Abstract
As the focus of this chapter, we discuss nonparametric item response theory for ordinal person scales, specifically the monotone homogeneity model and Mokken scale analysis, which is the data-analysis procedure used for investigating the compliance between the monotone homogeneity model and data. Next, we discuss the unrestricted latent class model as an even more liberal model for investigating the scalability of a set of items, producing nominal scales, but we also discuss an ordered latent class model that one can use to investigate assumptions about item response functions in the monotone homogeneity model and other nonparametric item response models. Finally, we discuss cognitive diagnostic models, which are the core of this volume, and which are a further deepening of latent class models, providing diagnostic information about the people who responded to a set of items. A data analysis example, using item scores of 1210 respondents on 44 items from the Millon Clinical Multiaxial Inventory III, demonstrates how the monotone homogeneity model, the latent class model, and two cognitive diagnostic models can be used jointly to understand one’s data.
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Andrews, R. L., & Currim, I. S. (2003). A comparison of segment retention criteria for finite mixture logit models. Journal of Marketing Research, 40, 235–243. https://doi.org/10.1509/jmkr.40.2.235.19225
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573. https://doi.org/10.1007/BF02293814
Bouwmeester, S., Vermunt, J. K., & Sijtsma, K. (2007). Development and individual differences in transitive reasoning: A fuzzy trace theory approach. Developmental Review, 27, 41–74. https://doi.org/10.1016/j.dr.2006.08.001
Brusco, M. J., Köhn, H.-F., & Steinley, D. (2015). An exact method for partitioning dichotomous items within the framework of the monotone homogeneity model. Psychometrika, 80, 949–967. https://doi.org/10.1007/s11336-015-9459-8
Chiu, C. Y., & Douglas, J. A. (2013). A nonparametric approach to cognitive diagnosis by proximity to ideal response patterns. Journal of Classification, 30, 225–250. https://doi.org/10.1007/s00357-013-9132-9
Croon, M. (1990). Latent class analysis with ordered latent classed. British Journal of Mathematical and Statistical Psychology, 43, 171–192. https://doi.org/10.1111/j.2044-8317.1990.tb00934.x
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. https://doi.org/10.1111/j.2044-8317.1991.tb00964.x
de la Torre, J. (2009). DINA model and parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34, 115–130. https://doi.org/10.3102/1076998607309474
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199. https://doi.org/10.1007/s11336-011-9207-7
de la Torre, J., & Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353. https://doi.org/10.1007/BF02295640
de la Torre, J, van der Ark, L. A., & Rossi, G. (2018). Analysis of clinical data from cognitive diagnosis modeling framework. Measurement and Evaluation in Counseling and Development, 51, 281–296. https://doi.org/10.1177/0748175615569110
Douglas, J. A. (2001). Asymptotic identifiability of nonparametric item response models. Psychometrika, 66, 531–540. https://doi.org/10.1007/BF02296194
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. Retrieved from http://www.jstor.org/stable/4355592
Ellis, J. L. (2014). An inequality for correlations in unidimensional monotone latent variable models for binary variables. Psychometrika, 79, 303–316. https://doi.org/10.1007/s11336-013-9341-5
Formann, A. K., & Kohlmann, T. (2002). Three-parameter linear logistic latent class analysis. In J. A. Hagenaars & A. L. McCutcheon (Eds.),. Applied latent class analysis (pp. 183–210). Cambridge, UK: Cambridge University Press.
Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231. https://doi.org/10.1093/biomet/61.2.215
Grayson, D. A. (1988). Two-group classification in latent trait theory: Scores with monotone likelihood ratio. Psychometrika, 53, 383–392. https://doi.org/10.1007/BF02294219
Haertel, E. H. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26, 301–321. https://doi.org/10.1111/j.1745-3984.1989.tb00336.x
Hagenaars, J. A., & McCutcheon, A. L. (2002). Applied latent class analysis. Cambridge, UK: Cambridge University Press.
Heinen, T. (1996). Latent class and discrete latent trait models. Similarities and differences. Thousand Oaks, CA: Sage.
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. https://doi.org/10.1007/BF02294555
Hemker, B. T., van der Ark, L. A., & Sijtsma, K. (2001). On measurement properties of continuation ratio models. Psychometrika, 66, 487–506. https://doi.org/10.1007/BF02296191
Hoijtink, H., & Molenaar, I. W. (1997). A multidimensional item response model: Constrained latent class analysis using Gibbs sampler and posterior predictive checks. Psychometrika, 62, 171–189. https://doi.org/10.1007/BF02295273
Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.. Retrieved from https://projecteuclid.org/euclid.aos/1176350174
Hubert, M., & Vandervieren, E. (2008). An adjusted boxplot for skewed distributions. Computational Statistics & Data Analysis, 52, 5186–5201. https://doi.org/10.1016/j.csda.2007.11.008
Junker, B. W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 21, 1359–1378. Retrieved from http://www.jstor.org/stable/2242199
Junker, B. W., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81. https://doi.org/10.1177/01466216000241004
Junker, B. W., & Sijtsma, K. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272. https://doi.org/10.1177/01466210122032064
Karabatsos, G., & Sheu, C.-F. (2004). Order-constrained Bayes inference for dichotomous models of unidimensional nonparametric IRT. Applied Psychological Measurement, 28, 110–125. https://doi.org/10.1177/0146621603260678
Leighton, J. A., & Gierl, M. J. (2007). Cognitive diagnostic assessment for education. Theory and applications. Cambridge, UK: Cambridge University Press.
