{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:23:01Z","timestamp":1760145781546,"version":"build-2065373602"},"reference-count":74,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,9,5]],"date-time":"2024-09-05T00:00:00Z","timestamp":1725494400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"national funds through FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, Portugal","award":["UIDB\/00006\/2020"],"award-info":[{"award-number":["UIDB\/00006\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AppliedMath"],"abstract":"<jats:p>The classical tests for combining p-values use suitable statistics T(P1,\u2026,Pn), which are based on the assumption that the observed p-values are genuine, i.e., under null hypotheses, are observations from independent and identically distributed Uniform(0,1) random variables P1,\u2026,Pn. However, the phenomenon known as publication bias, which generally results from the publication of studies that reject null hypotheses of no effect or no difference, can tempt researchers to replicate their experiments, generally no more than once, with the aim of obtaining \u201cbetter\u201d p-values and reporting the smallest of the two observed p-values, to increase the chances of their work being published. However, when such \u201cfake p-values\u201d exist, they tamper with the statistic T(P1,\u2026,Pn) because they are observations from a Beta(1,2) distribution. If present, the right model for the random variables Pk is described as a tilted Uniform distribution, also called a Mendel distribution, since it was underlying Fisher\u2019s critique of Mendel\u2019s work. Therefore, methods for combining genuine p-values are reviewed, and it is shown how quantiles of classical combining test statistics, allowing a small number of fake p-values, can be used to make an informed decision when jointly combining fake (from Two P) and genuine (from not Two P) p-values.<\/jats:p>","DOI":"10.3390\/appliedmath4030060","type":"journal-article","created":{"date-parts":[[2024,9,5]],"date-time":"2024-09-05T05:30:02Z","timestamp":1725514202000},"page":"1128-1142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Two P or Not Two P: Mendel Random Variables in Combining Fake and Genuine p-Values"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9276-7011","authenticated-orcid":false,"given":"M. F\u00e1tima","family":"Brilhante","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica e Estat\u00edstica, Faculdade de Ci\u00eancias e Tecnologia, Universidade dos A\u00e7ores, Rua da M\u00e3e de Deus, 9500-321 Ponta Delgada, Portugal"},{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2903-6993","authenticated-orcid":false,"given":"M. Ivette","family":"Gomes","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Departamento de Estat\u00edstica e Investiga\u00e7\u00e3o Operacional, Faculdade de Ci\u00eancias da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Academia das Ci\u00eancias de Lisboa, Rua da Academia das Ci\u00eancias 19, 1249-122 Lisboa, Portugal"},{"name":"Instituto de Investiga\u00e7\u00e3o Cient\u00edfica Bento da Rocha Cabral, Cal\u00e7ada Bento da Rocha Cabral 14, 1250-012 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3364-0357","authenticated-orcid":false,"given":"Sandra","family":"Mendon\u00e7a","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Departamento de Matem\u00e1tica\u2014FCEE, Campus Universit\u00e1rio da Penteada, Universidade da Madeira, 9020-105 Funchal, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8999-1354","authenticated-orcid":false,"given":"Dinis","family":"Pestana","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Departamento de Estat\u00edstica e Investiga\u00e7\u00e3o Operacional, Faculdade de Ci\u00eancias da Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Instituto de Investiga\u00e7\u00e3o Cient\u00edfica Bento da Rocha Cabral, Cal\u00e7ada Bento da Rocha Cabral 14, 1250-012 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7371-363X","authenticated-orcid":false,"given":"Rui","family":"Santos","sequence":"additional","affiliation":[{"name":"Centro de Estat\u00edstica e Aplica\u00e7\u00f5es, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal"},{"name":"Escola Superior de Tecnologia e Gest\u00e3o, Instituto Polit\u00e9cnico de Leiria, Apartado 4133, 2411-901 Leiria, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1080\/14786440009463897","article-title":"On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling","volume":"50","author":"Pearson","year":"1900","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci."},{"key":"ref_2","first-page":"186","article-title":"An argument for divine providence, taken from the constant regularity observ\u2019d in the births of both sexes","volume":"27","author":"Arbuthnot","year":"1710","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"87","DOI":"10.2307\/2340521","article-title":"On the interpretation of \u03c72 from contingency tables, and the calculation of P","volume":"85","author":"Fisher","year":"1922","journal-title":"J. R. Stat. Soc."},{"key":"ref_4","unstructured":"Fisher, R.A. (1925). Statistical Methods for Research Workers, Oliver and Boyd."},{"key":"ref_5","first-page":"503","article-title":"The arrangement of field experiments","volume":"33","author":"Fisher","year":"1926","journal-title":"J. Minist. Agric."