Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models

It is often considered desirable to have the same ordering of the items by difficulty across different levels of the trait or ability. Such an ordering is an invariant item ordering (IIO). An IIO facilitates the interpretation of test results. For dichotomously scored items, earlier research surveyed the theory and methods of an invariant ordering in a nonparametric IRT context. Here the focus is on polytomously scored items, and both nonparametric and parametric IRT models are considered.The absence of the IIO property in twononparametric polytomous IRT models is discussed, and two nonparametric models are discussed that imply an IIO. A method is proposed that can be used to investigate whether empirical data imply an IIO. Furthermore, only twoparametric polytomous IRT models are found to imply an IIO. These are the rating scale model (Andrich, 1978) and a restricted rating scale version of the graded response model (Muraki, 1990). Well-known models, such as the partial credit model (Masters, 1982) and the graded response model (Samejima, 1969), do no imply an IIO.

[1]  P. Rosenbaum.  Comparing item characteristic curves , 1987 .

[2]  E. Muraki.  A GENERALIZED PARTIAL CREDIT MODEL: APPLICATION OF AN EM ALGORITHM , 1992 .

[3]  G. Masters.  A rasch model for partial credit scoring , 1982 .

[4]  William G. Bulgren,et al.  Approximations to the Poisson, Binomial, and Hypergeometric Distribution Functions , 1971 .

[5]  Eiji Muraki,et al.  Fitting a Polytomous Item Response Model to Likert-Type Data , 1990 .

[6]  David Andrich,et al.  Distinctive and Incompatible Properties of Two Common Classes of IRT Models for Graded Responses , 1995 .

[7]  Cees A. W. Glas,et al.  The One Parameter Logistic Model , 1995 .

[8]  Paul R. Rosenbaum,et al.  Probability inequalities for latent scales , 1987 .

[9]  K Sijtsma,et al.  A survey of theory and methods of invariant item ordering. , 1996, The British journal of mathematical and statistical psychology.

[10]  Gideon J. Mellenbergh,et al.  Conceptual Notes on Models for Discrete Polytomous Item Responses , 1995 .

[11]  D. Andrich.  A rating formulation for ordered response categories , 1978 .

[12]  Klaas Sijtsma,et al.  Mokken scale analysis for polychotomous items: theory, a computer program and an empirical application , 1990 .

[13]  Ivo W. Molenaar,et al.  Nonparametric Models for Polytomous Responses , 1997 .

[14]  Hua-Hua Chang,et al.  The unique correspondence of the item response function and item category response functions in polytomously scored item response models , 1994 .

[15]  Brian W. Junker,et al.  Stochastic ordering using the latent trait and the sum score in polytomous IRT models , 1997 .

[16]  F. Samejima.  Estimation of latent ability using a response pattern of graded scores , 1968 .

[17]  Brian W. Junker,et al.  Essential independence and likelihood-based ability estimation for polytomous items , 1991 .

[18]  David Thissen,et al.  A taxonomy of item response models , 1986 .

[19]  Hartmann Scheiblechner,et al.  Isotonic ordinal probabilistic models (ISOP) , 1995 .

[20]  P. Rosenbaum,et al.  Conditional Association and Unidimensionality in Monotone Latent Variable Models , 1985 .

[21]  Brian W. Junker,et al.  Polytomous IRT models and monotone likelihood ratio of the total score , 1996 .

[22]  Klaas Sijtsma,et al.  Selection of Unidimensional Scales From a Multidimensional Item Bank in the Polytomous Mokken I RT Model , 1995 .

[23]  It's an ill wind that brings no good. Studies on odour annoyance and the dispersion of odorant concentrations from industries. , 1992 .