Ligtvoet, R., & Vermunt, J. K. (2012). Latent class models for testing monotonicity and invariant item ordering for polytomous items. British Journal of Mathematical and Statistical Psychology, 65, 237–250. https://doi.org/10.1111/j.2044-8317.2011.02019.x
Linzer, D. A. (2011). Reliable inference in highly stratified contingency tables: Using latent class models as density estimators. Political Analysis, 19, 173–187. https://doi.org/10.1093/pan/mpr006
Linzer, D. A., & Lewis, J. B. (2011). poLCA: An R package for polytomous variable latent class analysis. Journal of Statistical Software, 42(10), 1–29. https://doi.org/10.18637/jss.v042.i10
MacReady, G. B., & Dayton, C. M. (1977). The use of probabilistic models in the assessment of mastery. Journal of Educational Statistics, 2, 99–120. https://doi.org/10.3102/10769986002002099
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174. https://doi.org/10.1007/BF02296272
Michell, J. (1999). Measurement in psychology. A critical history of a methodological concept. Cambridge, UK: Cambridge University Press.
Millon, T., Millon, C., Davis, R., & Grossman, S. (2009). MCMI-III Manual (4th ed.). Minneapolis, MN: Pearson Assessments.
Mokken, R. J. (1971). A theory and procedure of scale analysis. The Hague, The Netherlands/Berlin, Germany: Mouton/De Gruyter.
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176. https://doi.org/10.1177/014662169201600206
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630. https://doi.org/10.1007/BF02294494
Ramsay, J. O. (2016). Functional approaches to modeling response data. In W. J. van der Linden (Ed.), Handbook of item response theory. Volume one. Models (pp. 337–350). Boca Raton, FL: Chapman & Hall/CRC.
Rossi, G., Elklit, A., & Simonsen, E. (2010). Empirical evidence for a four factor framework of personality disorder organization: Multigroup confirmatory factor analysis of the million clinical multiaxial inventory–III personality disorders scales across Belgian and Danish data samples. Journal of Personality Disorders, 24, 128–150. https://doi.org/10.1521/pedi.2010.24.1.128
Rossi, G., Sloore, H., & Derksen, J. (2008). The adaptation of the MCMI-III in two non-English-speaking countries: State of the art of the Dutch language version. In T. Millon & C. Bloom (Eds.), The Millon inventories: A practitioner’s guide to personalized clinical assessment (2nd ed., pp. 369–386). New York, NY: The Guilford Press.
Rupp, A. A., Templin, J., & Henson, R. A. (2010). Diagnostic measurement. Theory, methods, and applications. New York, NY: The Guilford Press.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores (Psychometric Monograph No. 17). Richmond, VA: Psychometric Society. Retrieved from http://www.psychometrika.org/journal/online/MN17.pdf
Schwarz, G. E. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464. https://doi.org/10.2307/2958889
Sijtsma, K., & Hemker, B. T. (1998). Nonparametric polytomous IRT models for invariant item ordering, with results for parametric models. Psychometrika, 63, 183–200. https://doi.org/10.1007/BF02294774
Sijtsma, K., & Molenaar, I. W. (2002). Introduction to nonparametric item response theory. Thousand Oaks, CA: Sage.
Sijtsma, K., & Molenaar, I. W. (2016). Mokken models. In W. J. van der Linden (Ed.), Handbook of item response theory. Volume one. Models (pp. 303–321). Boca Raton, FL: Chapman & Hall/CRC.
