Relationships between a mathematical measurement model and its real-world applications are discussed. A distinction is made between large data matrices commonly found in educational measurement and smaller matrices found in attitude and personality measurement. Nonparametric methods are evaluated for estimating item response functions and (unconditional and conditional) inter-item covariances.
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References
1.
Barlow, R. E. , Bartholomew, D. J., Bremner, J. M., & Brunk, H. D. (1972). Statistical inference under order restrictions. New York: Wiley.
2.
Bolt, D. M. (2001). Conditional covariance-based representation of multidimensional test structure. Applied Psychological Measurement, 25, 244–257.
3.
Croon, M. A. (1991). Investigating Mokken scalability of dichotomous items by means of ordinal latent class analysis. British Journal of Mathematical and Statistical Psychology, 44, 315–331.
4.
Douglas, J. , & Cohen, A. (2001). Nonparametric item response function estimation for assessing parametric model fit. Applied Psychological Measurement, 25, 234–243.
5.
Glas, C. A. W. , & Verhelst, N. D. (1995). Testing the Rasch model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 69–95). New York: Springer-Verlag.
6.
Habing, B. (2001). Nonparametric regression and the parametric bootstrap for local dependence assessment. Applied Psychological Measurement, 25, 221–233.
7.
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.
8.
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.
9.
Hoijtink, H. , & Molenaar, I. W. (1997). A multidimensional item response model: Constrained latent class analysis using the Gibbs sampler and posterior predictive checks. Psychometrika, 62, 171–189.
10.
Huisman, J. M. E. (1999). Item nonresponse: Occurrence, causes and imputation of missing answers to test items (M & T Series No. 32). Leiden, The Netherlands: DSWO Press.
11.
Huisman, J. M. E. , & Molenaar, I. W. (2001). Imputation of missing scale data with item response models. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 222–244). New York: Springer-Verlag.
12.
Junker, B. W. (2001). On the interplay between nonparametric and parametric IRT, with some thoughts about the future. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 247–276). New York: Springer-Verlag.
13.
Junker, B. , & Sijtsma, K. (2001a). Nonparametric item response theory in action: An overview of the Special Issue. Applied Psychological Measurement, 25, 211–220.
14.
Junker, B. , & Sijtsma, K. (2001b). Cognitive assessment models with few assumptions and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272.
15.
Laros, J. A. , & Tellegen, P. J. (1991). Construction and validation of the SON-R 5.5-17, the Snijders Oomen non-verbal intelligence test. Groningen, The Netherlands: Wolters-Noordhoff.
16.
Loevinger, J. (1948). The technique of homogeneous tests compared with some aspects of “scale analysis” and factor analysis. Psychological Bulletin, 45, 507–530.
17.
Lord, F. M. , & Novick, M. R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Wesley.
18.
Meijer, R. R. (1994). The number of Guttman errors as a simple and powerful person-fit statistic. Applied Psychological Measurement, 18, 311–314.
19.
Meijer, R. R. , Sijtsma, K., & Molenaar, I. W. (1995). Reliability estimation for single dichotomous items based on Mokken’s IRT model. Applied Psychological Measurement, 19, 323–335.
20.
Meijer, R. R. , & van Krimpen-Stoop, E. M. L. A. (2001). Person fit across subgroups: An achievement testing example. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 377–390). New York: Springer-Verlag.
21.
Mokken, R. J. (1971). A theory and procedure of scale analysis. The Hague: Mouton; Berlin, DeGruyter.
22.
Molenaar, I. W. (1982). Mokken scaling revisited. Kwantitatieve Methoden, 8, 145–164.
23.
Molenaar, I. W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.
24.
Molenaar, I. W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items. Kwantitatieve Methoden, 37, 97–117.
25.
Molenaar, I. W. , & Sijtsma, K. (2000). User’s manual MSP5 for Windows: A program for Mokken scale analysis for polytomous items (Version 5.0) [Software manual]. Groningen, The Netherlands: ProGAMMA.
26.
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.
27.
Ramsay, J. O. (1993). TESTGRAF: A computer program for nonparametric analysis of testing data. Unpublished manuscript, McGill University.
28.
Sijtsma, K. (1998). Methodology review: Nonparametric IRT approaches to the analysis of dichotomous item scores. Applied Psychological Measurement, 22, 3–32.
29.
Sijtsma, K. (2001). Developments in measurement of persons and items by means of item response models. Behaviormetrika, 28, 65–94.
30.
Sijtsma, K. , & Molenaar, I. W. (1987). Reliability of test scores in nonparametric item response theory. Psychometrika, 52, 79–97.
31.
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-Verlag.
32.
Snijders, T. A. B. (2001). Two-level nonparametric scaling for dichotomous data. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 319–338). New York: Springer-Verlag.
33.
Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality. Psychometrika, 52, 589–617.
34.
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.
35.
Stout, W. F. , Goodwin-Froelich, A., & Gao, F. (2001). Using resampling methods to produce an improved DIMTEST procedure. In A. Boomsma, M. A. J. van Duijn, & T. A. B. Snijders (Eds.), Essays on item response theory (pp. 357–375). New York: Springer-Verlag.
36.
van der Ark, L. A. (2001). Relationships and properties of polytomous item response theory models. Applied Psychological Measurement, 25, 273–282.
37.
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.
38.
Zhang, J. , & Stout, W. F. (1999a). Conditional covariance structure of generalized compensatory multidimensional items. Psychometrika, 64, 129–152.
39.
Zhang, J. , & Stout, W. F. (1999b). The theoretical DETECT index of dimensionality and its application to approximate simple structure. Psychometrika, 64, 213–249.
