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.
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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.
Blei, D. M., & Jordan, M. I. (2006). Variational inference for Dirichlet process mixtures. Journal of Bayesian Analysis, 1(1), 121–143.
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.
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443–459.
Bock, R. D., Gibbons, R., & Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12(3), 261–280.
Browne, M. W. (2001). An overview of analytic rotation in exploratory factor analysis. Multivariate Behavioral Research, 36(1), 111–150.
Cagnone, S., & Monari, P. (2013). Latent variable models for ordinal data by using the adaptive quadrature approximation. Computational Statistics, 28, 597–619.
Cai, L. (2010). High-dimensional exploratory item factor analysis by a Metropolis–Hastings Robbins–Monro algorithm. Psychometrika, 75(1), 33–57.
Chalmers, R. P. (2012). mirt: A multidimensional item response theory package for the R environment. Journal of Statistical Software, 48, 1–29.
Chang, H.-H., & Stout, W. (1993). The asymptotic posterior normality of the latent trait in an IRT model. Psychometrika, 58(1), 37–52.
Chen, Y., Li, X., & Zhang, S. (2019). Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis. Psychometrika, 84(1), 124–146.
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.
Cho, A. E., Xiao, J., Wang, C., & Xu, G. (2022). Regularized variational estimation for exploratory item factor analysis. Psychometrika.
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.
Efron, B., & Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, pp. 54–75.
Embretson, S. E., & Reise, S. P. (2013). Item response theory. Psychology Press.
Feuerstahler, L. M., & Waller, N. G. (2014). Estimation of the 4-parameter model with marginal maximum likelihood. Multivariate Behavioral Research, 49(3), 285.
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.
Goldberg, L. R. (1992). The development of markers for the Big-Five factor structure. Psychological Assessment, 4(1), 26–42.
Goldberg, L. R., et al. (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.
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.
Jaakkola, T. S., & Jordan, M. I. (2000). Bayesian parameter estimation via variational methods. Statistics and Computing, 10(1), 25–37.
Kim, J., & Wilson, M. (2020). Polytomous item explanatory item response theory models. Educational and Psychological Measurement, 80(4), 726–755.
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.
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.
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.
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149–174.
McCrae, R. R., Zonderman, A. B., Costa, P. T., 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.
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.
Mullis, I. V. & Martin, M. O. (2017). TIMSS 2019 assessment frameworks. ERIC.
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159–176.
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.
Opper, M., & Archambeau, C. (2009). The variational Gaussian approximation revisited. Neural Computation, 21(3), 786–792.
Ormerod, J. T., & Wand, M. P. (2010). Explaining variational approximations. The American Statistician, 64(2), 140–153.
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.
Reckase, M. D. (2009). Multidimensional item response theory models, in Multidimensional item response theory. Springer.
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.
Schilling, S., & Bock, R. D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70(3), 533–555.
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.
Tisais, M. (2016). One-vs-each approximation to softmax for scalable estimation of probabilities. Neural Information Processing Systems, 29, 4161–4169.
Titterington, D. (2004). Bayesian methods for neural networks and related models. Statistical Science, 19(1), 128–139.
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.
Wang, C. (2015). On latent trait estimation in multidimensional compensatory item response models. Psychometrika, 80(2), 428–449.
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.
Wolfinger, R., & O’connell, M. (1993). Generalized linear mixed models a pseudo-likelihood approach. Journal of Statistical Computation and Simulation, 48(3–4), 233–243.
Yamaguchi, K., & Okada, K. (2020). Variational Bayes inference algorithm for the saturated diagnostic classification model. Psychometrika, 85(4), 973–995.
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.
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.
Acknowledgements
This work is partially supported by IES grant R305D200015 and NSF grants SES-1846747 and SES-2150601.
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Cui, C., Wang, C. & Xu, G. Variational Estimation for Multidimensional Generalized Partial Credit Model. Psychometrika (2024). https://doi.org/10.1007/s11336-024-09955-8
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DOI: https://doi.org/10.1007/s11336-024-09955-8