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Bayes Factor Covariance Testing in Item Response Models

Author

Listed:
  • Jean-Paul Fox

    (University of Twente)

  • Joris Mulder

    (Tilburg University)

  • Sandip Sinharay

    (Educational Testing Service)

Abstract

Two marginal one-parameter item response theory models are introduced, by integrating out the latent variable or random item parameter. It is shown that both marginal response models are multivariate (probit) models with a compound symmetry covariance structure. Several common hypotheses concerning the underlying covariance structure are evaluated using (fractional) Bayes factor tests. The support for a unidimensional factor (i.e., assumption of local independence) and differential item functioning are evaluated by testing the covariance components. The posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal (Helmert) matrix. This posterior distribution is defined as a shifted-inverse-gamma, thereby introducing a default prior and a balanced prior distribution. Based on that, an MCMC algorithm is described to estimate all model parameters and to compute (fractional) Bayes factor tests. Simulation studies are used to show that the (fractional) Bayes factor tests have good properties for testing the underlying covariance structure of binary response data. The method is illustrated with two real data studies.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:82:y:2017:i:4:d:10.1007_s11336-017-9577-6
    DOI: 10.1007/s11336-017-9577-6
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    References listed on IDEAS

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    1. Klugkist, Irene & Hoijtink, Herbert, 2007. "The Bayes factor for inequality and about equality constrained models," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6367-6379, August.
    2. Benjamin R. Saville & Amy H. Herring, 2009. "Testing Random Effects in the Linear Mixed Model Using Approximate Bayes Factors," Biometrics, The International Biometric Society, vol. 65(2), pages 369-376, June.
    3. Martijn G. De Jong & Jan-Benedict E. M. Steenkamp & Jean-Paul Fox, 2007. "Relaxing Measurement Invariance in Cross-National Consumer Research Using a Hierarchical IRT Model," Journal of Consumer Research, Journal of Consumer Research Inc., vol. 34(2), pages 260-278, June.
    4. Perrakis, Konstantinos & Ntzoufras, Ioannis & Tsionas, Efthymios G., 2014. "On the use of marginal posteriors in marginal likelihood estimation via importance sampling," Computational Statistics & Data Analysis, Elsevier, vol. 77(C), pages 54-69.
    5. Michael J. Daniels, 2002. "Bayesian analysis of covariance matrices and dynamic models for longitudinal data," Biometrika, Biometrika Trust, vol. 89(3), pages 553-566, August.
    6. Bo Cai & David B. Dunson, 2006. "Bayesian Covariance Selection in Generalized Linear Mixed Models," Biometrics, The International Biometric Society, vol. 62(2), pages 446-457, June.
    7. David J. Spiegelhalter & Nicola G. Best & Bradley P. Carlin & Angelika Van Der Linde, 2002. "Bayesian measures of model complexity and fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(4), pages 583-639, October.
    8. Sinharay S. & Stern H.S., 2002. "On the Sensitivity of Bayes Factors to the Prior Distributions," The American Statistician, American Statistical Association, vol. 56, pages 196-201, August.
    9. Paul Boeck, 2008. "Random Item IRT Models," Psychometrika, Springer;The Psychometric Society, vol. 73(4), pages 533-559, December.
    10. 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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    Cited by:

    1. 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.
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    4. Konrad Klotzke & Jean-Paul Fox, 2019. "Modeling Dependence Structures for Response Times in a Bayesian Framework," Psychometrika, Springer;The Psychometric Society, vol. 84(3), pages 649-672, September.

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