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1111\/j.1469-8986.1996.tb02121.x","article-title":"Effect sizes and p values: What should be reported and what should be replicated?","volume":"33","author":"Greenwald","year":"1996","journal-title":"Psychophysiology"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"171085","DOI":"10.1098\/rsos.171085","article-title":"The reproducibility of research and the misinterpretation of p-values","volume":"4","author":"Colquhoun","year":"2017","journal-title":"R. Soc. Open Sci."},{"key":"ref_8","unstructured":"Tippett, L.H.C. (1931). The Methods of Statistics, Williams & Norgate."},{"key":"ref_9","unstructured":"Fisher, R.A. (1932). Statistical Methods for Research Workers, Oliver and Boyd. [4th ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Brilhante, M.F., Gomes, M.I., Mendon\u00e7a, S., Pestana, D., and Santos, R. (2024). Meta-analysis of genuine and fake p-values. Preprints.","DOI":"10.20944\/preprints202407.1927.v1"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1080\/00031305.2016.1154108","article-title":"The Asa statement on p-values: Context process, and purpose","volume":"70","author":"Wasserstein","year":"2016","journal-title":"Am. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1080\/00031305.2019.1583913","article-title":"Moving to a world beyond \u201cp < 0.05\u201d","volume":"73","author":"Wasserstein","year":"2019","journal-title":"Am. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1002\/sim.6342","article-title":"Statistical methods for dealing with publication bias in meta-analysis","volume":"34","author":"Jin","year":"2015","journal-title":"Stat. Med."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1111\/biom.12817","article-title":"Quantifying publication bias in meta-analysis","volume":"74","author":"Lin","year":"2018","journal-title":"Biometrics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1214\/ss\/1030037958","article-title":"Publication bias in meta-analysis: A Bayesian data-augmentation approach to account for issues exemplified in the passive smoking debate","volume":"12","author":"Givens","year":"1997","journal-title":"Stat. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1080\/00033793600200111","article-title":"Has Mendel\u2019s work been rediscovered?","volume":"1","author":"Fisher","year":"1936","journal-title":"Ann. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/BF00058647","article-title":"Some characterizations of the uniform distribution with applications to random number generation","volume":"44","author":"Deng","year":"1992","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1037\/h0059111","article-title":"A statistical consideration in psychological research","volume":"48","author":"Wilkinson","year":"1951","journal-title":"Psychol. Bull."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1093\/biomet\/73.3.751","article-title":"An improved Bonferroni procedure for multiple tests of significance","volume":"73","author":"Simes","year":"1986","journal-title":"Biometrika"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1080\/00223980.1972.9924813","article-title":"An additive method for combining probability values from independent experiments","volume":"80","author":"Edgington","year":"1972","journal-title":"J. Psychol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1093\/biomet\/25.3-4.379","article-title":"On a method of determining whether a sample of size n supposed to have been drawn from a parent population having a known probability integral has probably been drawn at random","volume":"25","author":"Pearson","year":"1933","journal-title":"Biometrika"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Brilhante, M.F., Gomes, M.I., Mendon\u00e7a, S., Pestana, D., and Pestana, P. (2023). Generalized Beta models and population growth, so many routes to chaos. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020194"},{"key":"ref_23","first-page":"425","article-title":"On a new method of determining \u201cgoodness of fit\u201d","volume":"26","author":"Pearson","year":"1934","journal-title":"Biometrika"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3867","DOI":"10.1214\/09-AOS697","article-title":"Karl Pearson\u2019s meta-analysis revisited","volume":"37","author":"Owen","year":"2009","journal-title":"Ann. Stat."},{"key":"ref_25","unstructured":"Stouffer, S.A., Schuman, E.A., DeVinney, L.C., Star, S., and Williams, R.M. (1949). The American Soldier: Adjustment during Army Life, Princeton University Press."},{"key":"ref_26","unstructured":"Rustagi, J. (1979). The logit method for combining probabilities. Symposium on Optimizing Methods in Statistics, Academic Press."},{"key":"ref_27","first-page":"559","article-title":"Combining independent tests of significance","volume":"49","author":"Birnbaum","year":"1954","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_28","unstructured":"Lidsey, G. (1954). Selected quantitative techniques. Handbook of Social Psychology: Theory and Methods, Addison-Wesley."