In this study, the authors develop a generalized multilevel facets model, which is not only a multilevel and two-parameter generalization of the facets model, but also a multilevel and facet generalization of the generalized partial credit model. Because the new model is formulated within a framework of nonlinear mixed models, no efforts are needed to develop parameter estimation procedures, and existing computer programs can be directly applied. Through simulations, the authors found that the parameters in the generalized multilevel facets model could be recovered fairly well using the SAS NLMIXED procedure. To illustrate applications of the new model, a real data set about ratings of household appliances was analyzed with gender, age, and education level as the Level 2 predictors. Further model generalization is discussed.
Adams, R.J., & Wilson, M.R. (1996). Formulating the Rasch model as a mixed coefficients multinomial logit. In G. Englhard & M. Wilson (Eds.), Objective measurement: Theory into practice (Vol. 3, pp. 143-166). Norwood, NJ: Ablex.
2.
Adams, R.J., Wilson, M., & Wang, W.-C. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21, 1-23.
3.
Adams, R.J., Wilson, M., & Wu, M. (1997). Multilevel item response models: An approach to errors in variables regression. Journal of Educational and Behavioral Statistics, 22, 47-76.
4.
Andersen, E.B. (2004). Latent regression analysis based on the rating scale model. PsychologyScience, 46, 209-226.
5.
Andrich, D. (1978). A rating formulation for ordered response categories . Psychometrika, 43, 561-573.
6.
Baker, F., & Kim, S.-H. (2004). Item response theory: Parameter estimation techniques (2nd ed.). New York: Marcel Dekker.
7.
Béguin, A.A., & Glas, C.A.W. (2001). MCMC estimation and model-fit analysis of multidimensional IRT models. Psychometrika, 66, 541-562.
8.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397-479). Reading, MA: Addison-Wesley.
9.
Bock, R.D., Gibbons, R., & Muraki, E.J. (1988). Full information item factor analysis. Applied Psychological Measurement, 12, 261-280.
10.
Bond, T., & Fox, C. (2001). Applying the Rasch model: Fundamental measurement in the human sciences. Mahwah, NJ: Lawrence Erlbaum.
11.
Chen, C.-T., & Wang, W.-C. (in press). Effects of ignoring item interaction on item parameter estimation and detection of interacting items. Applied Psychological Measurement.
12.
Christensen, K.B., Bjorner, J.B., Kreiner, S., & Petersen, J.H. (2004). Latent regression in loglinear Rasch models. Communications in Statistics: Theory and Methods , 33, 1341-1356.
13.
De Boeck, P., & Wilson, M. (Eds.). (2004). Explanatory item response models: A generalized linear and nonlinear approach. New York: Springer-Verlag.
14.
De Boeck, P., Wilson, M., & Acton, G.S. (2005). A conceptual and psychometric framework for distinguishing categories and dimensions. Psychological Review, 112, 129-158.
15.
Embretson, S. (1980). Multicomponent latent trait models for ability tests. Psychometrika, 45, 479-494.
16.
Embretson, S. (1984). A general latent trait model for response processes . Psychometrika, 49, 175-186.
17.
Embretson, S.E., & Reise, S.P. (2000). Item response theory for psychologists. Mahwah, NJ: Lawrence Erlbaum.
18.
Fischer, G.H. (1973). The linear logistic test model as instrument in educational research. Acta Psychologica, 37, 359-374.
19.
Fischer, G. H., & Molenaar, I. W. (Eds.). (1995). Rasch models: Foundations, recent developments, and applications. New York: Springer.
20.
Fischer, G.H., & Parzer, P. (1991). An extension of the rating scale model with an application to the measurement of treatment effects. Psychometrika , 56, 637-651.
21.
Fischer, G.H., & Pononcy, I. (1994). An extension of the partial credit model with an application to the measurement of change. Psychometrika , 59, 177-192.
22.
Fischer, G.H., & Seliger, E. (1997). Multidimensional linear logistic models for change . In. W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory. (pp. 323-346). New York: Springer-Verlag.
23.
Fox, J.P., & Glas, C.A.W. (2001). Bayesian estimation of a multilevel IRT model using Gibbs sampling. Psychometrika, 66, 271-288.
24.
Hambleton, R.K., & Swaminathan, H. (1985). Item response theory: Principles and applications. Boston: Kluwer-Nijhoff .
25.
Hedeker, D. (1999). MIXNO: A computer program for mixed-effects nominal logistic regression. Journal of Statistical Software, 4, 1-92.
26.
Hedeker, D. (2003). A mixed-effects multinomial logistic regression model. Statistics in Medicine, 22, 1433-1446.
27.
Hedeker, D., & Gibbons, R.D. (1996). MIXOR: A computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine , 49, 157-176.
28.
Hoogland, J.J., & Boomsma, A. (1998). Robustness studies in covariance structure modeling: An overview and a meta-analysis. Sociological Methods and Research, 26, 329-367.
29.
Kelderman, H. (1996). Multidimensional Rasch models for partial-credit scoring. Applied Psychological Measurement, 20, 155-168.
Lord, F.M. (1980). Applications of item response theory to practical testing problems. Hillside, NJ: Lawrence Erlbaum.
32.
Lunz, M.E., Wright, B.D., & Linacre, J.M. (1990). Measuring the impact of judge severity on examination scores. Applied Measurement in Education, 3, 331-345.
33.
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149-174.
34.
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models (2nd ed.). London: Chapman & Hall.
35.
McCulloch, C.E., & Searle, S.R. (2001). Generalized, linear, and mixed models. New York: John Wiley.
