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.
Correspondence concerning this article should be sent to Alexander Robitzsch, Leibniz Institute for Science and Mathematics Education (IPN), Olshausenstr. 62, 24118 Kiel, Germany. Email: robitzsch@ipn.uni-kiel.de
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Researchers von Davier and Haberman (2014) showed that a linear hierarchy among skills implies a reduced number of identifiable item parameters.
- 2.
Note that in the G-DINA model, larger regularization parameters were chosen because item parameters were estimated in the logit metric. In the regularized latent class model, item parameters are estimated in the metric of probabilities and, hence, smaller values have to be chosen. For smaller sample sizes, a wider range of λ values should be chosen.
- 3.
The standard deviation of the 2PL model cannot be directly compared with the 1PL model as the value depends on the choice of the reference item.
References
Asparouhov, T., & Muthen, B. (2014). Variable-specific entropy contribution (Technical appendix). http://www.statmodel.com/7_3_papers.shtml
Bartolucci, F. (2007). A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika, 72(2), 141–157.
Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122–4137.
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.
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.
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.
Chen, J., & Zhou, H. (2017). Test designs and modeling under the general nominal diagnosis model framework. PLoS One, 12(6), e0180016.
Chen, Y., Li, X., Liu, J., & Ying, Z. (2016). A fused latent and graphical model for multivariate binary data. arXiv:1606.08925.
Chen, Y., Li, X., Liu, J., & Ying, Z. (2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82(3), 660–692.
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.
Chiu, C. Y. (2013). Statistical refinement of the Q-matrix in cognitive diagnosis. Applied Psychological Measurement, 37(8), 598–618.
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.
Croon, M. (1990). Latent class analysis with ordered latent classes. British Journal of Mathematical and Statistical Psychology, 43(2), 171–192.
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.
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.
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.
de la Torre, J. (2009a). A cognitive diagnosis model for cognitively based multiple-choice options. Applied Psychological Measurement, 33(3), 163–183.
de la Torre, J. (2009b). DINA model and parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34(1), 115–130.
de la Torre, J. (2011). The generalized DINA model framework. Psychometrika, 76(2), 179–199.
de la Torre, J., & Douglas, J. (2004). Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69(3), 333–353.
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.
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.
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.
De Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11(3), 183–196.
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.
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.
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.
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.
Fan, J., & Lv, J. (2010). A selective overview of variable selection in high dimensional feature space. Statistica Sinica, 20(1), 101–148.
Fiacco, A. V., & McCormick, G. P. (1968). Nonlinear programming: Sequential unconstrained minimization techniques. New York, NY: Wiley.
Formann, A. K. (1985). Constrained latent class models: Theory and applications. British Journal of Mathematical and Statistical Psychology, 38(1), 87–111.
Formann, A. K. (1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 87(418), 476–486.
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.
Formann, A. K., & Kohlmann, T. (1998). Structural latent class models. Sociological Methods & Research, 26(4), 530–565.
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.
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.
George, A. C., & Robitzsch, A. (2015). Cognitive diagnosis models in R: A didactic. The Quantitative Methods for Psychology, 11(3), 189–205.
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.
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
Groß, J., & George, A. C. (2014). On prerequisite relations between attributes in noncompensatory diagnostic classification. Methodology, 10(3), 100–107.
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.
Hastie, T., Tibshirani, R., & Wainwright, M. (2015). Statistical learning with sparsity: The lasso and generalizations. Boca Raton, FL: CRC Press.
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.
Henson, R., Roussos, L., Douglas, J., & He, X. (2008). Cognitive diagnostic attribute-level discrimination indices. Applied Psychological Measurement, 32(4), 275–288.
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.
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.
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.
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.
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.
Huang, H. Y., & Wang, W. C. (2014). The random-effect DINA model. Journal of Educational Measurement, 51(1), 75–97.
Huang, P. H., Chen, H., & Weng, L. J. (2017). A penalized likelihood method for structural equation modeling. Psychometrika, 82(2), 329–354.
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.
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.
Kang, H.-A., Liu, J., & Ying, Z. (2017). A general diagnostic classification model. arXiv:1707.06318.
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.
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.
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.
