{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T22:18:35Z","timestamp":1757542715834,"version":"3.40.3"},"publisher-location":"Cham","reference-count":23,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783031159107"},{"type":"electronic","value":"9783031159114"}],"license":[{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,3]],"date-time":"2022-09-03T00:00:00Z","timestamp":1662163200000},"content-version":"vor","delay-in-days":245,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Stratified sampling can be useful in risk-limiting audits (RLAs), for instance, to accommodate heterogeneous voting equipment or laws that mandate jurisdictions draw their audit samples independently. We combine the union-intersection tests in SUITE, the reduction of RLAs to testing whether the means of a collection of lists are all<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\le 1\/2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mo>\u2264<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>of SHANGRLA, and the<jats:italic>nonnegative supermartingale<\/jats:italic>(NNSM) tests in ALPHA to improve the efficiency and flexibility of stratified RLAs. A simple, non-adaptive strategy for combining stratumwise NNSMs decreases the measured risk in the 2018 pilot hybrid audit in Kalamazoo, Michigan, USA by more than an order of magnitude, from 0.037 for SUITE to 0.003 for our method. We give a simple, computationally inexpensive, adaptive rule for deciding which stratum to sample next that reduces audit workload by as much as 74% in examples. We also present NNSM-based tests that are computationally tractable even when there are many strata, illustrated with a simulated audit stratified across California\u2019s 58\u00a0counties.<\/jats:p>","DOI":"10.1007\/978-3-031-15911-4_7","type":"book-chapter","created":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T14:04:29Z","timestamp":1662127469000},"page":"106-121","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Sweeter than SUITE: Supermartingale Stratified Union-Intersection Tests of\u00a0Elections"],"prefix":"10.1007","author":[{"given":"Jacob V.","family":"Spertus","sequence":"first","affiliation":[]},{"given":"Philip B.","family":"Stark","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,3]]},"reference":[{"unstructured":"Appel, A.W. and Stark, P.B.: Evidence-based elections: Create a meaningful paper trail, then audit. Georgetown Law Technol. Rev. 4(2), 523\u2013541 (2020). https:\/\/georgetownlawtechreview.org\/wp-content\/uploads\/2020\/07\/4.2-p523-541-Appel-Stark.pdf","key":"7_CR1"},{"doi-asserted-by":"crossref","unstructured":"Appel, A.W., DeMillo, R.A., Stark, P.B.: Ballot-marking devices cannot assure the will of the voters. Election Law J. Rules Polit. Policy 19(3), 432\u2013450 (2020). https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract_id=3375755","key":"7_CR2","DOI":"10.1089\/elj.2019.0619"},{"unstructured":"Baker, P., Haberman, M.: In torrent of falsehoods, trump claims election is being stolen. The New York Times, November 2020. ISSN 0362\u20134331. https:\/\/www.nytimes.com\/2020\/11\/05\/us\/politics\/trump-presidency.html","key":"7_CR3"},{"unstructured":"Chaitlin, D.: Sidney powell shares 270-page binder of documents buttressing election fraud claims, December 2020. https:\/\/www.washingtonexaminer.com\/news\/sidney-powell-shares-election-fraud-claims. Section: News","key":"7_CR4"},{"doi-asserted-by":"crossref","unstructured":"Hall, J., et al.: Implementing risk-limiting post-election audits in California. In: Proceedings of\u00a02009 Electronic Voting Technology Workshop\/Workshop on Trustworthy Elections (EVT\/WOTE\u00a02009), Montreal, Canada, August 2009. USENIX http:\/\/www.usenix.org\/event\/evtwote09\/tech\/full_papers\/hall.pdf","key":"7_CR5","DOI":"10.2139\/ssrn.1411219"},{"doi-asserted-by":"crossref","unstructured":"Higgins, M., Rivest, R., Stark, P.: Sharper