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The R Package CDM for Diagnostic Modeling

  • Alexander RobitzschEmail author
  • Ann Cathrice George
Chapter
First Online:
Part of the Methodology of Educational Measurement and Assessment book series (MEMA)

Abstract

In this chapter, the R (R Core Team, R: a language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, 2017) pack-age CDM (Robitzsch A, Kiefer T, George AC, Uenlue A, CDM: cognitive diagnosis modeling. R package version 6.0-101. https://CRAN.R-project.org/package=CDM, 2017; George AC, Robitzsch A, Kiefer T, Groß J, Ünlü A, J Stat Softw 74(2):1–24. 10.18637/jss.v074.i02, 2016) for estimating diagnostic classification models is introduced. First, the model classes that can be estimated with the CDM package are introduced. Second, the CDM package structure and some of its features are discussed. Third, the usage of the CDM package is demonstrated in a data application. Finally, potential future developments of the CDM package are discussed.

Keywords

Diagnostic classification models Software R Estimation Regularization Latent class models 
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References

  1. Asparouhov, T., & Muthen, B. (2014). Variable-specific entropy contribution (Technical appendix). http://www.statmodel.com/7_3_papers.shtml
  2. Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72(2), 141–157.CrossRefGoogle Scholar
  3. Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122–4137.CrossRefGoogle Scholar
  4. Breheny, P., & Huang, J. (2011). Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection. Annals of Applied Statistics, 5(1), 232–253.CrossRefGoogle Scholar
  5. Chalmers, R. P. (2018). Numerical approximation of the observed information matrix with Oakes’ identity. British Journal of Mathematical and Statistical Psychology, 71(3), 415–436.Google Scholar
  6. Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and absolute fit evaluation in cognitive diagnosis modeling. Journal of Educational Measurement, 50(2), 123–140.CrossRefGoogle Scholar
  7. Chen, J., & Zhou, H. (2017). Test designs and modeling under the general nominal diagnosis model framework. PLoS One, 12(6), e0180016.CrossRefGoogle Scholar
  8. Chen, Y., Li, X., Liu, J., & Ying, Z. (2016). A fused latent and graphical model for multivariate binary data. arXiv:1606.08925.Google Scholar
  9. Chen, Y., Li, X., Liu, J., & Ying, Z. (2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82(3), 660–692.CrossRefGoogle Scholar
  10. Chen, Y., Liu, J., Xu, G., & Ying, Z. (2015). Statistical analysis of Q-matrix based diagnostic classification models. Journal of the American Statistical Association, 110(510), 850–866.CrossRefGoogle Scholar
  11. Chiu, C. Y. (2013). Statistical refinement of the Q-matrix in cognitive diagnosis. Applied Psychological Measurement, 37(8), 598–618.CrossRefGoogle Scholar
  12. Chiu, C.-Y., & Köhn, H.-F. (this volume). Nonparametric methods in cognitively diagnostic assessment. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  13. Croon, M. (1990). Latent class analysis with ordered latent classes. British Journal of Mathematical and Statistical Psychology, 43(2), 171–192.CrossRefGoogle Scholar
  14. Cui, Y., Gierl, M. J., & Chang, H.-H. (2012). Estimating classification consistency and accuracy for cognitive diagnostic assessment. Journal of Educational Measurement, 49(1), 198–138.CrossRefGoogle Scholar
  15. Culpepper, S. A., & Hudson, A. (2018). An improved strategy for Bayesian estimation of the reduced reparametrized unified model. Applied Psychological Measurement, 42(2), 99–115.CrossRefGoogle Scholar
  16. de la Torre, J. (2008). An empirically based method of Q-matrix validation for the DINA model: Development and applications. Journal of Educational Measurement, 45(4), 343–362.CrossRefGoogle Scholar
  17. de la Torre, J. (2009a). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33(3), 163–183.CrossRefGoogle Scholar
  18. de la Torre, J. (2009b). DINA model and parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34(1), 115–130.CrossRefGoogle Scholar
  19. de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199.CrossRefGoogle Scholar
  20. de la Torre, J., & Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69(3), 333–353.CrossRefGoogle Scholar
