Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-07T18:08:12.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Unifying Differential Item Functioning in Factor Analysis for Categorical Data Under a Discretization of a Normal Variant

Published online by Cambridge University Press:  01 January 2025

Yu-Wei Chang ,
Nan-Jung Hsu  and
Rung-Ching Tsai
Yu-Wei Chang
Affiliation:
Feng Chia University
Nan-Jung Hsu
Affiliation:
National Tsing-Hua University
Rung-Ching Tsai*
Affiliation:
National Taiwan Normal University
*
Correspondence should be made to Rung-Ching Tsai, Department of Mathematics, National Taiwan Normal University, No. 88, Sec. 4, Ting-Zhou Road, Taipei 116, Taiwan. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The multiple-group categorical factor analysis (FA) model and the graded response model (GRM) are commonly used to examine polytomous items for differential item functioning to detect possible measurement bias in educational testing. In this study, the multiple-group categorical factor analysis model (MC-FA) and multiple-group normal-ogive GRM models are unified under the common framework of discretization of a normal variant. We rigorously justify a set of identified parameters and determine possible identifiability constraints necessary to make the parameters just-identified and estimable in the common framework of MC-FA. By doing so, the difference between categorical FA model and normal-ogive GRM is simply the use of two different sets of identifiability constraints, rather than the seeming distinction between categorical FA and GRM. Thus, we compare the performance on DIF assessment between the categorical FA and GRM approaches through simulation studies on the MC-FA models with their corresponding particular sets of identifiability constraints. Our results show that, under the scenarios with varying degrees of DIF for examinees of different ability levels, models with the GRM type of identifiability constraints generally perform better on DIF detection with a higher testing power. General guidelines regarding the choice of just-identified parameterization are also provided for practical use.

Keywords

differential item functioningidentifiabilitydiscretization of a normal variantgraded response models
Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 2 , June 2017 , pp. 382 - 406
Copyright
Copyright © 2017 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.)

Footnotes

We would like to dedicate this paper to Roger Millsap for his great contribution in the topic of measurement invariance. This work is simply another shot of the beautiful object he already showed us, just from a different angle.

