# zbMATH — the first resource for mathematics

Generalized fiducial inference for logistic graded response models. (English) Zbl 1402.62316
Summary: 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 62P15 Applications of statistics to psychology 62F15 Bayesian inference 62H25 Factor analysis and principal components; correspondence analysis
##### Software:
BUGS; flexMIRT; IRTPRO; JAGS; KernSmooth; ks; mirt; Mplus; OpenBUGS; rjags; Stan
Full Text:
##### References:
 [1] Agresti, A. (2002). Categorical data analysis. Hoboken, NJ: Wiley. · Zbl 1018.62002 [2] Bickel, P. J., & Doksum, K. A. (2015). Mathematical statistics: Basic ideas and selected topics (2nd ed., Vol. i). Boca Raton, FL: CRC Press. · Zbl 1380.62002 [3] Birnbaum, A; Lord, FM (ed.); Novick, MR (ed.), Some latent train models and their use in inferring an examinee’s ability, 395-479, (1968), Reading, MA [4] Bock, RD, Estimating item parameters and latent ability when responses are scored in two or more nominal categories, Psychometrika, 37, 29-51, (1972) · Zbl 0233.62016 [5] Bock, RD; Aitkin, M, Marginal maximum likelihood estimation of item parameters: application of an EM algorithm, Psychometrika, 46, 443-459, (1981) [6] Bock, RD; Lieberman, M, Fitting a response model for $$n$$ dichotomously scored items, Psychometrika, 35, 179-197, (1970) [7] Bradlow, ET; Wainer, H; Wang, X, A Bayesian random effects model for testlets, Psychometrika, 64, 153-168, (1999) · Zbl 1365.62451 [8] Cai, L, SEM of another flavour: two new applications of the supplemented EM algorithm, British Journal of Mathematical and Statistical Psychology, 61, 309-329, (2008) [9] Cai, L, High-dimensional exploratory item factor analysis by a metropolis-Hastings robbins-monro algorithm, Psychometrika, 75, 33-57, (2010) · Zbl 1272.62113 [10] Cai, L, Metropolis-Hastings robbins-monro algorithm for confirmatory item factor analysis, Journal of Educational and Behavioral Statistics, 35, 307-335, (2010) [11] Cai, L, A two-tier full-information item factor analysis model with applications, Psychometrika, 75, 581-612, (2010) · Zbl 1208.62183 [12] Cai, L., Thissen, D., & du Toit, S. H. C. (2011). IRTPRO for windows [Computer software manual]. Lincolnwood, IL: Scientific Software International. [13] Carpenter, B; Gelman, A; Hoffman, M; Lee, D; Goodrich, B; Betancourt, M; etal., Stan: A probabilistic programming language, Journal of Statistical Software, 76, 1-32, (2016) [14] 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/. [15] Cisewski, J; Hannig, J, Generalized fiducial inference for normal linear mixed models, The Annals of Statistics, 40, 2102-2127, (2012) · Zbl 1257.62075 [16] Curtis, SM, BUGS code for item response theory, Journal of Statistical Software, 36, 1-34, (2010) [17] Datta, G. S., & Mukerjee, R. (2004). Probability matching priors: Higher order asymptotics. New York: Springer. · Zbl 1044.62031 [18] Doucet, A., De Freitas, N., & Gordon, N. (2001). An introduction to sequential Monte Carlo methods. New York: Springer. · Zbl 0967.00022 [19] Duong, T. (2014). ks: Kernel smoothing [Computer software manual]. R package version 1.9.3. http://CRAN.R-project.org/package=ks. [20] Edwards, MC, A Markov chain Monte Carlo approach to confirmatory item factor analysis, Psychometrika, 75, 474-497, (2010) · Zbl 1208.62188 [21] Efron, B, R. A. Fisher in the 21st century, Statistical Science, 13, 95-114, (1998) · Zbl 1074.01536 [22] Efron, B., & Tibshirani, R. (1994). An Introduction to the bootstrap. Boca Raton, FL: CRC Press. Retrieved from https://books.google.com/books?id=gLlpIUxRntoC. [23] Fisher, RA, Inverse probability, Proceedings of the Cambridge Philosophical Society, 26, 528-535, (1930) · JFM 56.1083.05 [24] Fisher, RA, The concepts of inverse probability and fiducial probability referring to unknown parameters, Proceedings of the Royal Society of London Series A, 139, 343-348, (1933) · Zbl 0006.17401 [25] Fisher, RA, The fiducial argument in statistical inference, Annals of Eugenics, 6, 391-398, (1935) [26] Forero, CG; Maydeu-Olivares, A; Gallardo-Pujol, D, Factor analysis with ordinal indicators: A Monte Carlo study comparing DWLS and ULS estimation, Structural Equation Modeling, 16, 625-641, (2009) [27] Gelman, A; Jakulin, A; Pittau, MG; Su, Y-S, A weakly informative default prior distribution for logistic and other regression models, The Annals of Applied Statistics, 2, 1360-1383, (2008) · Zbl 1156.62017 [28] Ghosh, J; Bickel, PJ, A decomposition for the likelihood ratio statistic and the bartlett correction: A Bayesian argument, Annals of Statistics, 18, 1070-1090, (1990) · Zbl 0727.62035 [29] Haberman, SJ, Adaptive quadrature for item response models, ETS Research Report Series, 2006, 