Relationships among twenty polytomous item response theory (IRT) models (parametric and nonparametric) and eight measurement properties relevant to polytomous IRT are described. Three tables are provided to assist in choice of an appropriate model, and examples demonstrate the use of the models in test construction.
Get full access to this article
View all access options for this article.
References
1.
Agresti, A. (1990). Categorical data analysis. New York: Wiley.
2.
Akkermans, W. (1999). Polytomous item scores and Guttman dependence. British Journal of Mathematical and Statistical Psychology, 52, 39–62.
3.
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.
4.
Cavalini, P. M. (1992). It’s an ill wind that brings no good: Studies on odour annoyance and the dispersion of odour concentrations from industries. Unpublished doctoral dissertation. University of Groningen, Groningen, The Netherlands.
5.
Hemker, B. T. (1996). Unidimensional IRT models for polytomous items with results for Mokken scale analysis. Unpublished doctoral dissertation. Utrecht University, Utrecht, The Netherlands.
6.
Hemker, B. T. , Sijtsma, K., Molenaar, I. W., & Junker, B. W. (1996). Polytomous IRT models and monotone likelihood ratio of the total score. Psychometrika, 61, 679–693.
7.
Hemker, B. T. , Sijtsma, K., Molenaar, I. W., & Junker, B. W. (1997). Stochastic ordering using the latent trait and the sum score in polytomous IRT models. Psychometrika, 62, 331–347.
8.
Hemker, B. T. , van der Ark, L. A., & Sijtsma, K. (in press). On measurement properties of continuation ratio models. Psychometrika.
9.
Jansen, P. G. W. , & Roskam, E. E. (1986). Latent trait models and dichotomization of graded responses. Psychometrika, 51, 69–91.
10.
Junker, B. W. (1998). Some remarks on Scheiblechner’s treatment of ISOP models. Psychometrika, 63, 73–85.
11.
Lehmann, E. L. (1986). Testing statistical hypotheses. (2nd ed.). New York: Wiley.
12.
Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
13.
Mellenbergh, G. J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.
14.
Molenaar, I. W. (1983). Item steps (Heymans Bulletins HB-83-630-EX). Groningen, The Netherlands: University of Groningen.
15.
Molenaar, I. W. (1997). Nonparametric models for polytomous responses. In W. J. van der Linden, & R. K. Hambleton (Eds.), Handbook of modern item response theory. (pp. 369–380). New York: Springer.
16.
Molenaar, I. W. , & Sijtsma, K. (2000). MSP5 for Windows [Software manual]. Groningen, The Netherlands: ProGAMMA.
17.
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–177.
18.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometric Monograph No. 17.
19.
Samejima, F. (1972). A general model for free response data. Psychometric Monograph No. 18.
20.
Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572.
Sijtsma, K. , & Hemker, B. T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200.
23.
Sijtsma, K. , & Hemker, B. T. (2000). A taxonomy of IRT models for ordering persons and items using simple sum scores. Journal of Educational and Behavioral Statistics, 25, 391–415.
24.
Sijtsma, K. , & Junker, B. W. (1996). A survey of theory and methods of invariant item ordering. British Journal of Mathematical and Statistical Psychology, 49, 79–105.
25.
Sijtsma, K. , & van der Ark, L. A. (2001). Progress in NIRT analysis of polytomous item scores: Dilemmas and practical solutions. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 297–318). New York: Springer.
26.
Thissen, D. , & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.
27.
Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical and Statistical Psychology, 43, 39–55.
28.
van der Ark, L. A. (1999). A reference card for the relationships between IRT models for ordered polytomous items and some relevant properties (WORC Paper No. 99.10.02). Tilburg, The Netherlands: Tilburg University, WORC.
29.
Vermunt, J. K. (1997). EM: A general program for the analysis of categorical data. Unpublished manuscript, Tilburg University, Tilburg, The Netherlands. Available from http://www.kub.nl/faculteiten/fsw/organisatie/departementen/mto/software2.html.
30.
Vermunt, J. K. (2001). The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Applied Psychological Measurement, 25, 283–294.
31.
Von Davier, M. (1996). WINMIRA V1.68: A program system for analyses with the Rasch model, with the latent class analysis and with the mixed Rasch model [Software manual]. Kiel, Germany: IPN Institute for Science Education.
