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Automatic Bayes Factors for Testing Equality- and Inequality-Constrained Hypotheses on Variances

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  • Florian Böing-Messing

    (Jheronimus Academy of Data Science
    Tilburg University)

  • Joris Mulder

    (Tilburg University)

Abstract

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.

Suggested Citation

  • Florian Böing-Messing & Joris Mulder, 2018. "Automatic Bayes Factors for Testing Equality- and Inequality-Constrained Hypotheses on Variances," Psychometrika, Springer;The Psychometric Society, vol. 83(3), pages 586-617, September.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:3:d:10.1007_s11336-018-9615-z
    DOI: 10.1007/s11336-018-9615-z
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    References listed on IDEAS

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    1. Jean-Paul Fox & Joris Mulder & Sandip Sinharay, 2017. "Bayes Factor Covariance Testing in Item Response Models," Psychometrika, Springer;The Psychometric Society, vol. 82(4), pages 979-1006, December.
    2. Liang, Feng & Paulo, Rui & Molina, German & Clyde, Merlise A. & Berger, Jim O., 2008. "Mixtures of g Priors for Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 410-423, March.
    3. Raymond J. Carroll, 2003. "Variances Are Not Always Nuisance Parameters," Biometrics, The International Biometric Society, vol. 59(2), pages 211-220, June.
    4. Mulder, Joris & Hoijtink, Herbert & Leeuw, Christiaan de, 2012. "BIEMS: A Fortran 90 Program for Calculating Bayes Factors for Inequality and Equality Constrained Models," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 46(i02).
    5. Mulder, Joris, 2014. "Prior adjusted default Bayes factors for testing (in)equality constrained hypotheses," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 448-463.
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