{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T16:42:33Z","timestamp":1786725753835,"version":"3.56.0"},"reference-count":21,"publisher":"National Academy of Sciences","issue":"33","license":[{"start":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T00:00:00Z","timestamp":1786665600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"name":"The Wellcome Trust, SIr Henry Wellcome Fellowship","award":["222781\/Z\/21\/Z"],"award-info":[{"award-number":["222781\/Z\/21\/Z"]}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,8,18]]},"abstract":"<jats:p>\n                    Reporting comparisons with\n                    <jats:italic toggle=\"yes\">P<\/jats:italic>\n                    \u2265 0.05 as showing \u201cno effect\u201d or \u201cno difference\u201d remains one of the most widespread and problematic misinterpretations in the scientific literature. A statistically nonsignificant result shows only that the data do not provide strong evidence for a difference. This distinction matters because such findings can arise for two very different reasons: Either there is no meaningful difference, or a meaningful difference is present but cannot be detected reliably because of limited sample size or high variability. We highlight equivalence testing as a practical framework for distinguishing between these possibilities. Using the two one-sided tests (TOST) approach, investigators can formally test whether effects large enough to be scientifically, clinically, or practically meaningful can be ruled out. We explain the logic of TOST, show how it distinguishes practical equivalence from inconclusive evidence, and illustrate its use in paired and unpaired comparisons. To support broader adoption, we also introduce a freely available online calculator that enables researchers to perform common equivalence tests without coding.\n                  <\/jats:p>","DOI":"10.1073\/pnas.2611548123","type":"journal-article","created":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T16:25:05Z","timestamp":1786724705000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Beyond \u201cnon-significant\u201d results: Why and how to test for practical equivalence"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0157-4386","authenticated-orcid":false,"given":"Jakub","family":"Tomek","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/052gg0110","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Anatomy, Physiology, and Genetics, University of Oxford","place":["Oxford, United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aaron","family":"Caldwell","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00xcryt71","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Biostatistics, Fay W. Boozman College of Public Health, University of Arkansas for Medical Sciences Northwest","place":["Springdale"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7002-3748","authenticated-orcid":false,"given":"David A.","family":"Eisner","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/027m9bs27","id-type":"ROR","asserted-by":"publisher"}],"name":"Division of Cardiovascular Sciences, University of Manchester","place":["Manchester, United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,8,14]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1136\/bmj.311.7003.485"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1136\/bmj.328.7438.476"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/d41586-019-00857-9"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1198\/000313006X152649"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsos.242167"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1177\/1948550617697177"},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.1177\/2515245918770963"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01068419"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1214\/ss\/1032280304"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.4103\/picr.picr_55_22"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.5334\/irsp.82"},{"key":"e_1_3_4_12_2","doi-asserted-by":"crossref","unstructured":"J. Tomek D. Eisner Basic Statistics for Life Scientists: A Concise Handbook of Essential Techniques (Wiley 2025).","DOI":"10.1002\/9781394284993"},{"key":"e_1_3_4_13_2","doi-asserted-by":"publisher","DOI":"10.1177\/17456916221091565"},{"key":"e_1_3_4_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jesp.2021.104159"},{"key":"e_1_3_4_15_2","doi-asserted-by":"publisher","DOI":"10.1111\/ene.13941"},{"key":"e_1_3_4_16_2","doi-asserted-by":"publisher","DOI":"10.1586\/erp.11.9"},{"key":"e_1_3_4_17_2","unstructured":"FDA FDA guidance for industry: Statistical approaches to establishing bioequivalence. (2001). https:\/\/www.fda.gov\/media\/70958\/download. Accessed 27 May 2026."},{"key":"e_1_3_4_18_2","doi-asserted-by":"publisher","DOI":"10.1111\/sjos.12625"},{"key":"e_1_3_4_19_2","doi-asserted-by":"crossref","unstructured":"A. R. Caldwell Exploring equivalence testing with the updated TOSTER R package. PsyArXiv [Preprint] (2022). https:\/\/doi.org\/10.31234\/OSF.IO\/TY8DE (Accessed 15 January 2026).","DOI":"10.31234\/osf.io\/ty8de"},{"key":"e_1_3_4_20_2","doi-asserted-by":"publisher","DOI":"10.1098\/rsos.220346"},{"key":"e_1_3_4_21_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10654-016-0149-3"}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2611548123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T16:25:11Z","timestamp":1786724711000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2611548123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,14]]},"references-count":21,"journal-issue":{"issue":"33","published-print":{"date-parts":[[2026,8,18]]}},"alternative-id":["10.1073\/pnas.2611548123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2611548123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,14]]},"assertion":[{"value":"2026-08-14","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2611548123"}}