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Abstract
Rasch mixture models can be a useful tool when checking the assumption of measurement invariance for a single Rasch model. They provide advantages compared to manifest differential item functioning (DIF) tests when the DIF groups are only weakly correlated with the manifest covariates available. Unlike in single Rasch models, estimation of Rasch mixture models is sensitive to the specification of the ability distribution even when the conditional maximum likelihood approach is used. It is demonstrated in a simulation study how differences in ability can influence the latent classes of a Rasch mixture model. If the aim is only DIF detection, it is not of interest to uncover such ability differences as one is only interested in a latent group structure regarding the item difficulties. To avoid any confounding effect of ability differences (or impact), a new score distribution for the Rasch mixture model is introduced here. It ensures the estimation of the Rasch mixture model to be independent of the ability distribution and thus restricts the mixture to be sensitive to latent structure in the item difficulties only. Its usefulness is demonstrated in a simulation study, and its application is illustrated in a study of verbal aggression.
References
|
Ackerman, T. A. (1992). A didactic explanation of item bias, item impact, and item validity from a multidimensional perspective. Journal of Educational Measurement, 29:67-91. Google Scholar | Crossref | ISI | |
|
Andersen, E. B. (1972). A goodness of fit test for the Rasch model. Psychometrika, 38, 123-140. Google Scholar | Crossref | ISI | |
|
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-573. Google Scholar | Crossref | ISI | |
|
Ankenmann, R. D., Witt, E. A., Dunbar, S. B. (1999). An investigation of the power of the likelihood ratio goodness-of-fit statistic in detecting differential item functioning. Journal of Educational Measurement, 36, 277-300. Google Scholar | Crossref | ISI | |
|
Baghaei, P., Carstensen, C. H. (2013). Fitting the mixed Rasch model to a reading comprehension test: Identifying reader types. Practical Assessment, Research & Evaluation, 18(5), 1-13. Google Scholar | |
|
Cohen, A. S., Bolt, D. M. (2005). A mixture model analysis of differential item functioning. Journal of Educational Measurement, 42, 133-148. Google Scholar | Crossref | ISI | |
|
De Boeck, P., Wilson, M. (Eds.). (2004). Explanatory item response models: A generalized linear and nonlinear approach. New York, NY: Springer-Verlag. Google Scholar | Crossref | |
|
DeMars, C. E. (2010). Type I error inflation for detecting DIF in the presence of impact. Educational and Psychological Measurement, 70, 961-972. Google Scholar | SAGE Journals | ISI | |
|
DeMars, C. E., Lau, A. (2011). Differential item functioning detection with latent classes: How accurately can we detect who is responding differentially? Educational and Psychological Measurement, 71, 597-616. Google Scholar | SAGE Journals | ISI | |
|
Dempster, A., Laird, N., Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society B, 39(1), 1-38. Google Scholar | |
|
Frick, H., Strobl, C., Leisch, F., Zeileis, A. (2012). Flexible Rasch mixture models with package psychomix. Journal of Statistical Software, 48(7), 1-25. Google Scholar | Crossref | ISI | |
|
Gustafsson, J.-E. (1980). Testing and obtaining fit of data in the Rasch model. British Journal of Mathematical and Statistical Psychology, 33, 220. Google Scholar | Crossref | ISI | |
|
Holland, P. W., Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In Wainer, H., Braun, H. I. (Eds.), Test validity (pp. 129-145). Hillsdale, NJ: Lawrence Erlbaum. Google Scholar | |
|
Hong, S., Min, S.-Y. (2007). Mixed Rasch modeling of the self-rating depression scale: Incorporating latent class and Rasch rating scale models. Educational and Psychological Measurement, 67, 280-299. Google Scholar | SAGE Journals | ISI | |
|
Li, F., Cohen, A. S., Kim, S.-H., Cho, S.-J. (2009). Model selection methods for mixture dichotomous IRT models. Applied Psychological Measurement, 33, 353-373. Google Scholar | SAGE Journals | ISI | |
|
Li, Y., Brooks, G. P., Johanson, G. A. (2012). Item discrimination and type 1 error in the detection of differential item functioning. Educational and Psychological Measurement, 72, 847-861. Google Scholar | SAGE Journals | ISI | |
|
Maij-de Meij, A. M., Kelderman, H., van der Flier, H. (2010). Improvement in detection of differential item functioning using a mixture item response theory model. Multivariate Behavioral Research, 45, 975-999. Google Scholar | Crossref | Medline | ISI | |
|
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174. Google Scholar | Crossref | ISI | |
|
McLachlan, G., Peel, D. (2000). Finite mixture models. New York, NY: John Wiley. Google Scholar | Crossref | |
|
Molenaar, I. W. (1995). Estimation of item parameters. In Fischer, G. H., Molenaar, I. W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 39-51). New York, NY: Springer-Verlag. Google Scholar | Crossref | |
|
Nieweglowski, L. (2009). clv: Cluster validation techniques (R package version 0.3-2.1). Retrieved from http://CRAN.R-project.org/package=clv Google Scholar | |
|
Preinerstorfer, D., Formann, A. K. (2011). Parameter recovery and model selection in mixed Rasch models. British Journal of Mathematical and Statistical Psychology, 65, 251-262. Google Scholar | Crossref | Medline | ISI | |
|
R Core Team . (2013). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Google Scholar | |
|
Rand, W. M. (1971). Objective criteria for the evaluation of clustering methods. Journal of the American Statistical Association, 66, 846-850. Google Scholar | Crossref | ISI | |
|
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Chicago, IL: University of Chicago Press. Google Scholar | |
|
Rost, J. (1990). Rasch models in latent classes: An integration of two approaches to item analysis. Applied Psychological Measurement, 14, 271-282. Google Scholar | SAGE Journals | ISI | |
|
Rost, J. (1991). A logistic mixture distribution model for polychotomous item responses. British Journal of Mathematical and Statistical Psychology, 44, 75-92. Google Scholar | Crossref | ISI | |
|
Rost, J., von Davier, M. (1995). Mixture distribution Rasch models. In Fischer, G. H., Molenaar, I. W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 257-268). New York, NY: Springer-Verlag. Google Scholar | Crossref | |
|
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-464. Google Scholar | Crossref | ISI | |
|
Strobl, C., Kopf, J., Zeileis, A. (2013). Rasch trees: A new method for detecting differential item functioning in the Rasch model. Psychometrika. Advance online publication. doi:10.1007/s11336-013-9388-3 Google Scholar | Crossref | Medline | ISI | |
|
von Davier, M., Rost, J. (1995). Polytomous mixed Rasch models. In Fischer, G. H., Molenaar, I. W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 371-379). New York, NY: Springer-Verlag. Google Scholar | Crossref | |
|
Zickar, M. J., Gibby, R. E., Robie, C. (2004). Uncovering faking samples in applicant, incumbent, and experimental data sets: An application of mixed-model item response theory. Organizational Research Methods, 7, 168-190. Google Scholar | SAGE Journals | ISI |