  21. de la Torre, J., & Douglas, J. (2008). Model evaluation and multiple strategies in cognitive diagnosis: An analysis of fraction subtraction data. Psychometrika, 73(4), 595–624.CrossRefGoogle Scholar
  22. de la Torre, J., & Lee, Y.-S. (2013). Evaluating the Wald test for item-level comparison of saturated and reduced models in cognitive diagnosis. Journal of Educational Measurement, 50(4), 355–373.CrossRefGoogle Scholar
  23. de la Torre, J., & Minchen, N. D. (this volume). The G-DINA model framework. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  24. De Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11(3), 183–196.CrossRefGoogle Scholar
  25. Decarlo, L. T. (2012). Recognizing uncertainty in the Q-matrix via a Bayesian extension of the DINA model. Applied Psychological Measurement, 36(6), 447–468.CrossRefGoogle Scholar
  26. Desmarais, M. C., & Naceur, R. (2013). A matrix factorization method for mapping items to skills and for enhancing expert-based Q-matrices. In H. C. Lane, K. Yacef, J. Mostow, & P. Pavlik (Eds.), Artificial intelligence in education (pp. 441–450). Berlin, Germany: Springer.CrossRefGoogle Scholar
  27. Dibello, L. V., Roussos, L. A., & Stout, W. (2007). Review of cognitively diagnostic assessment and a summary of psychometric models. In C. R. Rao & S. Sinharay (Eds.), Handbook of statistics, volume 26, psychometrics (pp. 979–1030). Amsterdam, The Netherlands: Elsevier.Google Scholar
  28. Embretson, S. E. (this volume). Diagnostic modeling of skill hierarchies and cognitive processes with MLTM-D. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  29. Fan, J., & Lv, J. (2010). A selective overview of variable selection in high dimensional feature space. Statistica Sinica, 20(1), 101–148.Google Scholar
  30. Fiacco, A. V., & McCormick, G. P. (1968). Nonlinear programming: Sequential unconstrained minimization techniques. New York, NY: Wiley.Google Scholar
  31. Formann, A. K. (1985). Constrained latent class models: Theory and applications. British Journal of Mathematical and Statistical Psychology, 38(1), 87–111.CrossRefGoogle Scholar
  32. Formann, A. K. (1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 87(418), 476–486.CrossRefGoogle Scholar
  33. Formann, A. K. (2007). (Almost) equivalence between conditional and mixture maximum likelihood estimates for some models of the Rasch type. In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models (pp. 177–189). New York, NY: Springer.CrossRefGoogle Scholar
  34. Formann, A. K., & Kohlmann, T. (1998). Structural latent class models. Sociological Methods & Research, 26(4), 530–565.CrossRefGoogle Scholar
  35. 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.CrossRefGoogle Scholar
  36. George, A. C., & Robitzsch, A. (2014). Multiple group cognitive diagnosis models, with an emphasis on differential item functioning. Psychological Test and Assessment Modeling, 56(4), 405–432.Google Scholar
  37. George, A. C., & Robitzsch, A. (2015). Cognitive diagnosis models in R: A didactic. The Quantitative Methods for Psychology, 11(3), 189–205.CrossRefGoogle Scholar
  38. George, A. C., & Robitzsch, A. (2018). Focusing on interactions between content and cognition: A new perspective on gender differences in mathematical sub-competencies. Applied Measurement in Education, 31(1), 79–97.CrossRefGoogle Scholar
  39. George, A. C., Robitzsch, A., Kiefer, T., Groß, J., & Ünlü, A. (2016). The R package CDM for cognitive diagnosis models. Journal of Statistical Software, 74(2), 1–24.  https://doi.org/10.18637/jss.v074.i02 CrossRefGoogle Scholar
  40. Groß, J., & George, A. C. (2014). On prerequisite relations between attributes in noncompensatory diagnostic classification. Methodology, 10(3), 100–107.CrossRefGoogle Scholar
  41. Han, Z., & Johnson, M. S. (this volume). Global- and item-level model fit indices. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  42. Hastie, T., Tibshirani, R., & Wainwright, M. (2015). Statistical learning with sparsity: The lasso and generalizations. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
  43. Henson, R., & Templin, J. L. (this volume). Loglinear cognitive diagnostic model (LCDM). In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  44. Henson, R., Roussos, L., Douglas, J., & He, X. (2008). Cognitive diagnostic attribute-level discrimination indices. Applied Psychological Measurement, 32(4), 275–288.CrossRefGoogle Scholar