References

Asparouhov, T., & Muthén, B. O. (2006). Robust chi square difference testing with mean and variance adjusted test statistics. Mplus web notes: No. 10. Retrieved from http://www.statmodel.com/download/webnotes/webnote10Google Scholar
Bollen, K. A. Bollen, K. A. (1989). The general model, part II: Extensions. Structural equations with latent variables, New York: Wiley 395–448.CrossRefGoogle Scholar
Chang, H., & Mazzeo, J. (1994). The unique correspondence of the item response function and item category response functions in polytomously scored item response models. Psychometrika, 59, 391–404CrossRefGoogle Scholar
Chang, Y-W, & Huang, W-K, & Tsai, R. (2015). Detection using multiple-group categorical CFA with minimum free baseline approach. Journal of Educational Measurement, 52, 181–199CrossRefGoogle Scholar
Cohen, A. S., & Kim, S-H, & Baker, F. B. (1993). Detection of differential item functioning in the graded response model. Applied Psychological Measurement, 17, 335–350CrossRefGoogle Scholar
De Boeck, P., & Wilson, M. De Boeck, P., & Wilson, M. (2004). A framework for item response models. Explanatory item response models: A generalized and nonlinear approach, New York: Springer 3–41CrossRefGoogle Scholar
Elosua, P. (2011). Assessing measurement equivalence in ordered-categorical data. Psicológica, 32, 403–421.Google Scholar
Jöreskog, K. G., & Moustaki, I. (2001). Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347–387CrossRefGoogle ScholarPubMed
Kamata, A., & Bauer, D. J. (2008). A note on the relationship between factor analytic and item response theory models. Structural Equation Modeling, 15, 136–153CrossRefGoogle Scholar
Kim, E. S., & Yoon, M. (2011). Testing measurement invariance: A comparison of multiple-group categorical CFA and IRT. Structural Equation Modeling, 18, 212–228CrossRefGoogle Scholar
Kim, S. H., & Cohen, A. S. (1998). Detection of differential item functioning under the graded response model with the likelihood ratio test. Applied Psychological Measurement, 22, 345–355CrossRefGoogle Scholar
Kim, S. H., & Cohen, A. S. (2002). A comparison of linking and concurrent calibration under the graded response model. Applied Psychological Measurement, 26, 25–41CrossRefGoogle Scholar
Lee, J. (2009). Type I error and power of the mean and covariance structure confirmatory factor analysis for differential item functioning detection: methodological issues and resolutions. Unpublished doctoral thesis, University of Kansas, Kansas, USA.Google Scholar
Lee, S. Y., & Poon, W. Y., & Bentler, P. M. (1989). Simultaneous analysis of multivariate polychoric correlation model in several groups. Psychometrika, 54, 63–73CrossRefGoogle Scholar
Lopez Rivas, G. E., & Stark, S., & Chernyshenko, O. S. (2009). The effects of referent item parameters on differential item functioning detection using the free baseline likelihood ratio test. Applied Psychological Measurement, 33, 251–265CrossRefGoogle Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
McDonald, R. P. van der Linden, W. J., & Hambleton, R. K. (1997). Normal ogive multidimensional model. Handbook of modern item response theory, New York: Springer 257–269CrossRefGoogle Scholar
Meade, A. W., & Lautenschlager, G. J. (2004). A comparison of item response theory and confirmatory factor analytic methodologies for establishing measurement equivalence/invariance. Organizational Research Methods, 7, 361–388CrossRefGoogle Scholar
Millsap, R. E. (2011). Statistical approaches to measurement invariance, New York: Routledge.Google Scholar
Millsap, R. E., & Tein, Y. J. (2004). Assessing factorial invariance in ordered-categorical measures. Multivariate Behavioral Research, 39, 479–515CrossRefGoogle Scholar
Muthén, B. O. (2002). Beyond SEM: General latent variable modeling. Behaviormetrika, 29, 81–117CrossRefGoogle Scholar
Muthén, B. O., du Toit, S. H. C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Unpublished manuscript.Google Scholar
Muthén, L. K., & Muthén, B. O. (1998–2010). Mplus user’s guide (6th ed). Los Angeles, CA: Muthén & Muthén.Google Scholar
Rabe-Hesketh, S., & Skrondal, A. (2001). Parameterization of multivariate random effects models for categorical data. Biometrics, 57, 1256–1264CrossRefGoogle ScholarPubMed
Rensvold, R. B., & Cheung, G. W. Schriesheim, C. A., & Neider, L. L. (2001). Testing for metric invariance using structural equations models: Solving the standardization problem. Equivalence in measurement, research in management, Greenwich, CT: Information Age Publishers 25–50.Google Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 34, (4, Pt. 2), 1–100.CrossRefGoogle Scholar
San Martín, E., & González, J., & Tuerlinckx, F. (2009). Identified parameters, parameters of interest and their relationships. Measurement Interdisciplinary Research and Perspective, 7, 95–103CrossRefGoogle Scholar
San Martín, E., & Rolin, J. M. (2013). Identification of parametric Rasch-type models. Journal of Statistical Planning and Inference, 143, 116–130CrossRefGoogle Scholar
San Martín, E., & Rolin, J. M., & Castro, L. M. (2013). Identification of the 1PL model with guessing parameter: parametric and semi-parametric results. Psychometrika, 78, 341–379CrossRefGoogle ScholarPubMed
Satorra, A. Heijmails, D. D. H., & Pollock, D. S. G., & Satorra, A. (2000). Scaled and adjusted restricted tests in multisample analysis of moment structures. Innovations in multivariate statistical analysis: A Festschrift for Heinz Neudecker, Dordrecht: Kluwer Academic Publishers 233–247CrossRefGoogle Scholar
Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66, 507–514CrossRefGoogle Scholar
Stark, S., & Chernyshenko, O. S., & Drasgow, F. (2006). Detecting differential item functioning with confirmatory factor analysis and item response theory: Toward a unified strategy. Journal of Applied Psychology, 91, 1292–1306CrossRefGoogle Scholar
Stocking, M. L., & Lord, F. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7, 201–210CrossRefGoogle Scholar
Takane, Y., & de Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408CrossRefGoogle Scholar
Wang, W. C., & Yeh, Y. L. (2003). Effects of anchor item methods on differential item functioning detection with the likelihood ratio test. Applied Psychological Measurement, 27, 479–498CrossRefGoogle Scholar
Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12, 58–79CrossRefGoogle ScholarPubMed
Wu, H., & Estabrook, R. (2016). Identification of confirmatory factor analysis models of different levels of invariance for ordered categorical outcomes. Psychometrika, doi: 10.1007/s11336-016-9506-0.CrossRefGoogle ScholarPubMed