{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,11]],"date-time":"2026-09-11T10:04:09Z","timestamp":1789121049458,"version":"build-2803163510"},"reference-count":47,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"unspecified","delay-in-days":2314,"URL":"https:\/\/www.cambridge.org\/core\/terms"},{"start":{"date-parts":[[2018,9,1]],"date-time":"2018-09-01T00:00:00Z","timestamp":1535760000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Psychometrika"],"published-print":{"date-parts":[[2018,9]]},"abstract":"<jats:p>In comparing characteristics of independent populations, researchers frequently expect a certain structure of the population variances. These expectations can be formulated as hypotheses with equality and\/or inequality constraints on the variances. In this article, we consider the Bayes factor for testing such (in)equality-constrained hypotheses on variances. Application of Bayes factors requires specification of a prior under every hypothesis to be tested. However, specifying subjective priors for variances based on prior information is a difficult task. We therefore consider so-called automatic or default Bayes factors. These methods avoid the need for the user to specify priors by using information from the sample data. We present three automatic Bayes factors for testing variances. The first is a Bayes factor with equal priors on all variances, where the priors are specified automatically using a small share of the information in the sample data. The second is the fractional Bayes factor, where a fraction of the likelihood is used for automatic prior specification. The third is an adjustment of the fractional Bayes factor such that the parsimony of inequality-constrained hypotheses is properly taken into account. The Bayes factors are evaluated by investigating different properties such as information consistency and large sample consistency. Based on this evaluation, it is concluded that the adjusted fractional Bayes factor is generally recommendable for testing equality- and inequality-constrained hypotheses on variances.<\/jats:p>","DOI":"10.1007\/s11336-018-9615-z","type":"journal-article","created":{"date-parts":[[2018,5,3]],"date-time":"2018-05-03T22:32:06Z","timestamp":1525386726000},"page":"586-617","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Automatic Bayes Factors for Testing Equality- and Inequality-Constrained Hypotheses on Variances"],"prefix":"10.1017","volume":"83","author":[{"given":"Florian","family":"B\u00f6ing-Messing","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joris","family":"Mulder","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,1,1]]},"reference":[{"key":"S0033312300006906_CR18","doi-asserted-by":"publisher","DOI":"10.1201\/b11158"},{"key":"S0033312300006906_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmp.2014.09.004"},{"key":"S0033312300006906_CR43","doi-asserted-by":"publisher","DOI":"10.1111\/j.2517-6161.1982.tb01217.x"},{"key":"S0033312300006906_CR1","first-page":"267","volume-title":"2nd international symposium on information theory","author":"Akaike","year":"1973"},{"key":"S0033312300006906_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2009.09.022"},{"key":"S0033312300006906_CR44","doi-asserted-by":"publisher","DOI":"10.1111\/j.2044-8317.2012.02059.x"},{"key":"S0033312300006906_CR20","volume-title":"Theory of probability","author":"Jeffreys","year":"1961"},{"key":"S0033312300006906_CR41","doi-asserted-by":"publisher","DOI":"10.1037\/0894-4105.9.2.157"},{"key":"S0033312300006906_CR25","doi-asserted-by":"publisher","DOI":"10.1198\/016214507000001337"},{"key":"S0033312300006906_CR6","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1999.10474149"},{"key":"S0033312300006906_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/s11336-017-9577-6"},{"key":"S0033312300006906_CR46","doi-asserted-by":"publisher","DOI":"10.2307\/2532947"},{"key":"S0033312300006906_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-011-9295-3"},{"key":"S0033312300006906_CR7","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1996.10476668"},{"key":"S0033312300006906_CR42","volume-title":"Multilevel analysis: An introduction to basic and advanced multilevel modeling","author":"Snijders","year":"2012"},{"key":"S0033312300006906_CR5","doi-asserted-by":"publisher","DOI":"10.1214\/06-BA115"},{"key":"S0033312300006906_CR10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmp.2015.08.001"},{"key":"S0033312300006906_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmp.2016.01.002"},{"key":"S0033312300006906_CR39","doi-asserted-by":"publisher","DOI":"10.1186\/1744-9081-2-30"},{"key":"S0033312300006906_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmp.2009.09.003"},{"key":"S0033312300006906_CR16","first-page":"118","article-title":"Discussion of O\u2019Hagan","volume":"57","author":"Gilks","year":"1995","journal-title":"Journal of the Royal Statistical Society. 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