  45. Hojtink, H., & Molenaar, I. W. (1997). A multidimensional item response model: Constrained latent class analysis using the Gibbs sampler and posterior predictive checks. Psychometrika, 62(2), 171–189.CrossRefGoogle Scholar
  46. Hong, H., Wang, C., Lim, Y. C., & Douglas, J. (2015). Efficient models for cognitive diagnosis with continuous and mixed-type latent variables. Applied Psychological Measurement, 39(1), 31–43.CrossRefGoogle Scholar
  47. Hou, L., de la Torre, J., & Nandakumar, R. (2014). Differential item functioning assessment in cognitive diagnostic modeling: Application of the Wald test to investigate DIF in the DINA model. Journal of Educational Measurement, 51(1), 98–125.CrossRefGoogle Scholar
  48. Hsieh, C. A., Xu, X., & von Davier, M. (2010). Variance estimation for NAEP data using a resampling-based approach: An application of cognitive diagnostic models (RR-10-26). Princeton, NJ: Educational Testing Service.Google Scholar
  49. Hu, J., Miller, M. D., Huggins-Manley, A. C., & Chen, Y. H. (2016). Evaluation of model fit in cognitive diagnosis models. International Journal of Testing, 16(2), 119–141.CrossRefGoogle Scholar
  50. Huang, H. Y., & Wang, W. C. (2014). The random-effect DINA model. Journal of Educational Measurement, 51(1), 75–97.CrossRefGoogle Scholar
  51. Huang, P. H., Chen, H., & Weng, L. J. (2017). A penalized likelihood method for structural equation modeling. Psychometrika, 82(2), 329–354.CrossRefGoogle Scholar
  52. Huo, Y., & de la Torre, J. (2014). Estimating a cognitive diagnostic model for multiple strategies via the EM algorithm. Applied Psychological Measurement, 38(6), 464–485.CrossRefGoogle Scholar
  53. Jang, E. E. (2009). Cognitive diagnostic assessment of L2 reading comprehension ability: Validity arguments for fusion model application to LanguEdge assessment. Language Testing, 26(1), 31–73.CrossRefGoogle Scholar
  54. Kang, H.-A., Liu, J., & Ying, Z. (2017). A general diagnostic classification model. arXiv:1707.06318.Google Scholar
  55. Khorramdel, L., Shin, H. J., and von Davier, M. (this volume). GDM software mdltm including parallel EM algorithm. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  56. Kunina-Habenicht, O., Rupp, A. A., & Wilhelm, O. (2009). A practical illustration of multidimensional diagnostic skills profiling: Comparing results from confirmatory factor analysis and diagnostic classification models. Studies in Educational Evaluation, 35(2–3), 64–70.CrossRefGoogle Scholar
  57. Kuo, B. C., Chen, C. H., & de la Torre, J. (2018). A cognitive diagnosis model for identifying coexisting skills and misconceptions. Applied Psychological Measurement, 42(3), 179–191.CrossRefGoogle Scholar
  58. Leighton, J. P., Gierl, M. J., & Hunka, S. M. (2004). The attribute hierarchy method for cognitive assessment: A variation on Tatsuoka’s rule space approach. Journal of Educational Measurement, 41(3), 205–237.CrossRefGoogle Scholar
  59. Li, H., Hunter, C. V., & Lei, P. W. (2016). The selection of cognitive diagnostic models for a reading comprehension test. Language Testing, 33(3), 391–409.CrossRefGoogle Scholar
  60. Li, X., & Wang, W. C. (2015). Assessment of differential item functioning under cognitive diagnosis models: The DINA model example. Journal of Educational Measurement, 52(1), 28–54.CrossRefGoogle Scholar
  61. Liu, X., & Johnson, M. S. (this volume). Estimating CDMs using MCMC. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  62. Liu, J., & Kang, H.-A. (this volume). Q-matrix learning via latent variable selection and identifiability. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  63. Liu, J., Xu, G., & Ying, Z. (2013). Theory of the self-learning Q-matrix. Bernoulli, 19(5A), 1790–1817.CrossRefGoogle Scholar
  64. Liu, R., Huggins-Manley, A. C., & Bulut, O. (2018). Retrofitting diagnostic classification models to responses from IRT-based assessment forms. Educational and Psychological Measurement, 78(3), 357–383.CrossRefGoogle Scholar
  65. Liu, Y., Douglas, J. A., & Henson, R. A. (2009). Testing person fit in cognitive diagnosis. Applied Psychological Measurement, 33(8), 579–598.CrossRefGoogle Scholar
  66. Liu, Y., Xin, T., Andersson, B., & Tian, W. (2018). Information matrix estimation procedures for cognitive diagnostic models. British Journal of Mathematical and Statistical Psychology (in press).  https://doi.org/10.1111/bmsp.12134.