36.
McDonald, R.P. (1982). Linear versus nonlinear models in item response theory. Applied Psychological Measurement, 6, 379-396.
37.
McKinley, R.L., & Reckase, M.D. (1983). MAXLOG: A computer program for the estimation of the parameters of a multidimensional logistic model. Behavior Research Methods and Instrumentation, 15, 389-390.
38.
Mislevy, R.J. (1985). Estimation of latent group effects. Journal of the American Statistical Association,80, 993-997.
39.
Mislevy, R.J., & Bock, R.D. (1989). A hierarchical item-response model for educational testing. In R. D. Bock (Ed.), Multilevel analysis of educational data (pp. 57-74). San Diego, CA : Academic Press.
40.
Muraki, E. (1992). A generalized partial credit model: Application of an EM-algorithm. Applied Psychological Measurement, 16, 159-176.
41.
Muraki, E., & Carlson, J.E. (1995). Full-information factor analysis for polytomous item responses. Applied Psychological Measurement, 19, 73-90.
42.
Myford, C.M., & Wolfe, E.W. (2003). Detecting and measuring rater effects using many-facet Rasch measurement: Part IJournal of Applied Measurement, 4, 386-422.
43.
Myford, C.M., & Wolfe, E.W. (2004). Detecting and measuring rater effects using many-facet Rasch measurement: Part II. Journal of Applied Measurement , 5, 189-227.
44.
Patz, R., & Junker, B. (1999). Applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses. Journal of Educational and Behavioral Statistics, 24, 342-366.
45.
Pinheiro, J.C., & Bates, D.M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal ofComputational and Graphical Statistics, 4, 12-35.
46.
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2003). Maximum likelihood estimation of generalized linear models with covariate measurement error. Stata Journal, 3, 385-410.
47.
Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2004). Generalized multilevel structural equation modeling . Psychometrika, 69, 167-190.
48.
Rasbash, J., Browne, W.J., Goldstein, H., Yang, M., Plewis, I., Healy, M., et al. (2000). A user's guide to MLwiN (Version 2.1) [Computer software and manual]. London: Institute of Education, University of London.
49.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Danmarks Paedogogische Institut.
50.
Raudenbush, S.W., Bryk, A.S., Cheong, Y.-F., & Congdon, R. (2004). HLM 6: Hierarchical linear and nonlinear modeling [Computer software and manual]. Lincolnwood, IL: Scientific Software International.
51.
Rijmen, F., Tuerlinckx, F., De Boeck, P., & Kuppens, P. (2003). A nonlinear mixed model framework for item response theory. Psychological Methods, 8, 185-205.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement , 17, 1-100.
54.
SAS Institute. (1999). The NLMIXED procedure [Computer software]. Cary, NC: Author.
55.
Sheu, C.-F., Chen, C.-T., Su, Y.-H., & Wang, W.-C. (2005). Using SAS PROC NLMIXED to fit item response theory models. Behavior Research Methods, 37, 202-218.
56.
Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. Boca Raton, FL: Chapman & Hall/CRC Press.
57.
Smith, E. V., Jr., & Smith, R. M. (Eds.). (2004). Introduction to Rasch measurement. Maple Grove, MN: JAM Press.
58.
Smits, D.J.M., & De Boeck, P. (2003). A componential IRT model for guilt. Multivariate Behavioral Research, 38, 161-188.
59.
Smits D.J.M., De Boeck, P., & Verhelst N.D. (2003). Estimation of the MIRID: A program and a SAS-based approach. Behavior Research Methods, Instruments, and Computers , 35, 537-549.
60.
StataCorp. (2003). Stata statistical software: Release 8.0 [Computer software]. College Station, TX: Author.
61.
Tuerlinckx, F., Rijmen, F., Molenberghs, G., Verbeke, G., Briggs, D., Van den Noortgate, W., et al. (2004). Estimation and software. In P. De Boeck & M. Wilson (Eds.), Explanatory item response models: A generalized linear and nonlinear approach (pp. 343-373). New York: Springer-Verlag.
62.
Van den Noortgate, W., De Boeck, P., & Meulders, M. (2003). Cross-classification multilevel logistic models in psychometrics. Journal of Educational and Behavioral Statistics , 28, 369-386.
63.
van der Linden, W. J., & Hambleton, R. K. (Eds.). (1997). Handbook of modern item response theory. New York: Springer.
64.
Wang, W.-C. (2003). Ranking academic journals of psychology in Taiwan (NSC Research Report No. NSC91-2413-H-194-032). Taipei , Taiwan: National Science Council.
65.
Wang, W.-C., & Wilson, M. (2005). Exploring local item dependence using a random-effects facet model. Applied Psychological Measurement, 29, 296-318.
66.
Wilson, M. (1992). The ordered partition model: An extension of the partial credit model. Applied Psychological Measurement, 16, 309-425.
67.
Wolfinger, R.D., & SAS Institute. (n.d.). Fitting nonlinear mixed models with the new NLMIXED procedure. Retrieved June 6, 2005, from http://support.sas.com/rnd/app/papers/nlmixedsugi.pdf
Yang, C. (2000). A study on the evaluation on the long-life products and its relationship with life styles in Taiwan (NSC Research Report No. NSC89-2411-H224-005-SSS). Taipei, Taiwan: National Science Council.
70.
Yen, W.M. (1984). Effect of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8, 125-145.
71.
Yen, W.M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement , 30, 187-213.
72.
Zwinderman, A.H. (1991). A generalized Rasch model for manifest predictors . Psychometrika, 56, 589—600.