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.
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.
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.
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.
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.
Liu, J., Xu, G., & Ying, Z. (2013). Theory of the self-learning Q-matrix. Bernoulli, 19(5A), 1790–1817.
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.
Liu, Y., Douglas, J. A., & Henson, R. A. (2009). Testing person fit in cognitive diagnosis. Applied Psychological Measurement, 33(8), 579–598.
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.
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.
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.
Ma, W., Iaconangelo, C., & de la Torre, J. (2016). Model similarity, model selection, and attribute classification. Applied Psychological Measurement, 40(3), 200–217.
Maydeu-Olivares, A., & Joe, H. (2014). Assessing approximate fit in categorical data analysis. Multivariate Behavioral Research, 49(4), 305–328.
McDonald, R. P., & Mok, M. M. C. (1995). Goodness of fit in item response models. Multivariate Behavioral Research, 30(1), 23–40.
Mislevy, R. J., & Wilson, M. (1996). Marginal maximum likelihood estimation for a psychometric model of discontinuous development. Psychometrika, 61(1), 41–71.
Nussbeck, F. W., & Eid, M. (2015). Multimethod latent class analysis. Frontiers in Psychology | Quantitative Psychology and Measurement, 6, 1332.
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.
Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24(1), 50–64.
Ozaki, K. (2015). DINA models for multiple-choice items with few parameters: Considering incorrect answers. Applied Psychological Measurement, 39(6), 431–447.
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.
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.
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.
Pritikin, J. N. (2017). A comparison of parameter covariance estimation methods for item response models in an expectation-maximization framework. Cogent Psychology, 4, 1279435.
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.
R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
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.
Ravand, H., & Robitzsch, A. (2015). Cognitive diagnostic modeling using R. Practical Assessment, Research & Evaluation, 20(11), 1–12.
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
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.
San Martin, E. S., del Pino, G., & De Boeck, P. (2006). IRT models for ability-based guessing. Applied Psychological Measurement, 30(3), 183–203.
Sen, S., & Bradshaw, L. (2017). Comparison of relative fit indices for diagnostic model selection. Applied Psychological Measurement, 41(6), 422–438.
Shen, X., Pan, W., & Zhu, Y. (2012). Likelihood-based selection and sharp parameter estimation. Journal of the American Statistical Association, 107(497), 223–232.
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.
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.
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.
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.
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.
Templin, J., & Bradshaw, L. (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies. Psychometrika, 79(2), 317–339.
Templin, J., & Hoffman, L. (2013). Obtaining diagnostic classification model estimates using Mplus. Educational Measurement: Issues and Practice, 32(2), 37–50.
Templin, J. L., & Henson, R. A. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11(3), 287–305.
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.
Tutz, G., & Schauberger, G. (2015). A penalty approach to differential item functioning in Rasch models. Psychometrika, 80(1), 21–43.
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.
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.
von Davier, M. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61(2), 287–307.
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.
von Davier, M. (2010). Hierarchical mixtures of diagnostic models. Psychological Test and Assessment Modeling, 52(1), 8–28.
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.
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.
von Davier, M., & Haberman, S. J. (2014). Hierarchical diagnostic classification models morphing into unidimensional ‘diagnostic’ classification models − A commentary. Psychometrika, 79(2), 340–346.
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.
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.
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.
White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25.
Wilson, M. (2009). Measuring progressions: Assessment structures underlying a learning progression. Journal of Research in Science Teaching, 46(6), 716–730.
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.
Xu, G., & Shang, Z. (2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113(523), 1284–1295.
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.
Xu, X., & von Davier, M. (2008b). Fitting the structured general diagnostic model to NAEP data (RR-08-27). Educational Testing Service. Princeton, NJ.
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.
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.
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.
Zhan, P. (2017). Using JAGS for Bayesian cognitive diagnosis models: A tutorial. arXiv:1708.02632.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Robitzsch, A., George, A.C. (2019). The R Package CDM for Diagnostic Modeling. 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_26
Download citation
DOI: https://doi.org/10.1007/978-3-030-05584-4_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05583-7
Online ISBN: 978-3-030-05584-4
eBook Packages: EducationEducation (R0)