{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:25:09Z","timestamp":1760243109921,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T00:00:00Z","timestamp":1438128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000888","name":"W.M. Keck Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000888","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Statistical modeling is often used to measure the strength of evidence for or against hypotheses about given data. We have previously proposed an information-dynamic framework in support of a properly calibrated measurement scale for statistical evidence, borrowing some mathematics from thermodynamics, and showing how an evidential analogue of the ideal gas equation of state could be used to measure evidence for a one-sided binomial hypothesis comparison (\u201ccoin is fair\u201d vs. \u201ccoin is biased towards heads\u201d). Here we take three important steps forward in generalizing the framework beyond this simple example, albeit still in the context of the binomial model. We: (1) extend the scope of application to other forms of hypothesis comparison; (2) show that doing so requires only the original ideal gas equation plus one simple extension, which has the form of the Van der Waals equation; (3) begin to develop the principles required to resolve a key constant, which enables us to calibrate the measurement scale across applications, and which we find to be related to the familiar statistical concept of degrees of freedom. This paper thus moves our information-dynamic theory substantially closer to the goal of producing a practical, properly calibrated measure of statistical evidence for use in general applications.<\/jats:p>","DOI":"10.3390\/e17085333","type":"journal-article","created":{"date-parts":[[2015,7,29]],"date-time":"2015-07-29T10:25:31Z","timestamp":1438165531000},"page":"5333-5352","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Statistical Evidence Measured on a Properly Calibrated Scale across Nested and Non-nested Hypothesis Comparisons"],"prefix":"10.3390","volume":"17","author":[{"given":"Veronica","family":"Vieland","sequence":"first","affiliation":[{"name":"Battelle Center for Mathematical Medicine, The Research Institute at Nationwide Children's Hospital, 575 Children's Crossroad, Columbus, OH 43215, USA"},{"name":"Departments of Pediatrics and Statistics, The Ohio State University, Columbus, OH 43215, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sang-Cheol","family":"Seok","sequence":"additional","affiliation":[{"name":"Battelle Center for Mathematical Medicine, The Research Institute at Nationwide Children's Hospital, 575 Children's Crossroad, Columbus, OH 43215, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1159\/000093775","article-title":"Thermometers: Something for statistical geneticists to think about","volume":"61","author":"Vieland","year":"2006","journal-title":"Hum. Hered."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1159\/000324838","article-title":"Where\u2019s the Evidence?","volume":"71","author":"Vieland","year":"2011","journal-title":"Hum. Hered."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Vieland, V.J., and Hodge, S.E. (2011). Measurement of Evidence and Evidence of Measurement (Invited Commentary). Stat. Appl. Genet. Mol. Biol., 10.","DOI":"10.2202\/1544-6115.1682"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s12064-013-0180-9","article-title":"Measurement of statistical evidence on an absolute scale following thermodynamic principles","volume":"132","author":"Vieland","year":"2013","journal-title":"Theory Biosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1159\/000367599","article-title":"Evidence, temperature, and the laws of thermodynamics","volume":"78","author":"Vieland","year":"2014","journal-title":"Hum. Hered."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chang, H. (2004). Inventing Temperature: Measurement and Scientific Progress, Oxford University Press.","DOI":"10.1093\/0195171276.001.0001"},{"key":"ref_7","unstructured":"Zhang, Z. (2009). A Law of Likelihood for Composite Hypotheses. arXiv:0901.0463v1."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Taper, M.L., and Lele, S.R. (2004). The Nature of Scientific Evidence: Statistical, Philosophical, and Empirical Considerations, University of Chicago Press.","DOI":"10.7208\/chicago\/9780226789583.001.0001"},{"key":"ref_9","unstructured":"Royall, R. (1997). Statistical Evidence: A likelihood Paradigm, Chapman & Hall."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1080\/01621459.1995.10476572","article-title":"Bayes Factors","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1159\/000330634","article-title":"KELVIN: A software package for rigorous measurement of statistical evidence in human genetics","volume":"72","author":"Vieland","year":"2011","journal-title":"Hum. Hered."},{"key":"ref_12","unstructured":"Kullback, S. (1997). Information Theory and Statistics, Dover."},{"key":"ref_13","unstructured":"Fermi, E. (1956). Thermodynamics, Dover Publications."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1080\/00031305.1973.10479042","article-title":"What are degrees of freedom?","volume":"27","author":"Good","year":"1973","journal-title":"Am. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Sober, E. (2008). Evidence and Evolution: The Logic Behind the Science, Cambridge University Press.","DOI":"10.1017\/CBO9780511806285"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1159\/000094709","article-title":"An alternative foundation for the planning and evaluation of linkage analysis I. Decoupling \u201cerror probabilities\u201d from \u201cmeasures of evidence\u201d","volume":"61","author":"Strug","year":"2006","journal-title":"Hum. Hered."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Krantz, D.H., Luce, R.D., Suppes, P., and Tversky, A. (2007). Foundations of Measurement Volume 1 1971, Dover.","DOI":"10.1016\/B978-0-12-425401-5.50009-X"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1214\/13-BA824","article-title":"Hypothesis assessment and inequalities for Bayes factors and relative belief ratios","volume":"8","author":"Baskurt","year":"2013","journal-title":"Bayesian Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1349","DOI":"10.1080\/01621459.2000.10474346","article-title":"Principal Information Theoretic Approaches","volume":"95","author":"Soofi","year":"2000","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","unstructured":"Hacking, I. (1965). Logic of Statistical Inference, Cambridge University Press."},{"key":"ref_21","unstructured":"Edwards, A. (1992). Likelihood, Johns Hopkins University Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/8\/5333\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:49:53Z","timestamp":1760215793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/8\/5333"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,7,29]]},"references-count":21,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2015,8]]}},"alternative-id":["e17085333"],"URL":"https:\/\/doi.org\/10.3390\/e17085333","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,7,29]]}}}