p-values for stratified post-election audits. Stat. Polit. Policy 2(1) (2011). http:\/\/www.bepress.com\/spp\/vol2\/iss1\/7","key":"7_CR6","DOI":"10.2202\/2151-7509.1031"},{"unstructured":"Howard, L., Rivest, R., Stark, P.: A review of robust post-election audits: various methods of risk-limiting audits and Bayesian audits. Technical report, Brennan Center for Justice (2019). https:\/\/www.brennancenter.org\/sites\/default\/files\/2019-11\/2019_011_RLA_Analysis_FINAL_0.pdf","key":"7_CR7"},{"doi-asserted-by":"publisher","unstructured":"Howard, S.R., Ramdas, A., McAuliffe, J., Sekhon, J.: Time-uniform, nonparametric, nonasymptotic confidence sequences. Ann. Stat. 49(2) (2021). https:\/\/doi.org\/10.1214\/20-aos1991","key":"7_CR8","DOI":"10.1214\/20-aos1991"},{"unstructured":"Kahn, C.: Half of republicans say Biden won because of a \u2019Rigged\u2019 election: reuters\/Ipsos poll. Reuters, November 2020. https:\/\/www.reuters.com\/article\/us-usa-election-poll-idUSKBN27Y1AJ","key":"7_CR9"},{"unstructured":"Levine, A.: Donald Trump\u2019s favorite voting machines, September 2020. http:\/\/washingtonmonthly.com\/2020\/09\/23\/donald-trumps-favorite-voting-machines\/","key":"7_CR10"},{"key":"7_CR11","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1007\/978-3-030-00419-4_12","volume-title":"Electronic Voting","author":"K Ottoboni","year":"2018","unstructured":"Ottoboni, K., Stark, P.B., Lindeman, M., McBurnett, N.: Risk-limiting audits by stratified union-intersection tests of elections (SUITE). In: Krimmer, R., Volkamer, M., Cortier, V., Gor\u00e9, R., Hapsara, M., Serd\u00fclt, U., Duenas-Cid, D. (eds.) E-Vote-ID 2018. LNCS, vol. 11143, pp. 174\u2013188. Springer, Cham (2018). https:\/\/doi.org\/10.1007\/978-3-030-00419-4_12"},{"key":"7_CR12","doi-asserted-by":"publisher","DOI":"10.1002\/9780470689516","volume-title":"Permutation Tests for Complex Data: Theory, Applications, and Software","author":"F Pesarin","year":"2010","unstructured":"Pesarin, F., Salmaso, L.: Permutation Tests for Complex Data: Theory, Applications, and Software. Wiley, West Sussex (2010)"},{"doi-asserted-by":"crossref","unstructured":"Stark, P.: Conservative statistical post-election audits. Ann. Appl. Stat. 2, 550\u2013581 (2008). http:\/\/arxiv.org\/abs\/0807.4005","key":"7_CR13","DOI":"10.1214\/08-AOAS161"},{"doi-asserted-by":"crossref","unstructured":"Stark, P.: A sharper discrepancy measure for post-election audits. Ann. Appl. Stat. 2, 982\u2013985 (2008). http:\/\/arxiv.org\/abs\/0811.1697","key":"7_CR14","DOI":"10.1214\/08-AOAS171"},{"key":"7_CR15","doi-asserted-by":"publisher","first-page":"708","DOI":"10.1109\/TIFS.2009.2034210","volume":"4","author":"P Stark","year":"2009","unstructured":"Stark, P.: CAST: canvass audits by sampling and testing. IEEE Trans. Inf. Forensics Secur. Spec. Issue Electron. Voting 4, 708\u2013717 (2009)","journal-title":"IEEE Trans. Inf. Forensics Secur. Spec. Issue Electron. Voting"},{"unstructured":"Stark, P.: Auditing a collection of races simultaneously. Technical report. arXiv.org (2009). http:\/\/arxiv.org\/abs\/0905.1422v1","key":"7_CR16"},{"unstructured":"Stark, P.: Delayed stratification for timely risk-limiting audits. https:\/\/www.stat.berkeley.edu\/~stark\/Preprints\/delayed19.pdf (2019)","key":"7_CR17"},{"key":"7_CR18","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/978-3-030-54455-3_23","volume-title":"Financial Cryptography and Data Security","author":"PB Stark","year":"2020","unstructured":"Stark, P.B.: Sets of half-average nulls generate risk-limiting audits: SHANGRLA. In: Bernhard, M., et al. (eds.) FC 2020. LNCS, vol. 12063, pp. 319\u2013336. Springer, Cham (2020). https:\/\/doi.org\/10.1007\/978-3-030-54455-3_23"},{"doi-asserted-by":"crossref","unstructured":"Stark, P.: ALPHA: audit that learns from previously hand-audited ballots. Annals of Applied Statistics, Conditionally accepted 