{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T11:10:29Z","timestamp":1762081829015,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,22]],"date-time":"2022-12-22T00:00:00Z","timestamp":1671667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The full Bayesian significance test (FBST) for precise hypotheses is a Bayesian alternative to the traditional significance tests based on p-values. The FBST is characterized by the e-value as an evidence index in favor of the null hypothesis (H). An important practical issue for the implementation of the FBST is to establish how small the evidence against H must be in order to decide for its rejection. In this work, we present a method to find a cutoff value for the e-value in the FBST by minimizing the linear combination of the averaged type-I and type-II error probabilities for a given sample size and also for a given dimensionality of the parameter space. Furthermore, we compare our methodology with the results obtained from the test with adaptive significance level, which presents the capital-P P-value as a decision-making evidence measure. For this purpose, the scenario of linear regression models with unknown variance under the Bayesian approach is considered.<\/jats:p>","DOI":"10.3390\/e25010019","type":"journal-article","created":{"date-parts":[[2022,12,23]],"date-time":"2022-12-23T02:09:03Z","timestamp":1671761343000},"page":"19","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Adaptive Significance Levels in Tests for Linear Regression Models: The e-Value and P-Value Cases"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8128-1307","authenticated-orcid":false,"given":"Alejandra E. Pati\u00f1o","family":"Pati\u00f1o Hoyos","sequence":"first","affiliation":[{"name":"Facultad de Ingenier\u00eda, Instituci\u00f3n Universitaria Pascual Bravo, Medell\u00edn 050034, Colombia"},{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade de S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9328-0210","authenticated-orcid":false,"given":"Victor","family":"Fossaluza","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade de S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu\u00eds Gustavo","family":"Esteves","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade de S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6315-7484","authenticated-orcid":false,"given":"Carlos Alberto","family":"de Bragan\u00e7a Pereira","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade de S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.3390\/e1040099","article-title":"Evidence and Credibility: Full Bayesian Significance Test for Precise Hypotheses","volume":"1","author":"Pereira","year":"1999","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1093\/jigpal\/jzm032","article-title":"The Rules of Logic Composition for the Bayesian Epistemic e-Values","volume":"15","author":"Wagner","year":"2007","journal-title":"Logic J. IGPL"},{"key":"ref_3","first-page":"9","article-title":"Cognitive constructivism, eigen-solutions, and sharp statistical hypotheses","volume":"14","author":"Stern","year":"2007","journal-title":"Cybern. Hum. Knowing"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF02595698","article-title":"On the Bayesianity of Pereira-Stern tests","volume":"10","author":"Madruga","year":"2001","journal-title":"TEST"},{"key":"ref_5","unstructured":"Kempthorne, O., and Folks, L. (1971). Probability, Statistics, and Data Analysis, Iowa State University Press."},{"key":"ref_6","first-page":"49","article-title":"The Role of Significance Tests [with Discussion and Reply]","volume":"4","author":"Cox","year":"1978","journal-title":"Scand. J. Stat."},{"key":"ref_7","first-page":"1","article-title":"The Bayesian Approach [with Discussion and Reply]","volume":"5","author":"Lindley","year":"1978","journal-title":"Scand. J. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1111\/j.1365-2125.1982.tb01987.x","article-title":"Statistical significance tests","volume":"14","author":"Cox","year":"1982","journal-title":"Br. J. Clin. Pharmacol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF01083168","article-title":"Bayesian inference given data \u2018significant at\u03b1\u2019: Tests of point hypotheses","volume":"38","author":"Johnstone","year":"1995","journal-title":"Theor. Decis."},{"key":"ref_10","unstructured":"Oliveira, M.C. (2014). Definition of the Level of Significance as a Function of the Sample Size. [Masters Dissertation, Universidade de S\u00e3o Paulo, Instituto de Matem\u00e1tica e Estat\u00edstica, Departamento de Estat\u00edstica]. (In Portuguese)."