  67. Ma, W. (this volume). Cognitive diagnosis modeling using the GDINA R package. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  68. Ma, W., & de la Torre, J. (2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253–275.CrossRefGoogle Scholar
  69. Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection, and attribute classification. Applied Psychological Measurement, 40(3), 200–217.CrossRefGoogle Scholar
  70. Maydeu-Olivares, A., & Joe, H. (2014). Assessing approximate fit in categorical data analysis. Multivariate Behavioral Research, 49(4), 305–328.CrossRefGoogle Scholar
  71. McDonald, R. P., & Mok, M. M. C. (1995). Goodness of fit in item response models. Multivariate Behavioral Research, 30(1), 23–40.CrossRefGoogle Scholar
  72. Mislevy, R. J., & Wilson, M. (1996). Marginal maximum likelihood estimation for a psychometric model of discontinuous development. Psychometrika, 61(1), 41–71.CrossRefGoogle Scholar
  73. Nussbeck, F. W., & Eid, M. (2015). Multimethod latent class analysis. Frontiers in Psychology | Quantitative Psychology and Measurement, 6, 1332.Google Scholar
  74. Oakes, D. (1999). Direct calculation of the information matrix via the EM algorithm. Journal of the Royal Statistical Society: Series B, 61(2), 479–482.CrossRefGoogle Scholar
  75. Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24(1), 50–64.CrossRefGoogle Scholar
  76. Ozaki, K. (2015). DINA models for multiple-choice items with few parameters: Considering incorrect answers. Applied Psychological Measurement, 39(6), 431–447.CrossRefGoogle Scholar
  77. Park, Y. S., & Lee, Y.-S. (this volume). Explanatory cognitive diagnostic models. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  78. Park, J. Y., Lee, Y.-S., & Johnson, M. S. (2017). An efficient standard error estimator of the DINA model parameters when analysing clustered data. International Journal of Quantitative Research in Education, 4(1–2), 159–190.CrossRefGoogle Scholar
  79. Philipp, M., Strobl, C., de la Torre, J., & Zeileis, A. (2018). On the estimation of standard errors in cognitive diagnosis models. Journal of Educational and Behavioral Statistics, 43(1), 88–115.CrossRefGoogle Scholar
  80. Pritikin, J. N. (2017). A comparison of parameter covariance estimation methods for item response models in an expectation-maximization framework. Cogent Psychology, 4, 1279435.CrossRefGoogle Scholar
  81. Qiu, X.-L., Li, X., & Wang, W.-C. (this volume). Differential item functioning in diagnostic classification models. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  82. R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar
  83. Raiche, G., Magis, D., Blais, J. G., & Brochu, P. (2012). Taking atypical response patterns into account. In M. Simon, K. Ercikan, & M. Rousseau (Eds.), Improving large scale assessment in education: Theory, issues and practice (pp. 238–259). New York, NY: Routledge.CrossRefGoogle Scholar
  84. Ravand, H., & Robitzsch, A. (2015). Cognitive diagnostic modeling using R. Practical Assessment, Research & Evaluation, 20(11), 1–12.Google Scholar
  85. Robitzsch, A., Kiefer, T., George, A. C., & Uenlue, A. (2017). CDM: Cognitive diagnosis modeling. R package version 6.0-101. https://CRAN.R-project.org/package=CDM
  86. Rupp, A., & Templin, J. (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art. Measurement: Interdisciplinary Research and Perspectives, 6(4), 219–262.Google Scholar
  87. San Martin, E. S., del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30(3), 183–203.CrossRefGoogle Scholar
  88. Sen, S., & Bradshaw, L. (2017). Comparison of relative fit indices for diagnostic model selection. Applied Psychological Measurement, 41(6), 422–438.CrossRefGoogle Scholar
  89. Shen, X., Pan, W., & Zhu, Y. (2012). Likelihood-based selection and sharp parameter estimation. Journal of the American Statistical Association, 107(497), 223–232.CrossRefGoogle Scholar
  90. Shin, H. J., Wilson, M., & Choi, I. H. (2017). Structured constructs models based on change-point analysis. Journal of Educational Measurement, 54(3), 306–332.CrossRefGoogle Scholar
  91. Sinharay, S., & Johnson, M. S. (this volume). Measures of agreement: Reliability, classification accuracy, and classification consistency. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  92. Sorrel, M. A., Abad, F. J., Olea, J., de la Torre, J., & Barrada, J. R. (2017). Inferential item-fit evaluation in cognitive diagnosis modeling. Applied Psychological Measurement, 41(8), 614–631.CrossRefGoogle Scholar
  93. Stout, W., Henson, R., DiBello, L., & Shear, B. (this volume). The reparameterized unified model system: a diagnostic assessment modeling approach. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  94. Sun, J., Chen, Y., Liu, J., Ying, Z., & Xin, T. (2016). Latent variable selection for multidimensional item response theory models via L 1 regularization. Psychometrika, 81(4), 921–939.CrossRefGoogle Scholar
