Abstract
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.
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References
Agresti, A. (1990). Categorical data analysis. New York: Wiley.
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573.
Bleichrodt, N., Drenth, P.J.D., Zaal, J.N., & Resing, W.C.M. (1987). Revisie Amsterdamse kinder intelligentie test. Handleiding (Revision Amsterdam child intelligence test). Lisse, The Netherlands: Swets & Zeitlinger.
Cavalini, P.M. (1992). It’s an ill wind that brings no good. Studies on odour annoyance and the dispersion of odorant concentrations from industries. Unpublished doctoral dissertation, University of Groningen, The Netherlands.
Chang, H., & Mazzeo, J. (1994). The unique correspondence of the item response function and item category response function in polytomously scored item response models. Psychometrika, 59, 391–404.
Douglas, R., Fienberg, S.E., Lee, M.-L.T., Sampson, A.R., & Whitaker, L.R. (1991). Positive dependence concepts for ordinal contingency tables. In H.W. Block, A.R. Sampson, & T.H. Savits (Eds.), Topics in statistical dependence (pp. 189–202). Hayward, CA: Institute of Mathematical Statistics.
Emons, W.H.M., Sijtsma, K., & Meijer, R.R. (2007). On the consistency of individual classification using short scales. Psychological Methods, 12, 105–120.
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. 69–96). New York: Springer.
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.
Hemker, B.T., Van der Ark, L.A., & Sijtsma, K. (2001). On measurement properties of continuation ratio models. Psychometrika, 66, 487–506.
Hollander, M., Proschan, F., & Sethuraman, J. (1977). Functions decreasing in transposition and their applications in ranking problems. The Annals of Statistics, 5, 722–733.
Jansen, B.R.J., & Van der Maas, H.L.J. (1997). Statistical test of the rule assessment methodology by latent class analysis. Developmental Review, 17, 321–357.
Ligtvoet, R., Van der Ark, L.A., Te Marvelde, J.M., & Sijtsma, K. (2010a). Investigating an invariant item ordering for polytomously scored items. Educational and Psychological Measurement, 70, 578–595.
Ligtvoet, R., Van der Ark, L.A., Bergsma, W.P., & Sijtsma, K. (2010b). Examples concerning the relationships between latent/manifest scales (unpublished manuscript). Retrieved from http://spitswww.uvt.nl/~avdrark/research/LABSexamples.pdf.
Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
McNemar, Q. (1947). Note on the sampling error of the difference between correlated proportions or percentages. Psychometrika, 12, 153–157.
Mellenbergh, G.J. (1995). Conceptual notes on models for discrete polytomous item responses. Applied Psychological Measurement, 19, 91–100.
Mokken, R.J. (1971). A theory and procedure of scale analysis. The Hague/Berlin: Mouton/De Gruyter.
Molenaar, I.W. (1983). Item steps (Heymans Bulletin 83-630-EX). Groningen, The Netherlands: University of Groningen, Department of Statistics and Measurement Theory.
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.
Molenaar, I.W. (2004). About handy, handmade and handsome models. Statistica Neerlandica, 58, 1–20.
Molenaar, I.W., & Sijtsma, K. (2000). User’s Manual MSP5 for Windows. Groningen, The Netherlands: iec ProGAMMA.
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.
Muraki, E. (1992). A generalized partial credit model: applications for an EM algorithm. Applied Psychological Measurement, 16, 159–177.
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Nielsen and Lydiche.
Rosenbaum, P.R. (1987a). Probability inequalities for latent scales. British Journal of Mathematical & Statistical Psychology, 40, 157–168.
Rosenbaum, P.R. (1987b). Comparing item characteristic curves. Psychometrika, 52, 217–233.
Samejima, F. (1969). Estimation of latent trait ability using a response pattern of graded scores. Psychometrika Monograph (No. 17).
Samejima, F. (1997). Graded response model. In W.J. van der Linden, & R.K. Hambleton (Eds.), Handbook of modern item response theory (pp. 85–100). New York: Springer.
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.
Scheiblechner, H. (2003). Nonparametric IRT: testing the bi-isotonicity of Isotonic Probabilistic Models (ISOP). Psychometrika, 68, 79–96.
Shaked, M., & Shantikumar, J.G. (1994). Stochastic orders and their applications. San Diego, CA: Academic Press.
Sijtsma, K., & Hemker, B.T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200.
Sijtsma, K., & Junker, B.W. (1996). A survey of theory and methods of invariant item ordering. British Journal of Mathematical & Statistical Psychology, 49, 79–105.
Sijtsma, K., Meijer, R.R., & Van der Ark, L.A. (2011). Mokken Scale Analysis as time goes by: an update for scaling practitioners. Personality and Individual Differences, 50, 31–37.
Sijtsma, K., & Molenaar, I.W. (2002). Introduction to nonparametric item response theory. Thousand Oaks, CA: Sage.
Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical & Statistical Psychology, 43, 39–55.
Van der Ark, L.A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 1–19.
Van der Ark, L.A., Croon, M.A., & Sijtsma, K. (2008). Mokken scale analysis for dichotomous items using marginal models. Psychometrika, 73, 183–208.
Van der Ark, L.A., Hemker, B.T., & Sijtsma, K. (2002). Hierarchically related nonparametric IRT models, and practical data analysis methods. In G.A. Marcoulides, & I. Moustaki (Eds.), Latent variable and latent structure models (pp. 41–62). Mahwah, NJ: Erlbaum.
Van Engelenburg, G. (1997). On psychometric models for polytomous items with ordered categories within the framework of item response theory. Unpublished doctoral dissertation, University of Amsterdam.
Van Schuur, W.H. (2003). Mokken scale analysis: between the Guttman scale and parametric item response theory. Political Analysis, 11, 139–163.
Watson, R., Deary, I.J., & Shipley, B. (2008). A hierarchy of distress: Mokken scaling of the GHQ-30. Psychological Medicine, 38, 575–579.
Wechsler, D. (2003). Wechsler intelligence scale for children (4th ed.). San Antonio, TX: The Psychological Corporation.
Weekers, A.M., Brown, G.T.L., & Veldkamp, B.P. (2009). Analyzing the dimensionality of the Student’s Conceptions of Assessment inventory. In D.M. McInerney, G.T.L. Brown, & G.A.D. Liem (Eds.), Student perspectives on assessment: what students can tell us about assessment for learning Charlotte, NC: Information Age.
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Ligtvoet, R., van der Ark, L.A., Bergsma, W.P. et al. Polytomous Latent Scales for the Investigation of the Ordering of Items. Psychometrika 76, 200–216 (2011). https://doi.org/10.1007/s11336-010-9199-8
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DOI: https://doi.org/10.1007/s11336-010-9199-8