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Generalized Fiducial Inference for Logistic Graded Response Models

Published online by Cambridge University Press:  01 January 2025

Yang Liu  and
Jan Hannig
Yang Liu*
Affiliation:
University of California, Merced
Jan Hannig
Affiliation:
The University of North Carolina, Chapel Hill
*
Correspondence should be made to Yang Liu, Psychological Sciences, School of Social Sciences, Humanities, and Arts, University of California, Merced, 5200 North Lake Road, Merced, CA 95343, USA. Email: yliu85@ucmerced.edu
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Abstract

Samejima’s graded response model (GRM) has gained popularity in the analyses of ordinal response data in psychological, educational, and health-related assessment. Obtaining high-quality point and interval estimates for GRM parameters attracts a great deal of attention in the literature. In the current work, we derive generalized fiducial inference (GFI) for a family of multidimensional graded response model, implement a Gibbs sampler to perform fiducial estimation, and compare its finite-sample performance with several commonly used likelihood-based and Bayesian approaches via three simulation studies. It is found that the proposed method is able to yield reliable inference even in the presence of small sample size and extreme generating parameter values, outperforming the other candidate methods under investigation. The use of GFI as a convenient tool to quantify sampling variability in various inferential procedures is illustrated by an empirical data analysis using the patient-reported emotional distress data.

Keywords

generalized fiducial inferenceconfidence intervalMarkov chain Monte CarloBernstein–von Mises theoremitem response theorygraded response modelbifactor model
Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 4 , December 2017 , pp. 1097 - 1125
Copyright
Copyright © 2017 The Psychometric Society

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Footnotes

Electronic supplementary material The online version of this article (doi:10.1007/s11336-017-9554-0) contains supplementary material, which is available to authorized users.

