114]

    , Dafoe, A

    Caughey, D. , Dafoe, A. , Li, X. and Miratrix, L. (2023). Randomisation inference beyond the sharp null: bounded null hypotheses and quantiles of individual treatment effects. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85 1471--1491

    2023

  105. [115]

    and Romano, J

    Chung, E. and Romano, J. P. (2013). Exact and asymptotically robust permutation tests . The Annals of Statistics, 41 484 -- 507

    2013

  106. [116]

    , Tortarolo, D

    Cruces, G. , Tortarolo, D. and Vazquez-Bare, G. (2025). Design of partial population experiments with an application to spillovers in tax compliance. The Review of Economics and Statistics 1--45

    2025

  107. [117]

    DiCiccio, C. J. and Romano, J. P. (2017). Robust permutation tests for correlation and regression coefficients. Journal of the American Statistical Association, 112 1211--1220

    2017

  108. [118]

    Fisher, R. A. (1953). The Design of Experiments. Oliver and Boyd, Edinburgh

    1953

  109. [119]

    , Airoldi, E

    Forastiere, L. , Airoldi, E. M. and Mealli, F. (2021). Identification and estimation of treatment and interference effects in observational studies on networks. Journal of the American Statistical Association, 116 901--918

    2021

  110. [120]

    Huang, S. , Li, X. and Toulis, P. (2025). Randomization tests for monotone spillover effects. arXiv preprint arXiv:2501.02454

    2025 arXiv

  111. [121]

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

    2008

  112. [122]

    , Jiang, Z

    Imai, K. , Jiang, Z. and Malani, A. (2021). Causal inference with interference and noncompliance in two-stage randomized experiments. Journal of the American Statistical Association, 116 632--644

    2021

  113. [123]

    Imbens, G. W. and Rubin, D. B. (2015). Causal inference in statistics, social, and biomedical sciences. Cambridge university press

    2015

  114. [124]

    Liu, J. (2026). Inference for two-stage experiments under covariate-adaptive randomization. Journal of Econometrics, 253 106189

    2026

  115. [125]

    , Shaikh, A

    Liu, J. , Shaikh, A. M. and Toulis, P. (2026). Randomization inference in two-sided market experiments. 2504.06215 , ://arxiv.org/abs/2504.06215

    2026

  116. [126]

    and Zhong, L

    Liu, J. and Zhong, L. (2026). Randomization tests in switchback experiments. 2602.23257 , ://arxiv.org/abs/2602.23257

    2026

  117. [127]

    , Basse, G

    Puelz, D. , Basse, G. , Feller, A. and Toulis, P. (2021). A Graph-Theoretic Approach to Randomization Tests of Causal Effects under General Interference . Journal of the Royal Statistical Society Series B: Statistical Methodology, 84 174--204

    2021

  118. [128]

    Toulis, P. (2025). Asymptotic validity and finite-sample properties of approximate randomization tests. Biometrika asaf085

    2025

  119. [129]

    and Kao, E

    Toulis, P. and Kao, E. (2013). Estimation of causal peer influence effects. In International conference on machine learning. PMLR, 1489--1497

    2013

  120. [130]

    and Ding, P

    Wu, J. and Ding, P. (2021). Randomization tests for weak null hypotheses in randomized experiments. Journal of the American Statistical Association, 116 1898--1913

    2021

  121. [131]

    and Ding, P

    Zhao, A. and Ding, P. (2021). Covariate-adjusted fisher randomization tests for the average treatment effect. Journal of Econometrics, 225 278--294

    2021

  122. [132]

    Zhong, L. (2024). Unconditional randomization tests for interference. arXiv preprint arXiv:2409.09243

    2024 arXiv

Pith tools

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