Polytomous Latent Scales for the Investigation of the Ordering of Items

We propose three latent scales within the framework of nonparametric item response theory for polytomously scored items. Latent scales are models that imply an invariant item ordering, meaning that the order of the items is the same for each measurement value on the latent scale. This ordering property may be important in, for example, intelligence testing and person-fit analysis. We derive observable properties of the three latent scales that can each be used to investigate in real data whether the particular model adequately describes the data. We also propose a methodology for analyzing test data in an effort to find support for a latent scale, and we use two real-data examples to illustrate the practical use of this methodology.

[1]  I. Deary,et al.  A hierarchy of distress: Mokken scaling of the GHQ-30 , 2008, Psychological Medicine.

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

[3]  G. Tutz.  Sequential item response models with an ordered response , 1990 .

[4]  Stephen E. Fienberg,et al.  Positive dependence concepts for ordinal contingency tables , 1990 .

[5]  Cees A. W. Glas,et al.  Testing the Rasch Model , 1995 .

[6]  Klaas Sijtsma,et al.  Mokken scale analysis as time goes by: An update for scaling practitioners , 2011 .

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

[8]  van der Ark,et al.  Mokken Scale Analysis in R , 2007 .

[9]  Ivo W. Molenaar.  About handy, handmade and handsome models , 2004 .

[10]  Alan Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[11]  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 .

[12]  I. W. Molenaar,et al.  Rasch models: foundations, recent developments and applications , 1995 .

[13]  Moshe Shaked,et al.  Stochastic orders and their applications , 1994 .

[14]  Brenda R. J. Jansen,et al.  Statistical Test of the Rule Assessment Methodology by Latent Class Analysis , 1997 .

[15]  R. Hambleton,et al.  Handbook of Modern Item Response Theory , 1997 .

[16]  Klaas Sijtsma,et al.  On the consistency of individual classification using short scales. , 2007, Psychological methods.

[17]  Hartmann Scheiblechner.  Nonparametric IRT: Testing the bi-isotonicity of isotonic probabilistic models (ISOP) , 2003 .

[18]  Robert J. Mokken,et al.  A Theory and Procedure of Scale Analysis. , 1973 .

[19]  E. Muraki.  A Generalized Partial Credit Model , 1997 .

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

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

[22]  R. J. Mokken,et al.  Handbook of modern item response theory , 1997 .

[23]  Klaas Sijtsma,et al.  Introduction to Nonparametric Item Response Theory , 2002 .

[24]  Frank Proschan,et al.  Functions Decreasing in Transposition and Their Applications in Ranking Problems , 1977 .

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

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

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

[28]  L. A. van der Ark,et al.  Hierarchically Related Nonparametric IRT Models, and Practical Data Analysis Methods , 2001 .

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

[30]  Allan R. Sampson,et al.  Topics in statistical dependence , 1990 .

[31]  Rudy Ligtvoet,et al.  Investigating an Invariant Item Ordering for Polytomously Scored Items , 2010 .

[32]  Klaas Sijtsma,et al.  Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models , 1998 .

[33]  George A. Marcoulides,et al.  Latent variable and latent structure models , 2002 .

[34]  G. van Engelenburg.  On psychometric models for polytomous items with ordered categories within the framework of item response theory , 1997 .

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

[36]  Klaas Sijtsma,et al.  On measurement properties of continuation ratio models , 2001 .

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

[38]  F. Samejima.  Graded Response Model , 1997 .

[39]  D. Wechsler.  Wechsler Intelligence Scale for Children , 2020, Definitions.

[40]  W. H. Schuur,et al.  Mokken Scale Analysis: Between the Guttman Scale and Parametric Item Response Theory , 2003, Political Analysis.

[41]  E. Muraki.  A Generalized Partial Credit Model: Application of an EM Algorithm , 1992 .

[42]  Klaas Sijtsma,et al.  Mokken Scale Analysis for Dichotomous Items Using Marginal Models , 2007, Psychometrika.

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

[44]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

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

[46]  Q. Mcnemar.  Note on the sampling error of the difference between correlated proportions or percentages , 1947, Psychometrika.