  95. Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79(2), 317–339.CrossRefGoogle Scholar
  96. Templin, J., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32(2), 37–50.CrossRefGoogle Scholar
  97. Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287–305.CrossRefGoogle Scholar
  98. Tutz, G. (1997). Sequential models for ordered responses. In W. van der Linden & R. K. Hambleton (Eds.), Handbook of modern item response theory (pp. 139–152). New York, NY: Springer.CrossRefGoogle Scholar
  99. Tutz, G., & Schauberger, G. (2015). A penalty approach to differential item functioning in Rasch models. Psychometrika, 80(1), 21–43.CrossRefGoogle Scholar
  100. van der Ark, L. A., Rossi, G., & Sijtsma, K. (this volume). Nonparametric item response theory and mokken scale analysis, with relations to latent class models and cognitive diagnostic models. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  101. 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(3), 283–294.CrossRefGoogle Scholar
  102. von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61(2), 287–307.CrossRefGoogle Scholar
  103. von Davier, M. (2009). Some notes on the reinvention of latent structure models as diagnostic classification models. Measurement: Interdisciplinary Research & Perspectives, 7(1), 67–74.Google Scholar
  104. von Davier, M. (2010). Hierarchical mixtures of diagnostic models. Psychological Test and Assessment Modeling, 52(1), 8–28.Google Scholar
  105. von Davier, M. (2014). The log-linear cognitive diagnostic model (LCDM) as a special case of the general diagnostic model (GDM) (RR-14-40). Educational Testing Service. Princeton, NJ.Google Scholar
  106. von Davier, M. (this volume). The general diagnostic model. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  107. von Davier, M., & Haberman, S. J. (2014). Hierarchical diagnostic classification models morphing into unidimensional ‘diagnostic’ classification models − A commentary. Psychometrika, 79(2), 340–346.CrossRefGoogle Scholar
  108. von Davier, M., & Lee, Y.-S. (this volume). Introduction: From latent class analysis to DINA and beyond. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  109. von Davier, M., Naemi, B., & Roberts, R. D. (2012). Factorial versus typological models: A comparison of methods for personality data. Measurement: Interdisciplinary Research and Perspectives, 10(4), 185–208.Google Scholar
  110. Wang, W., Song, L., Chen, P., Meng, Y., & Ding, S. (2015). Attribute-level and pattern-level classification consistency and accuracy indices for cognitive diagnostic assessment. Journal of Educational Measurement, 52(4), 457–476.CrossRefGoogle Scholar
  111. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25.CrossRefGoogle Scholar
  112. Wilson, M. (2009). Measuring progressions: Assessment structures underlying a learning progression. Journal of Research in Science Teaching, 46(6), 716–730.CrossRefGoogle Scholar
  113. Xu, G. (this volume). Identifiability and cognitive diagnosis models. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  114. Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113(523), 1284–1295.CrossRefGoogle Scholar
  115. Xu, X., & von Davier, M. (2008a). Comparing multiple-group multinomial log-linear models for multidimensional skill distributions in the general diagnostic model (RR-08-35). Princeton, NJ: Educational Testing Service.Google Scholar
  116. Xu, X., & von Davier, M. (2008b). Fitting the structured general diagnostic model to NAEP data (RR-08-27). Educational Testing Service. Princeton, NJ.Google Scholar
  117. Xu, X., & von Davier, M. (this volume). Applying the general diagnostic model to proficiency data from a national skills survey. In M. von Davier & Y.-S. Lee (Eds.), Handbook of diagnostic classification models. Cham, Switzerland: Springer.Google Scholar
  118. Yamamoto, K. (1995). Estimating the effects of test length and test time on parameter estimation using the HYBRID model (TOEFL TR-10). Educational Testing Service. Princeton, NJ.Google Scholar
  119. Yamamoto, K., Khorramdel, L., & von Davier, M. (2013). Chapter 17: Scaling PIAAC cognitive data. In OECD (Ed.), Technical report of the survey of adult skills (PIAAC). Paris, France: OECD.Google Scholar
  120. Zhan, P. (2017). Using JAGS for Bayesian cognitive diagnosis models: A tutorial. arXiv:1708.02632.Google Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Educational MeasurementIPN Leibniz Institute for Science and Mathematics EducationKielGermany
  2. 2.Centre for International Student AssessmentMunichGermany
  3. 3.Federal Institute for Educational Research, Innovation and Development of the Austrian School SystemSalzburgAustria

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