Sijtsma, K., & van der Ark, L. A. (2017). A tutorial on how to do a Mokken scale analysis on your test and questionnaire data. British Journal of Mathematical and Statistical Psychology, 70, 137–158. https://doi.org/10.1111/bmsp.12078
Sorrel, M. A., Olea, J., Abad, F. J., de la Torre, J., Aguado, D., & Lievens, F. (2016). Validity and reliability of situational judgement test scores: A new approach based on cognitive diagnosis models. Organizational Research Methods, 19, 506–532. https://doi.org/10.1177/1094428116630065
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensional assessment and ability estimation. Psychometrika, 55, 293–326. https://doi.org/10.1007/BF02295289
Stout, W. F. (2002). Psychometrics: From practice to theory and back. Psychometrika, 67, 485–518. https://doi.org/10.1007/BF02295128
Straat, J. H., van der Ark, L. A., & Sijtsma, K. (2013). Comparing optimization algorithms for item selection in Mokken scale analysis. Journal of Classification, 30, 75–99. https://doi.org/10.1007/s00357-013-9122-y
Straat, J. H., van der Ark, L. A., & Sijtsma, K. (2016). Using conditional association to identify locally independent item sets. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 12, 117–123. https://doi.org/10.1027/1614-2241/a000115
Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199. https://doi.org/10.1007/BF01063886
Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305. https://doi.org/10.1037/1082-989X.11.3.287
Tijmstra, J., Hessen, D. J., van der Heijden, P. G. M., & Sijtsma, K. (2013). Testing manifest monotonicity using order-constrained statistical inference. Psychometrika, 78, 83–97. https://doi.org/10.1007/s11336-012-9297-x
van der Ark, L. A. (2005). Stochastic ordering of the latent trait by the sum score under various polytomous IRT models. Psychometrika, 70, 283–304. https://doi.org/10.1007/s11336-000-0862-3
van der Ark, L. A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 1–19. https://doi.org/10.18637/jss.v020.i11
van der Ark, L. A. (2012). New developments in Mokken scale analysis in R. Journal of Statistical Software, 48(5), 1–27. https://doi.org/10.18637/jss.v048.i05
van der Ark, L. A., & Bergsma, W. P. (2010). A note on stochastic ordering of the latent trait using the sum of polytomous item scores. Psychometrika, 75, 272–279. https://doi.org/10.1007/S11336-010-9147-7
van der Ark, L. A., van der Palm, D. W., & Sijtsma, K. (2011). A latent class approach to estimating test-score reliability. Applied Psychological Measurement, 35, 380–392.
van der Linden, W. J. (Ed.). (2016). Handbook of item response theory. Volume one. Models. Boca Raton, FL: Chapman & Hall/CRC.
van Onna, M. J. H. (2002). Bayesian estimation and model selection in ordered latent class models for polytomous items. Psychometrika, 67, 519–538. https://doi.org/10.1007/BF02295129
van Schuur, W. H. (2011). Ordinal item response theory. Mokken scale analysis. Thousand Oaks, CA: Sage.
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. https://doi.org/10.1177/01466210122032082
Vermunt, J. K., van Ginkel, J. R., van der Ark, L. A., & Sijtsma, K. (2008). Multiple imputation of incomplete categorical data using latent class analysis. Sociological Methodology, 38, 369–397.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–307. https://doi.org/10.1348/000711007X193957
von Davier, M. (2010). Hierarchical mixtures of diagnostic models. Psychological Test and Assessment Modeling, 52, 8–28. Retrieved from http://www.psychologie-aktuell.com/fileadmin/download/ptam/1-2010/02_vonDavier.pdf
von Davier, M. (2014). The DINA model as a constrained general diagnostic model: Two variants of a model equivalency. British Journal of Mathematical and Statistical Psychology, 67, 49–71. https://doi.org/10.1111/bmsp.12003
von Davier, M., & Haberman, S. (2014). Hierarchical diagnostic classification models morphing into unidimensional ‘diagnostic’ classification models – A commentary. Psychometrika, 79, 340–346. https://doi.org/10.1007/s11336-013-9363-z
Wetzel, E., Xu, X., & Von Davier, M. (2015). An alternative way to model population ability distributions in large-scale educational surveys. Educational and Psychological Measurement, 75, 739–763.
Yang, C.-C., & Yang, C.-C. (2007). Separating latent classes by information criteria. Journal of Classification, 24, 183–203. https://doi.org/10.1007/s00357-007-0010-1
Yang, X., & Embretson, S. E. (2007). Construct validity and cognitive diagnostic assessment. In J. A. Leighton & M. J. Gierl (Eds.), Cognitive diagnostic assessment for education. Theory and applications (pp. 119–145). Cambridge, UK: Cambridge University Press.
Yen, W. (1981). Using simulation results to choose a latent trait model. Applied Psychological Measurement, 5, 245–262. https://doi.org/10.1177/014662168100500212
Zheng, Y., & Chiu, C.-Y. (2016). NPCD: Nonparametric methods for cognitive diagnosis. R package version 1.0–10 [computer software]. Retrieved from https://CRAN.R-project.org/package=NPCD
Zijlstra, W. P., van der Ark, L. A., & Sijtsma, K. (2007). Outlier detection in test and questionnaire data. Multivariate Behavioral Research, 42(3), 531–555.
Zijlmans, E. A. O., van der Ark, L. A., Tijmstra, J., & Sijtsma, K. (2018). Methods for estimating item-score reliability. Applied Psychological Measurement, 42, 553–570. https://doi.org/10.1177/0146621618758290.
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van der Ark, L.A., Rossi, G., Sijtsma, K. (2019). Nonparametric Item Response Theory and Mokken Scale Analysis, with Relations to Latent Class Models and Cognitive Diagnostic Models. In: von Davier, M., Lee, YS. (eds) Handbook of Diagnostic Classification Models. Methodology of Educational Measurement and Assessment. Springer, Cham. https://doi.org/10.1007/978-3-030-05584-4_2
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