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1080\/01621459.1971.10482347","article-title":"Asymptotic optimality of Fisher\u2019s method of combining independent tests, I","volume":"66","author":"Littell","year":"1971","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1080\/01621459.1973.10481362","article-title":"Asymptotic optimality of Fisher\u2019s method of combining independent tests, II","volume":"68","author":"Littell","year":"1973","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/j.csda.2003.11.020","article-title":"A systematic comparison of methods for combining p-values from independent tests","volume":"47","author":"Loughin","year":"2004","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hartung, J., Knapp, G., and Sinha, B.K. (2008). Statistical Meta-Analysis with Applications, Wiley.","DOI":"10.1002\/9780470386347"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Kulinskaya, E., Morgenthaler, S., and Staudte, R.G. (2008). Meta Analysis. A Guide to Calibrating and Combining Statistical Evidence, Wiley.","DOI":"10.1002\/9780470985533"},{"key":"ref_34","first-page":"602","article-title":"Generalized p-values in significance testing of hypothesis in the presence of nuisance parameters","volume":"84","author":"Tsui","year":"1989","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis, Springer.","DOI":"10.1007\/978-1-4612-0825-9"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"11","DOI":"10.2307\/2533093","article-title":"The behavior of the p-value when the alternative is true","volume":"53","author":"Hung","year":"1997","journal-title":"Biometrics"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Oliveira, P., Temido, M., Henriques, C., and Vichi, M. (2013). Generalized p-values and random p-values when the alternative to uniformity is a mixture of a Beta(1,2) and uniform. Recent Developments in Modeling and Applications in Statistics, Springer.","DOI":"10.1007\/978-3-642-32419-2"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1214\/10-STS342","article-title":"A statistical model to explain the Mendel-Fisher controversy","volume":"25","author":"Pires","year":"2010","journal-title":"Stat. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Franklin, A., Edwards, A.W., Fairbanks, D.J., and Hartl, D.L. (2008). Ending the Mendel-Fisher Controversy, University of Pittsburgh Press.","DOI":"10.2307\/j.ctv10tq47g"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gomes, M.I., Pestana, D., Sequeira, F., Mendon\u00e7a, S., and Velosa, S. (2009, January 22\u201325). Uniformity of offsprings from uniform and non-uniform parents. Proceedings of the ITI 2009, 31st International Conference on Information Technology Interfaces, Cavtat\/Dubrovnik, Croatia.","DOI":"10.1109\/ITI.2009.5196087"},{"key":"ref_41","unstructured":"Brilhante, M., Pestana, D., and Sequeira, F. (2010, January 21\u201324). Combining p-values and random p-values. Proceedings of the ITI 2010, 32nd International Conference on Information Technology Interfaces, Cavtat\/Dubrovnik, Croatia."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1111\/j.2517-6161.1955.tb00201.x","article-title":"On the weighted combination of significance tests","volume":"17","author":"Good","year":"1955","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0167-7152(92)90282-A","article-title":"On the distribution of the weighted combination of independent probabilities","volume":"15","author":"Bhoj","year":"1992","journal-title":"Stat. Probab. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/j.jspi.2011.08.004","article-title":"Approximated distributions of the weighted sum of correlated chi-squared random variables","volume":"142","author":"Chuang","year":"2012","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.spl.2004.11.028","article-title":"A simple approximation for the distribution of the weighted combination of non-independent or independent probabilities","volume":"73","author":"Hou","year":"2005","journal-title":"Stat. Probab. Lett."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/0266476022000023767","article-title":"Weighted inverse chi-square method for correlated significance tests","volume":"30","author":"Makambi","year":"2003","journal-title":"J. Appl. Stat."},{"key":"ref_47","unstructured":"Yang, T.S. (2012). A New Weighted Combination Procedure. [Master\u2019s Thesis, Fu Jen Catholic University]."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Alves, G., and Yu, Y.K. (2011). Combining independent weighted P-values: Achieving computational stability by a systematic expansion with controllable accuracy. PLoS ONE, 6.","DOI":"10.1371\/journal.pone.0022647"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1111\/j.1467-842X.1961.tb00058.x","article-title":"The combination of probabilities: An application of orthonormal functions","volume":"3","author":"Lancaster","year":"1961","journal-title":"Aust. J. Stat."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"926","DOI":"10.1111\/j.1420-9101.2010.02226.x","article-title":"Is the weighted z-test the best method for combining probabilities from independent tests?","volume":"24","author":"Chen","year":"2011","journal-title":"J. Evol. Biol."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Dai, H., Leeder, J.S., and Cui, Y. (2014). A modified generalized Fisher method for combining probabilities from dependent tests. Front. Genet., 5.","DOI":"10.3389\/fgene.2014.00032"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"7442","DOI":"10.1080\/03610926.2022.2046088","article-title":"Distribution of weighted Lancaster\u2019s statistic for combining independent or dependent P-values, with applications to human genetic studies","volume":"52","author":"Hou","year":"2023","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1159","DOI":"10.1111\/biom.13634","article-title":"The generalized Fisher\u2019s combination and accurate p-value calculation under dependence","volume":"79","author":"Zhang","year":"2022","journal-title":"Biometrics"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v101.i01","article-title":"The poolr package for combining independent and dependent p values","volume":"101","author":"Cinar","year":"2022","journal-title":"J. Stat. Softw."