References

Agresti, A. (2002). Categorical data analysis Hoboken, NJ: WileyCrossRefGoogle Scholar
Bickel, P. J., & Doksum, K. A. (2015). Mathematical statistics: Basic ideas and selected topics (2nd ed., Vol. i). Boca Raton, FL: CRC Press.Google Scholar
Birnbaum, A. Lord, F. M. & Novick, M. R. (1968). Some latent train models and their use in inferring an examinee’s ability. Statistical theories of mental test scores Reading, MA: Addison-Wesley 395–479Google Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika 37 (1), 29–51CrossRefGoogle Scholar
Bock, R. D. & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika 46 (4), 443–459CrossRefGoogle Scholar
Bock, R. D. & Lieberman, M. (1970). Fitting a response model for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} dichotomously scored items. Psychometrika 35 (2), 179–197Google Scholar
Bradlow, E. T. Wainer, H. & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika 64 (2), 153–168CrossRefGoogle Scholar
Cai, L. (2008). SEM of another flavour: Two new applications of the supplemented EM algorithm. British Journal of Mathematical and Statistical Psychology 61 (2), 309–329CrossRefGoogle ScholarPubMed
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis-Hastings Robbins-Monro algorithm. Psychometrika 75 (1), 33–57CrossRefGoogle Scholar
Cai, L. (2010). Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis. Journal of Educational and Behavioral Statistics 35 (3), 307–335CrossRefGoogle Scholar
Cai, L. (2010). A two-tier full-information item factor analysis model with applications. Psychometrika 75 (4), 581–612CrossRefGoogle Scholar
Cai, L. Thissen, D. & du Toit, SHC (2011). IRTPRO for windows [Computer software manual] Lincolnwood, IL: Scientific Software InternationalGoogle Scholar
Carpenter, B. Gelman, A. Hoffman, M. Lee, D. Goodrich, B. & Betancourt, M. (2016). et al. Stan: A probabilistic programming language. Journal of Statistical Software 76 (1), 1–32Google Scholar
Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1–29. http://www.jstatsoft.org/v48/i06/.CrossRefGoogle Scholar
Cisewski, J. & Hannig, J. (2012). Generalized fiducial inference for normal linear mixed models. The Annals of Statistics 40 (4), 2102–2127CrossRefGoogle Scholar
Curtis, S. M. (2010). BUGS code for item response theory. Journal of Statistical Software 36 (1), 1–34CrossRefGoogle Scholar
Datta, G. S. & Mukerjee, R. (2004). Probability matching priors: Higher order asymptotics New York: SpringerCrossRefGoogle Scholar
Doucet, A. De Freitas, N. & Gordon, N. (2001). An introduction to sequential Monte Carlo methods New York: SpringerCrossRefGoogle Scholar
Duong, T. (2014). ks: Kernel smoothing [Computer software manual]. R package version 1.9.3. http://CRAN.R-project.org/package=ks.Google Scholar
Edwards, M. C. (2010). A Markov chain Monte Carlo approach to confirmatory item factor analysis. Psychometrika 75 (3), 474–497CrossRefGoogle Scholar
Efron, B. (1998). R. A. Fisher in the 21st century. Statistical Science 13 (2), 95–114Google Scholar
Efron, B., & Tibshirani, R. (1994). An Introduction to the bootstrap. Boca Raton, FL: CRC Press. Retrieved from https://books.google.com/books?id=gLlpIUxRntoC.CrossRefGoogle Scholar
Fisher, R. A. (1930). Inverse probability. Proceedings of the Cambridge Philosophical Society 26, 528–535CrossRefGoogle Scholar
Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to unknown parameters. Proceedings of the Royal Society of London Series A 139 (838), 343–348Google Scholar
Fisher, R. A. (1935). The fiducial argument in statistical inference. Annals of Eugenics 6 (4), 391–398CrossRefGoogle Scholar
Forero, C. G. Maydeu-Olivares, A. & Gallardo-Pujol, D. (2009). Factor analysis with ordinal indicators: A Monte Carlo study comparing DWLS and ULS estimation. Structural Equation Modeling 16 (4), 625–641CrossRefGoogle Scholar
Gelman, A. Jakulin, A. Pittau, M. G. & Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics 2 (4), 1360–1383CrossRefGoogle Scholar
Ghosh, J. & Bickel, P. J. (1990). A decomposition for the likelihood ratio statistic and the bartlett correction: A Bayesian argument. Annals of Statistics 18 (3), 1070–1090Google Scholar
Haberman, S. J. (2006). Adaptive quadrature for item response models. ETS Research Report Series 2006 (2), 1–10CrossRefGoogle Scholar
Haberman, S. J. (2013). A general program for item-response analysis that employs the stabilized newton-raphson algorithm. ETS Research Report Series, 2013(2). doi:10.1002/j.2333-8504.2013.tb02339.x.CrossRefGoogle Scholar
Hamilton, M. (1960). A rating scale for depression. Journal of Neurology, Neurosurgery, and Psychiatry 23 (1), 56–62CrossRefGoogle ScholarPubMed
Hannig, J. (2009). On generalized fiducial inference. Statistica Sinica 19 (2), 491Google Scholar
Hannig, J. (2013). Generalized fiducial inference via discretization. Statistica Sinica 23 (2), 489–514Google Scholar