},{"key":"ref_11","unstructured":"Pereira, C.A.B. (1985). Test of Hypotheses Defined in Spaces of Different Dimensions: Bayesian View and Classical Interpretation. [Ph.D. Thesis, Universidade de S\u00e3o Paulo, Instituto de Matem\u00e1tica e Estat\u015btica, Departamento de Estat\u015btica]. (In Portuguese)."},{"key":"ref_12","unstructured":"DeGroot, M.H. (1986). Probability and Statistics, Addison-Whesley Publishing Company. [2nd ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Pereira, C.A.B., Nakano, E.Y., Fossaluza, V., Esteves, L.G., Gannon, M.A., and Polpo, A. (2017). Hypothesis tests for bernoulli experiments: Ordering the sample space by bayes factors and using adaptive significance levels for decisions. Entropy, 19.","DOI":"10.20944\/preprints201709.0117.v1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1080\/00031305.2018.1518268","article-title":"Blending bayesian and classical tools to define optimal sample-size-dependent significance levels","volume":"73","author":"Gannon","year":"2019","journal-title":"Am. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1093\/genetics\/158.2.875","article-title":"An unconditional exact test for the Hardy-Weinberg equilibrium law: Sample-space ordering using the Bayes factor","volume":"158","author":"Irony","year":"2001","journal-title":"Genetics"},{"key":"ref_16","first-page":"27","article-title":"Bayesian hypothesis test: Using Surface Integrals to Distribute Prior Information among the Hypotheses","volume":"2","author":"Irony","year":"1995","journal-title":"Resenhas Inst. Matem\u00e1tica Estat\u00edstica Univ. Paulo"},{"key":"ref_17","first-page":"159","article-title":"On the concept of P-value","volume":"7","author":"Pereira","year":"1993","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Esteves, L.G., Izbicki, R., Stern, J.M., and Stern, R.B. (2019). Pragmatic Hypotheses in the Evolution of Science. Entropy, 21.","DOI":"10.3390\/e21090883"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Schervish, M.J. (1995). Theory of Statistics, Springer.","DOI":"10.1007\/978-1-4612-4250-5"},{"key":"ref_20","unstructured":"O\u2019Hagan, A., and Forster, J.J. (2004). Kendall\u2019s Advanced Theory of Statistics, Volume 2B: Bayesian Inference, Arnold."},{"key":"ref_21","unstructured":"Box, G.E.P., and Tiao, G.C. (1973). Bayesian Inference in Statistical Analysis, Wiley."},{"key":"ref_22","unstructured":"DeGroot, M.H. (1970). Optimal Statistical Decisions, McGraw-Hill."},{"key":"ref_23","first-page":"1","article-title":"Power of FBST: Standard examples","volume":"57","author":"Madruga","year":"2005","journal-title":"Inst. Interam. Estad\u00edstica. Estad\u00edstic"},{"key":"ref_24","first-page":"317","article-title":"Pragmatic Testing Precise Hypotheses","volume":"22","author":"Berger","year":"1987","journal-title":"Stat. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1002\/(SICI)1099-1255(1998090)13:5<435::AID-JAE506>3.0.CO;2-K","article-title":"Semiparametric estimation and consumer demand","volume":"13","author":"Blundell","year":"1998","journal-title":"J. Appl. Econom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0095-0696(78)90006-2","article-title":"Hedonic Housing Prices and the Demand for Clean Air","volume":"5","author":"Harrison","year":"1978","journal-title":"J. Environ. Econ. Manag."},{"key":"ref_27","unstructured":"Croissant, Y., and Graves, S. (2022, September 26). Ecdat: Data Sets for Econometrics. R Package Version 0.4-2. Available online: https:\/\/CRAN.R-project.org\/package=Ecdat."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Venables, W.N., and Ripley, B.D. (2002). Modern Applied Statistics with S, Springer. [4th ed.]. Available online: https:\/\/www.stats.ox.ac.uk\/pub\/MASS4\/.","DOI":"10.1007\/978-0-387-21706-2"},{"key":"ref_29","unstructured":"R Core Team (2022). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: https:\/\/www.R-project.org\/."},{"key":"ref_30","unstructured":"Mardia, K.V., Kent, J.T., and Bibby, J.M. (1979). Multivariate Analysis, Academic Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:47:50Z","timestamp":1760147270000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,22]]},"references-count":30,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010019"],"URL":"https:\/\/doi.org\/10.3390\/e25010019","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,12,22]]}}}