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1016\/j.ajhg.2010.06.009","article-title":"A versatile gene-based test for genome-wide association studies","volume":"87","author":"Liu","year":"2010","journal-title":"Am. J. Hum. Genet."},{"key":"ref_56","first-page":"65","article-title":"A simple sequentially multiple test procedure","volume":"6","author":"Holm","year":"1979","journal-title":"Scand. J. Stat."},{"key":"ref_57","unstructured":"Dewey, M. (2024, September 02). metap: Meta-Analysis of Significance Values, Available online: https:\/\/www.rdocumentation.org\/packages\/metap\/versions\/1.9."},{"key":"ref_58","unstructured":"Cooper, H., and Hedges, L.V. (1994). Combining significance levels. The Handbook of Research Synthesis, Russell Sage Foundation."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1080\/01621459.2018.1554485","article-title":"Cauchy combination test: A powerful test with analytic p-value calculation under arbitrary dependency structures","volume":"115","author":"Liu","year":"2020","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Ham, H., and Park, T. (2022). Combining p-values from various statistical methods for microbiome data. Front. Microbiol., 13.","DOI":"10.3389\/fmicb.2022.990870"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/2529826","article-title":"A method for combining non-independent, one-sided tests of significance","volume":"3","author":"Brown","year":"1975","journal-title":"Biometrics"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00949650802412607","article-title":"Distribution of Fisher\u2019s combination statistic when the tests are dependent","volume":"80","author":"Yang","year":"2010","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/S0167-7152(02)00310-3","article-title":"Combining dependent p-values","volume":"60","author":"Kost","year":"2002","journal-title":"Stat. Probab. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1093\/biostatistics\/kxt045","article-title":"Fisher\u2019s method of combining dependent statistics using generalizations of the gamma distribution with applications to genetic pleiotropic associations","volume":"15","author":"Li","year":"2014","journal-title":"Biostatistics"},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Chen, Z. (2022). Optimal tests for combining p-values. Appl. Sci., 12.","DOI":"10.1038\/s41598-022-07094-7"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1195","DOI":"10.1073\/pnas.1814092116","article-title":"The harmonic mean p-value for combining dependent tests","volume":"116","author":"Wilson","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Chien, L.C. (2020). Combining dependent p-values by gamma distributions. Stat. Appl. Genet. Mol. Biol., 19.","DOI":"10.1515\/sagmb-2019-0057"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1093\/biomet\/asaa027","article-title":"Combining p-values via averaging","volume":"107","author":"Vovk","year":"2020","journal-title":"Biometrika"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1214\/10-STS347","article-title":"Test martingales, Bayes factors and p-values","volume":"26","author":"Shafer","year":"2011","journal-title":"Stat. Sci."},{"key":"ref_70","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnwald, P., De Heide, R., and Koolen, W.M. (2020). Safe testing. Information Theory and Applications Workshop (ITA). J. R. Stat. Soc. Ser. B, 1\u201354.","DOI":"10.1109\/ITA50056.2020.9244948"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1111\/rssa.12647","article-title":"Testing by betting: A strategy for statistical and scientific communication","volume":"184","author":"Shafer","year":"2021","journal-title":"J. R. Stat. Soc. Ser. A (Stat. Soc.)"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1736","DOI":"10.1214\/20-AOS2020","article-title":"E-values: Calibration, combination and applications","volume":"49","author":"Vovk","year":"2021","journal-title":"Ann. Stat."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1214\/21-AOS2109","article-title":"Admissible ways of merging p-values under arbitrary dependence","volume":"50","author":"Vovk","year":"2022","journal-title":"Ann. Stat."},{"key":"ref_74","unstructured":"Vuursteen, L., Szab\u00f3, B., van der Vaart, A., and van Zanten, H. (2023, January 10\u201316). Optimal testing using combined test statistics across independent studies. Proceedings of the Neural Information Processing Systems, New Orleans, LA, USA."}],"container-title":["AppliedMath"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2673-9909\/4\/3\/60\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:49:03Z","timestamp":1760111343000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2673-9909\/4\/3\/60"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,5]]},"references-count":74,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["appliedmath4030060"],"URL":"https:\/\/doi.org\/10.3390\/appliedmath4030060","relation":{},"ISSN":["2673-9909"],"issn-type":[{"type":"electronic","value":"2673-9909"}],"subject":[],"published":{"date-parts":[[2024,9,5]]}}}