Hannig, J., Iyer, H., Lai, R. C. S., & Lee, T.C.M. (2015). Generalized fiducial inference: A review (Unpublished manuscript).Google Scholar
Hill, C. D. (2004). Precision of parameter estimates for the graded item response model (Unpublished master’s thesis). The University of North Carolina at Chapel Hill.Google Scholar
Houts, C. R. & Cai, L. (2013). flexMIRT user’s manual version 2: Flexible multilevel multidimensional item analysis and test scoring [Computer software manual] Chapel Hill, NC: Vector Psychometric GroupGoogle Scholar
Irwin, D. E. Stucky, B. Langer, M. M. Thissen, D. DeWitt, E. M. & Lai, J. S. (2010). et al. An item response analysis of the pediatric PROMIS anxiety and depressive symptoms scales. Quality of Life Research 19 (4), 595–607CrossRefGoogle ScholarPubMed
Kieftenbeld, V. & Natesan, P. (2012). Recovery of graded response model parameters: A comparison of marginal maximum likelihood and Markov chain Monte Carlo estimation. Applied Psychological Measurement 36 (5), 399–419CrossRefGoogle Scholar
Lehmann, E. (1999). Elements of large-sample theory. New York, NY: Springer. Retrieved from https://books.google.com/books?id=geIoxvgTXlEC.CrossRefGoogle Scholar
Liu, Y. (2015). Generalized fiducial inference for graded response models. (Doctoral dissertation), Retrieved from ProQuest Dissertations and Theses (Accession No. UNC15157)Google Scholar
Liu, Y. & Hannig, J. (2016). Generalized fiducial inference for binary logistic item response models. Psychometrika 81 (2), 290–324CrossRefGoogle ScholarPubMed
Liu, Y. & Thissen, D. (2014). Comparing score tests and other local dependence diagnostics for the graded response model. British Journal of Mathematical and Statistical Psychology 67 (3), 496–513CrossRefGoogle ScholarPubMed
Meng, X. L. & Schilling, S. (1996). Fitting full-information item factor models and an empirical investigation of bridge sampling. Journal of the American Statistical Association 91 (435), 1254–1267CrossRefGoogle Scholar
Muthén, L. K. & Muthén, B. O. (2012). Mplus user’s guide [Computer software manual] Los Angeles, CA: Muthén & MuthénGoogle Scholar
Pal Majumder, A., & Hannig, J. (2016). Higher order asymptotics of Generalized Fiducial Distribution (Unpublished manuscript).Google Scholar
Plummer, M. (2013a). Jags version 3.4.0 user manual [Computer software manual]. http://sourceforge.net/mcmc-jags/files/Manuals/3.x/.Google Scholar
Plummer, M., (2013b). rjags: Bayesian graphical models using MCMC [Computer software manual]. R package version 3-10. http://CRAN.R-project.org/package=rjags.Google Scholar
Reckase, M. (2009). Multidimensional item response theory New York: SpringerCrossRefGoogle Scholar
Rupp, A. A. Templin, J. & Henson, R. A. (2010). Diagnostic assessment: Theory, methods, and applications New York: GuilfordGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika monograph (Vol. 17). Richmond, VA: Psychometric Society.Google Scholar
Schilling, S. & Bock, R. D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika 70 (3), 533–555Google Scholar
Schweder, T. & Hjort, N. L. (2002). Confidence and likelihood. Scandinavian Journal of Statistics 29 (2), 309–332CrossRefGoogle Scholar
Spiegelhalter, D., Thomas, A., & Best, N. D. L. (2010). OpenBUGS version 3.1.1 user manual. http://www.openbugs.info/.Google Scholar
Thissen, D., & Hill, C. D. (2004). Infinite slope estimates in item response theory. Presentation at the annual meeting of the Psychometric Society, Monterey, CA, June 14–17.Google Scholar
Thissen, D. & Steinberg, L. (1988). Data analysis using item response theory. Psychological Bulletin 104 (3), 385–395CrossRefGoogle Scholar
Thissen, D. Steinberg, L. & Embretson, S. (2010). Using item response theory to disentangle constructs at different levels of generality. Measuring psychological constructs: Advances in model-based approaches Washington, DC: American Psychological Association 123–144CrossRefGoogle Scholar
van der Vaart, A. W. (2000). Asymptotic statistics New York: Cambridge University PressGoogle Scholar
Wand, M. P. & Jones, M. C. (1994). Kernel smoothing London: Chapman and HallCrossRefGoogle Scholar
Wirth, R. & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods 12 (1), 58CrossRefGoogle ScholarPubMed
Xie, M. & Singh, K. (2013). Confidence distribution, the frequentist distribution estimator of a parameter: A review. International Statistical Review 81 (1), 3–39CrossRefGoogle Scholar
Yang, J. S. Hansen, M. & Cai, L. (2012). Characterizing sources of uncertainty in item response theory scale scores. Educational and Psychological Measurement 72 (2), 264–290CrossRefGoogle Scholar
Yuan, K. H. Cheng, Y. & Patton, J. (2014). Information matrices and standard errors for MLEs of item parameters in IRT. Psychometrika 79 (2), 232–254CrossRefGoogle ScholarPubMed
Zabell, S. L. (1992). R. A. Fisher and fiducial argument. Statistical Science 7 (3), 369–387CrossRefGoogle Scholar
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