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Lord–Wingersky Algorithm Version 2.0 for Hierarchical Item Factor Models with Applications in Test Scoring, Scale Alignment, and Model Fit Testing

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  • Li Cai

Abstract

Lord and Wingersky’s (Appl Psychol Meas 8:453–461, 1984 ) recursive algorithm for creating summed score based likelihoods and posteriors has a proven track record in unidimensional item response theory (IRT) applications. Extending the recursive algorithm to handle multidimensionality is relatively simple, especially with fixed quadrature because the recursions can be defined on a grid formed by direct products of quadrature points. However, the increase in computational burden remains exponential in the number of dimensions, making the implementation of the recursive algorithm cumbersome for truly high-dimensional models. In this paper, a dimension reduction method that is specific to the Lord–Wingersky recursions is developed. This method can take advantage of the restrictions implied by hierarchical item factor models, e.g., the bifactor model, the testlet model, or the two-tier model, such that a version of the Lord–Wingersky recursive algorithm can operate on a dramatically reduced set of quadrature points. For instance, in a bifactor model, the dimension of integration is always equal to 2, regardless of the number of factors. The new algorithm not only provides an effective mechanism to produce summed score to IRT scaled score translation tables properly adjusted for residual dependence, but leads to new applications in test scoring, linking, and model fit checking as well. Simulated and empirical examples are used to illustrate the new applications. Copyright The Psychometric Society 2015

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  • Li Cai, 2015. "Lord–Wingersky Algorithm Version 2.0 for Hierarchical Item Factor Models with Applications in Test Scoring, Scale Alignment, and Model Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 535-559, June.
    • Handle: RePEc:spr:psycho:v:80:y:2015:i:2:p:535-559
    DOI: 10.1007/s11336-014-9411-3
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    References listed on IDEAS

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    1. Yiu-Fai Yung & David Thissen & Lori McLeod, 1999. "On the relationship between the higher-order factor model and the hierarchical factor model," Psychometrika, Springer;The Psychometric Society, vol. 64(2), pages 113-128, June.
    2. Li Cai, 2010. "A Two-Tier Full-Information Item Factor Analysis Model with Applications," Psychometrika, Springer;The Psychometric Society, vol. 75(4), pages 581-612, December.
    3. John Ross, 1966. "An empirical study of a logistic mental test model," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 325-340, September.
    4. Karl Holzinger & Frances Swineford, 1937. "The Bi-factor method," Psychometrika, Springer;The Psychometric Society, vol. 2(1), pages 41-54, March.
    5. Robert Gibbons & Donald Hedeker, 1992. "Full-information item bi-factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 57(3), pages 423-436, September.
    6. Michael Edwards, 2010. "A Markov Chain Monte Carlo Approach to Confirmatory Item Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 75(3), pages 474-497, September.
    7. Frank Rijmen & Kristof Vansteelandt & Paul Boeck, 2008. "Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations," Psychometrika, Springer;The Psychometric Society, vol. 73(2), pages 167-182, June.
    8. John Schmid & John Leiman, 1957. "The development of hierarchical factor solutions," Psychometrika, Springer;The Psychometric Society, vol. 22(1), pages 53-61, March.
    9. Li Cai, 2010. "High-dimensional Exploratory Item Factor Analysis by A Metropolis–Hastings Robbins–Monro Algorithm," Psychometrika, Springer;The Psychometric Society, vol. 75(1), pages 33-57, March.
    10. Stephen Schilling & R. Bock, 2005. "High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature," Psychometrika, Springer;The Psychometric Society, vol. 70(3), pages 533-555, September.
    11. Minjeong Jeon & Frank Rijmen & Sophia Rabe-Hesketh, 2013. "Modeling Differential Item Functioning Using a Generalization of the Multiple-Group Bifactor Model," Journal of Educational and Behavioral Statistics, , vol. 38(1), pages 32-60, February.
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    1. Sijia Huang & Li Cai, 2021. "Lord–Wingersky Algorithm Version 2.5 with Applications," Psychometrika, Springer;The Psychometric Society, vol. 86(4), pages 973-993, December.
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