Hostname: page-component-669899f699-rg895 Total loading time: 0 Render date: 2025-04-25T13:07:27.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Variational Estimation for Multidimensional Generalized Partial Credit Model

Published online by Cambridge University Press:  01 January 2025

Chengyu Cui ,
Chun Wang  and
Gongjun Xu Open the ORCID record for Gongjun Xu [Opens in a new window]
Chengyu Cui
Affiliation:
University of Michigan
Chun Wang*
Affiliation:
University of Washington
Gongjun Xu*
Affiliation:
University of Michigan
*
Correspondence should be made to ChunWang, College of Education, University ofWashington, 312 EMiller Hall, 2012 Skagit Lane, Seattle, WA98105, USA. Email: [email protected]
Correspondence should be made to Gongjun Xu, Department of Statistics, University of Michigan, 456 West Hall, 1085 South University, Ann Arbor, MI48109, USA. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Multidimensional item response theory (MIRT) models have generated increasing interest in the psychometrics literature. Efficient approaches for estimating MIRT models with dichotomous responses have been developed, but constructing an equally efficient and robust algorithm for polytomous models has received limited attention. To address this gap, this paper presents a novel Gaussian variational estimation algorithm for the multidimensional generalized partial credit model. The proposed algorithm demonstrates both fast and accurate performance, as illustrated through a series of simulation studies and two real data analyses.

Keywords

marginal maximum likelihood estimationvariational methodmultidimensional item response theoryexpectation-maximization algorithm
Type
Theory & Methods
Information
Psychometrika , Volume 89 , Issue 3 , September 2024 , pp. 929 - 957
Copyright
Copyright © 2024 The Author(s), under exclusive licence to 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

Footnotes

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s11336-024-09955-8.

