Hostname: page-component-6898cfcb55-kwdml Total loading time: 0 Render date: 2026-11-11T14:18:02.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Fiducial Inference for Binary Logistic Item 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 at Chapel Hill
*
Correspondence should be made to Yang Liu, School of Social Sciences, Humanities and Arts, University of California, Merced, 5200 North Lake Rd, Merced, CA 95343, USA. Email: yliu85@ucmerced.edu
Get accessRights & Permissions [Opens in a new window]

Abstract

Generalized fiducial inference (GFI) has been proposed as an alternative to likelihood-based and Bayesian inference in mainstream statistics. Confidence intervals (CIs) can be constructed from a fiducial distribution on the parameter space in a fashion similar to those used with a Bayesian posterior distribution. However, no prior distribution needs to be specified, which renders GFI more suitable when no a priori information about model parameters is available. In the current paper, we apply GFI to a family of binary logistic item response theory models, which includes the two-parameter logistic (2PL), bifactor and exploratory item factor models as special cases. Asymptotic properties of the resulting fiducial distribution are discussed. Random draws from the fiducial distribution can be obtained by the proposed Markov chain Monte Carlo sampling algorithm. We investigate the finite-sample performance of our fiducial percentile CI and two commonly used Wald-type CIs associated with maximum likelihood (ML) estimation via Monte Carlo simulation. The use of GFI in high-dimensional exploratory item factor analysis was illustrated by the analysis of a set of the Eysenck Personality Questionnaire data.

Keywords

generalized fiducial inferenceconfidence intervalMarkov chain Monte Carloitem response theorytwo-parameter logistic modelexploratory item factor analysis

Information

Type
Article
Information
Psychometrika , Volume 81 , Issue 2 , June 2016, pp. 290 - 324
Copyright
Copyright © 2016 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Albert, J.H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational and Behavioral Statistics, 17(3), 251–269CrossRefGoogle Scholar
Asparouhov, T., & Muthén, B. (2012). Comparison of computational methods for high dimensional item factor analysis. Unpublished manuscript retrieved from www.statmodel.com.Google Scholar
Benjamini, Y., & Hochberg, Y. (1995). Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of the Royal Statistical Society, Series B: Methodological, 57(1), 289–300CrossRefGoogle Scholar
Benjamini, Y., Yekutieli, D. (2005). False discovery rate-adjusted multiple confidence intervals for selected parameters. Journal of the American Statistical Association, 100(469), 71–81CrossRefGoogle Scholar
Bernaards, C.A., Jennrich, R.I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65, 676–696CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent train models and their use in inferring an examinee’s ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley.Google 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 \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), 179–197CrossRefGoogle Scholar
Browne, M.W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111–150CrossRefGoogle 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
Crawford, C.B., Ferguson, G.A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika, 35(3), 321–332CrossRefGoogle Scholar
Dempster, A.P. (1968). A generalization of bayesian inference. Journal of the Royal Statistical Society. Series B (Methodological), 20(2), 205–247CrossRefGoogle Scholar
Dempster, A.P. (2008). The dempster-shafer calculus for statisticians. International Journal of Approximate Reasoning, 48(2), 365–377CrossRefGoogle Scholar
Edwards, M.C. (2010). A Markov chain Monte Carlo approach to confirmatory item factor analysis. Psychometrika, 75(3), 474–497CrossRefGoogle Scholar
Eysenck, S.B., Eysenck, H.J., Barrett, P. (1985). A revised version of the psychoticism scale. Personality and individual differences, 6(1), 21–29CrossRefGoogle 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
Fraser, D.A.S. (1968). The structure of inference, New York: John Wiley & SonsGoogle Scholar
Ghosh, J.K., Ramamoorthi, R. (2003). Bayesian nonparametrics, New York: SpringerGoogle Scholar
Gunsjö, A. (1994). Faktoranalys av ordinala variabler. Studia statistica Upsaliensia, Stockholm, Sweden: Acta Universitatis UpsaliensisGoogle 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), 1–98CrossRefGoogle Scholar
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
Jennrich, R.I. (1973). Standard errors for obliquely rotated factor loadings. Psychometrika, 38(4), 593–604CrossRefGoogle Scholar
Le Cam, L., Yang, G.L. (2000). Asymptotics in Statistics: Some Basic Concepts. Springer Series in Statistics, New York: Springer-VerlagCrossRefGoogle Scholar
Liu, Y., & Maydeu-Olivares, A. (2013). Identifying the source of misfit in item response theory models. Multivariate Behavioral Research. In press.Google Scholar
Liu, Y., Thissen, D. (2012). Identifying local dependence with a score test statistic based on the bifactor logistic model. Applied Psychological Measurement, 36(8), 670–688CrossRefGoogle Scholar
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, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43(4), 551–560CrossRefGoogle Scholar
Muthén, L. K., & Muthén, B. O. (1998–2012). Mplus User’s Guide. Los Angeles, CA: Muthén & Muthén.Google Scholar
Neale, M.C., Miller, M.B. (1997). The use of likelihood-based confidence intervals in genetic models. Behavior genetics, 27(2), 113–120CrossRefGoogle ScholarPubMed
Patz, R.J., Junker, B.W. (1999). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24(4), 342–366CrossRefGoogle Scholar
Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applies statistician. The Annals of Statistics (pp. 1151–1172).CrossRefGoogle Scholar
Schilling, S., Bock, R.D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70(3), 533–555CrossRefGoogle Scholar
Shafer, G. (1976). A mathematical theory of evidence, Princeton, NJ: Princeton University PressCrossRefGoogle Scholar
Van der Vaart, A. W. (2000). Asymptotic statistics. Cambridge University Press.Google Scholar
Weerahandi, S. (1993). Generalized confidence intervals. Journal of the American Statistical Association, 88(423), 899–905CrossRefGoogle Scholar
Wirth, R., Edwards, M.C. (2007). Item factor analysis: current approaches and future directions. Psychological methods, 12(1), 58CrossRefGoogle ScholarPubMed
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