1-10, (2006) [30] 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. · JFM 56.1083.05 [31] Hamilton, M, A rating scale for depression, Journal of Neurology, Neurosurgery, and Psychiatry, 23, 56-62, (1960) [32] Hannig, J, On generalized fiducial inference, Statistica Sinica, 19, 491, (2009) · Zbl 1168.62004 [33] Hannig, J, Generalized fiducial inference via discretization, Statistica Sinica, 23, 489-514, (2013) · Zbl 1379.62002 [34] Hannig, J., Iyer, H., Lai, R. C. S., & Lee, T.C.M. (2015). Generalized fiducial inference: A review (Unpublished manuscript). [35] 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. [36] 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 Group. [37] Irwin, DE; Stucky, B; Langer, MM; Thissen, D; DeWitt, EM; Lai, JS; etal., An item response analysis of the pediatric PROMIS anxiety and depressive symptoms scales, Quality of Life Research, 19, 595-607, (2010) [38] Kieftenbeld, V; Natesan, P, Recovery of graded response model parameters: A comparison of marginal maximum likelihood and Markov chain Monte Carlo estimation, Applied Psychological Measurement, 36, 399-419, (2012) [39] Lehmann, E. (1999). Elements of large-sample theory. New York, NY: Springer. Retrieved from https://books.google.com/books?id=geIoxvgTXlEC. [40] Liu, Y. (2015). Generalized fiducial inference for graded response models. (Doctoral dissertation), Retrieved from ProQuest Dissertations and Theses (Accession No. UNC15157) [41] Liu, Y; Hannig, J, Generalized fiducial inference for binary logistic item response models, Psychometrika, 81, 290-324, (2016) · Zbl 1345.62155 [42] Liu, Y; Thissen, D, Comparing score tests and other local dependence diagnostics for the graded response model, British Journal of Mathematical and Statistical Psychology, 67, 496-513, (2014) · Zbl 1406.91367 [43] Meng, XL; Schilling, S, Fitting full-information item factor models and an empirical investigation of bridge sampling, Journal of the American Statistical Association, 91, 1254-1267, (1996) · Zbl 0925.62220 [44] Muthén, L. K., & Muthén, B. O. (2012). Mplus user’s guide [Computer software manual]. Los Angeles, CA: Muthén & Muthén. [45] Pal Majumder, A., & Hannig, J. (2016). Higher order asymptotics of Generalized Fiducial Distribution (Unpublished manuscript). · Zbl 1257.62075 [46] Plummer, M. (2013a). Jags version 3.4.0 user manual [Computer software manual]. http://sourceforge.net/mcmc-jags/files/Manuals/3.x/. [47] Plummer, M., (2013b). rjags: Bayesian graphical models using MCMC [Computer software manual]. R package version 3-10. http://CRAN.R-project.org/package=rjags. · Zbl 0233.62016 [48] Reckase, M. (2009). Multidimensional item response theory. New York: Springer. · Zbl 1291.62023 [49] Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic assessment: Theory, methods, and applications. New York: Guilford. [50] Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika monograph (Vol. 17). Richmond, VA: Psychometric Society. [51] Schilling, S; Bock, RD, High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature, Psychometrika, 70, 533-555, (2005) · Zbl 1306.62497 [52] Schweder, T; Hjort, NL, Confidence and likelihood, Scandinavian Journal of Statistics, 29, 309-332, (2002) · Zbl 1017.62026 [53] Spiegelhalter, D., Thomas, A., & Best, N. D. L. (2010). OpenBUGS version 3.1.1 user manual. http://www.openbugs.info/. [54] 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. · Zbl 1017.62026 [55] Thissen, D; Steinberg, L, Data analysis using item response theory, Psychological Bulletin, 104, 385-395, (1988) [56] Thissen, D; Steinberg, L; Embretson, S (ed.), Using item response theory to disentangle constructs at different levels of generality, 123-144, (2010), Washington, DC [57] van der Vaart, A. W. (2000). Asymptotic statistics. New York: Cambridge University Press. [58] Wand, M. P., & Jones, M. C. (1994). Kernel smoothing. London: Chapman and Hall. [59] Wirth, R; Edwards, MC, Item factor analysis: current approaches and future directions, Psychological Methods, 12, 58, (2007) [60] Xie, M; Singh, K, Confidence distribution, the frequentist distribution estimator of a parameter: A review, International Statistical Review, 81, 3-39, (2013) · Zbl 1416.62170 [61] Yang, JS; Hansen, M; Cai, L, Characterizing sources of uncertainty in item response theory scale scores, Educational and Psychological Measurement, 72, 264-290, (2012) [62] Yuan, KH; Cheng, Y; Patton, J, Information matrices and standard errors for MLEs of item parameters in IRT, Psychometrika, 79, 232-254, (2014) · Zbl 1288.62192 [63] Zabell, SL, R. A. Fisher and fiducial argument, Statistical Science, 7, 369-387, (1992) · Zbl 0955.62521
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.