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

References

Andersson, B., Xin, T. (2021). Estimation of latent regression item response theory models using a second-order Laplace approximation. Journal of Educational and Behavioral Statistics, 46(2), 244–265.CrossRefGoogle Scholar
Blei, D. M., Jordan, M. I. (2006). Variational inference for Dirichlet process mixtures. Journal of Bayesian Analysis, 1(1), 121–143.Google Scholar
Blei, D. M., Kucukelbir, A., McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518), 859–877.CrossRefGoogle Scholar
Bock, R. D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443–459.CrossRefGoogle Scholar
Bock, R. D., Gibbons, R., Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12(3), 261–280.CrossRefGoogle Scholar
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111–150.CrossRefGoogle Scholar
Cagnone, S., Monari, P. (2013). Latent variable models for ordinal data by using the adaptive quadrature approximation. Computational Statistics, 28, 597–619.CrossRefGoogle Scholar
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75(1), 33–57.CrossRefGoogle Scholar
Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48, 1–29.CrossRefGoogle Scholar
Chang, H.-H., Stout, W. (1993). The asymptotic posterior normality of the latent trait in an IRT model. Psychometrika, 58(1), 37–52.CrossRefGoogle Scholar
Chen, Y., Li, X., Zhang, S. (2019). Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis. Psychometrika, 84(1), 124–146.CrossRefGoogle ScholarPubMed
Cho, A. E., Wang, C., Zhang, X., Xu, G. (2021). Gaussian variational estimation for multidimensional item response theory. British Journal of Mathematical and Statistical Psychology, 74, 52–85.CrossRefGoogle ScholarPubMed
Cho, A. E., Xiao, J., Wang, C., & Xu, G. (2022). Regularized variational estimation for exploratory item factor analysis. Psychometrika.Google Scholar
Costa, P. T., McCrae, R. R. (2008). The revised NEO personality inventory (NEO-PI-R). The SAGE Handbook of Personality Theory and Assessment, 2(2), 179–198.Google Scholar
Efron, B., & Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, pp. 54–75.CrossRefGoogle Scholar
Embretson, S. E., & Reise, S. P. (2013). Item response theory. Psychology Press.CrossRefGoogle Scholar
Feuerstahler, L. M., Waller, N. G. (2014). Estimation of the 4-parameter model with marginal maximum likelihood. Multivariate Behavioral Research, 49(3), 285.CrossRefGoogle ScholarPubMed
Fishbein, B., Martin, M. O., Mullis, I. V., Foy, P. (2018). The TIMSS 2019 item equivalence study: Examining mode effects for computer-based assessment and implications for measuring trends. Large-Scale Assessments in Education, 6(1), 1–23.CrossRefGoogle Scholar
Goldberg, L. R. (1992). The development of markers for the Big-Five factor structure. Psychological Assessment, 4(1), 26–42.CrossRefGoogle Scholar
Goldberg, L. R. etal (1999). A broad-bandwidth, public domain, personality inventory measuring the lower-level facets of several five-factor models. Personality Psychology in Europe, 7(1), 7–28.Google Scholar
Hendrickson, A. E., White, P. O. (1964). Promax: A quick method for rotation to oblique simple structure. British Journal of Statistical Psychology, 17(1), 65–70.CrossRefGoogle Scholar
Jaakkola, T. S., Jordan, M. I. (2000). Bayesian parameter estimation via variational methods. Statistics and Computing, 10(1), 25–37.CrossRefGoogle Scholar
Kim, J., Wilson, M. (2020). Polytomous item explanatory item response theory models. Educational and Psychological Measurement, 80(4), 726–755.CrossRefGoogle ScholarPubMed
Lindstrom, M. J., Bates, D. M. (1988). Newton–Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. Journal of the American Statistical Association, 83(404), 1014–1022.Google Scholar
Martin, M. O., Mullis, I. V. (2019). TIMSS 2015: Illustrating advancements in large-scale international assessments. Journal of Educational and Behavioral Statistics, 44(6), 752–781.CrossRefGoogle Scholar
Martin, M. O., von Davier, M., & Mullis, I. V. (2020). Methods and procedures: TIMSS 2019 technical report. International Association for the Evaluation of Educational Achievement.Google Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149–174.CrossRefGoogle Scholar
McCrae, R. R., Zonderman, A. B., Costa, PT Jr, Bond, M. H., Paunonen, S. V. (1996). Evaluating replicability of factors in the revised NEO personality inventory: Confirmatory factor analysis versus procrustes rotation. Journal of Personality and Social Psychology, 70(3), 552–566.CrossRefGoogle Scholar
Meng, X., Xu, G., Zhang, J., Tao, J. (2020). Marginalized maximum a posteriori estimation for the four-parameter logistic model under a mixture modelling framework. British Journal of Mathematical and Statistical Psychology, 73, 51–82.CrossRefGoogle Scholar
Mullis, I. V. & Martin, M. O. (2017). TIMSS 2019 assessment frameworks. ERIC.Google Scholar
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159–176.CrossRefGoogle Scholar
Oka, M., Okada, K. (2023). Scalable Bayesian approach for the DINA Q-matrix estimation combining stochastic optimization and variational inference. Psychometrika, 88(1), 302–331.CrossRefGoogle ScholarPubMed
Opper, M., Archambeau, C. (2009). The variational Gaussian approximation revisited. Neural Computation, 21(3), 786–792.CrossRefGoogle ScholarPubMed
Ormerod, J. T., Wand, M. P. (2010). Explaining variational approximations. The American Statistician, 64(2), 140–153.CrossRefGoogle Scholar
Ormerod, J. T., Wand, M. P. (2012). Gaussian variational approximate inference for generalized linear mixed models. Journal of Computational and Graphical Statistics, 21(1), 2–17.CrossRefGoogle Scholar
Reckase, M. D. (2009). Multidimensional item response theory models, in Multidimensional item response theory. Springer.Google Scholar
Rijmen, F., Jeon, M. (2013). Fitting an item response theory model with random item effects across groups by a variational approximation method. Annals of Operations Research, 206(1), 647–662.CrossRefGoogle Scholar
Schilling, S., Bock, R. D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70(3), 533–555.CrossRefGoogle Scholar
Tian, W., Cai, L., Thissen, D., Xin, T. (2013). Numerical differentiation methods for computing error covariance matrices in item response theory modeling: An evaluation and a new proposal. Educational and Psychological Measurement, 73(3), 412–439.CrossRefGoogle Scholar
Tisais, M. (2016). One-vs-each approximation to softmax for scalable estimation of probabilities. Neural Information Processing Systems, 29, 4161–4169.Google Scholar
Titterington, D. (2004). Bayesian methods for neural networks and related models. Statistical Science, 19(1), 128–139.CrossRefGoogle Scholar
von Davier, M., Sinharay, S. (2010). Stochastic approximation methods for latent regression item response models. Journal of Educational and Behavioral Statistics, 35(2), 174–193.CrossRefGoogle Scholar
Wang, C. (2015). On latent trait estimation in multidimensional compensatory item response models. Psychometrika, 80(2), 428–449.CrossRefGoogle ScholarPubMed
Wiggins, J. S., & Trapnell, P. D. (1997). Personality structure: The return of the Big Five. In Handbook of personality psychology, pp. 737–765. Academic Press.CrossRefGoogle Scholar
Wolfinger, R., O’connell, M. (1993). Generalized linear mixed models a pseudo-likelihood approach. Journal of Statistical Computation and Simulation, 48 3–4233–243.CrossRefGoogle Scholar
Yamaguchi, K., Okada, K. (2020). Variational Bayes inference algorithm for the saturated diagnostic classification model. Psychometrika, 85(4), 973–995.CrossRefGoogle ScholarPubMed
Yao, L., Schwarz, R. D. (2006). A multidimensional partial credit model with associated item and test statistics: An application to mixed-format tests. Applied Psychological Measurement, 30(6), 469–492.CrossRefGoogle Scholar
Zhang, S., Chen, Y., Liu, Y. (2020). An improved stochastic EM algorithm for large-scale full-information item factor analysis. British Journal of Mathematical and Statistical Psychology, 73(1), 44–71.CrossRefGoogle ScholarPubMed
Supplementary material: File

Cui et al. Supplementary material

Cui et al. Supplementary material 1
Download Cui et al. Supplementary material(File)
File 605.2 KB
Supplementary material: File

Cui et al. Supplementary material

Cui et al. Supplementary material 2
Download Cui et al. Supplementary material(File)
File 5.2 KB