2022. https:\/\/arxiv.org\/abs\/2201.02707","key":"7_CR19","DOI":"10.1214\/22-AOAS1646"},{"doi-asserted-by":"crossref","unstructured":"Stark, P., Wagner, D.: Evidence-based elections. IEEE Secur. Priv. 10, 33\u201341 (2012). https:\/\/www.stat.berkeley.edu\/~stark\/Preprints\/evidenceVote12.pdf","key":"7_CR20","DOI":"10.1109\/MSP.2012.62"},{"unstructured":"Ville, J.: \u00c9tude critique de la notion de collectif (1939). http:\/\/eudml.org\/doc\/192893","key":"7_CR21"},{"unstructured":"Waudby-Smith, I., Ramdas, A.: Estimating means of bounded random variables by betting (2020). https:\/\/arxiv.org\/abs\/2010.09686","key":"7_CR22"},{"key":"7_CR23","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"124","DOI":"10.1007\/978-3-030-86942-7_9","volume-title":"Electronic Voting","author":"I Waudby-Smith","year":"2021","unstructured":"Waudby-Smith, I., Stark, P.B., Ramdas, A.: RiLACS: risk limiting audits via confidence sequences. In: Krimmer, R., et al. (eds.) E-Vote-ID 2021. LNCS, vol. 12900, pp. 124\u2013139. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-86942-7_9"}],"container-title":["Lecture Notes in Computer Science","Electronic Voting"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-15911-4_7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T23:25:02Z","timestamp":1676676302000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-15911-4_7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022]]},"ISBN":["9783031159107","9783031159114"],"references-count":23,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-15911-4_7","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2022]]},"assertion":[{"value":"3 September 2022","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"E-Vote-ID","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Joint Conference on Electronic Voting","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Bregenz","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Austria","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2022","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"4 October 2022","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"7 October 2022","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"7","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"evoteid2022","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/e-vote-id.org\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Double-blind","order":1,"name":"type","label":"Type","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"EasyChair","order":2,"name":"conference_management_system","label":"Conference Management System","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"30","order":3,"name":"number_of_submissions_sent_for_review","label":"Number of Submissions Sent for Review","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"10","order":4,"name":"number_of_full_papers_accepted","label":"Number of Full Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"0","order":5,"name":"number_of_short_papers_accepted","label":"Number of Short Papers Accepted","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"33% - The value is computed by the equation \"Number of Full Papers Accepted \/ Number of Submissions Sent for Review * 100\" and then rounded to a whole number.","order":6,"name":"acceptance_rate_of_full_papers","label":"Acceptance Rate of Full Papers","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"4","order":7,"name":"average_number_of_reviews_per_paper","label":"Average Number of Reviews per Paper","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"4","order":8,"name":"average_number_of_papers_per_reviewer","label":"Average Number of Papers per Reviewer","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}},{"value":"No","order":9,"name":"external_reviewers_involved","label":"External Reviewers Involved","group":{"name":"ConfEventPeerReviewInformation","label":"Peer Review Information (provided by the conference organizers)"}}]}}