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Regularized Latent Class Analysis with Application in Cognitive Diagnosis

Published online by Cambridge University Press:  01 January 2025

Yunxiao Chen ,
Xiaoou Li ,
Jingchen Liu  and
Zhiliang Ying
Yunxiao Chen
Affiliation:
Emory University
Xiaoou Li
Affiliation:
University of Minnesota
Jingchen Liu*
Affiliation:
Columbia University
Zhiliang Ying
Affiliation:
Columbia University
*
Correspondence should be made to Jingchen Liu , Department of Statistics, Columbia University, New York, NY, USA. Email: jcliu@stat.columbia.edu
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Abstract

Diagnostic classification models are confirmatory in the sense that the relationship between the latent attributes and responses to items is specified or parameterized. Such models are readily interpretable with each component of the model usually having a practical meaning. However, parameterized diagnostic classification models are sometimes too simple to capture all the data patterns, resulting in significant model lack of fit. In this paper, we attempt to obtain a compromise between interpretability and goodness of fit by regularizing a latent class model. Our approach starts with minimal assumptions on the data structure, followed by suitable regularization to reduce complexity, so that readily interpretable, yet flexible model is obtained. An expectation–maximization-type algorithm is developed for efficient computation. It is shown that the proposed approach enjoys good theoretical properties. Results from simulation studies and a real application are presented.

Keywords

diagnostic classification modelslatent class analysisregularizationconsistencyEM algorithm
Type
Original paper
Information
Psychometrika , Volume 82 , Issue 3 , September 2017 , pp. 660 - 692
Copyright
Copyright © 2016 The Psychometric Society

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References

Allman, E. S., Matias, C., & Rhodes, J. A.. (2009). Identifiability of parameters in latent structure models with many observed variables. The Annals of Statistics, 37, 3099–3132. doi:10.1214/09-AOS689.CrossRefGoogle Scholar
American Psychiatric Association Diagnostic and statistical manual of mental disorders 1994 4Washington, DC: American Psychiatric Association.Google Scholar
Chen, J., & Chen, Z.. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95, 759–771. doi:10.1093/biomet/asn034.CrossRefGoogle Scholar
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, 850–866. doi:10.1080/01621459.2014.934827.CrossRefGoogle Scholar
Chen, Y., Liu, J., & Ying, Z.. (2015). Online item calibration for Q-matrix in CD-CAT. Applied Psychological Measurement, 39, 5–15. doi:10.1177/0146621613513065.CrossRefGoogle ScholarPubMed
Croon, M.. (1990). Latent class analysis with ordered latent classes. British Journal of Mathematical and Statistical Psychology, 43, 171–192. doi:10.1111/j.2044-8317.1990.tb00934.x.CrossRefGoogle Scholar
Croon, M.. (1991). Investigating mokken scalability of dichotomous items by means of ordinal latent class analysis. British Journal of Mathematical and Statistical Psychology, 44, 315–331. doi:10.1111/j.2044-8317.1991.tb00964.x.CrossRefGoogle Scholar
Dalrymple, K., & D’Avanzato, C.. (2013). Differentiating the subtypes of social anxiety disorder. Expert Review of Neurotherapeutics, 13, 1271–1283. doi:10.1586/14737175.2013.853446.CrossRefGoogle ScholarPubMed
de la Torre, J.. (2011). The generalized DINA model framework. Psychometrika, 76, 179–199. doi:10.1007/s11336-011-9207-7.CrossRefGoogle Scholar
de la Torre, J., & Douglas, J.. (2004). Higher order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353. doi:10.1007/BF02295640.CrossRefGoogle Scholar
DiBello, L. V., Stout, W. F., Roussos, L. A.,Paul, R. L. B., Nichols, D., & Chipman, Susan F. (1995). Unified cognitive/psychometric diagnostic assessment likelihood-based classification techniques. Cognitively diagnostic assessment. Hillsdale, NJ: Erlbaum 361–389.Google Scholar
Fan, J., & Li, R.. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96, 1348–1360. doi:10.1198/016214501753382273.CrossRefGoogle Scholar
Fan, Y., & Tang, C. Y.. (2013). Tuning parameter selection in high dimensional penalized likelihood. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75, 531–552. doi:10.1111/rssb.12001.CrossRefGoogle Scholar
Friedman, J., Hastie, T., & Tibshirani, R.. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33, 1–22. doi:10.18637/jss.v033.i01 2929880.CrossRefGoogle ScholarPubMed
Goodman, L. A.. (1974). The analysis of systems of qualitative variables when some of the variables are unobservable. Part I—a modified latent structure approach. American Journal of Sociology, 79, 1179–1259. doi:10.1086/225676.CrossRefGoogle Scholar
Goodman, L. A.. (1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231. doi:10.1093/biomet/61.2.215.CrossRefGoogle Scholar
Grant, B. F., Kaplan, K., Shepard, J., & Moore, T.Source and accuracy statement for Wave 1 of the 2001–2002 National Epidemiologic Survey on Alcohol and Related Conditions 2003 Bethesda, MD: National Institute on Alcohol Abuse and Alcoholism.Google Scholar
Haberman, S. J., von Davier, M. & Lee, Y.-H. (2008) Comparison of multidimensional item response models: Multivariate normal ability distributions versus multivariate polytomous ability distributions. ETS Research Rep. No. RR-08-45. Princeton, NJ: ETS..CrossRefGoogle Scholar
Haertel, E. H.. (1989). Using restricted latent class models to map the skill structure of achievement items. Journal of Educational Measurement, 26, 301–321. doi:10.1111/j.1745-3984.1989.tb00336.x.CrossRefGoogle Scholar
Henson, R. A., Templin, J. L., & Willse, J. T.. (2009). Defining a family of cognitive diagnosis models using log-linear models with latent variables. Psychometrika, 74, 191–210. doi:10.1007/s11336-008-9089-5.CrossRefGoogle Scholar
Junker, B., & Sijtsma, K.. (2001). Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Applied Psychological Measurement, 25, 258–272. doi:10.1177/01466210122032064.CrossRefGoogle Scholar
Kessler, R. C., Berglund, P., Demler, O., Jin, R., Merikangas, K. R., & Walters, E. E.. (2005). Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the national comorbidity survey replication. Archives of General Psychiatry, 62, 593–602. doi:10.1001/archpsyc.62.6.593.CrossRefGoogle ScholarPubMed
Lazarsfeld, P. F., Henry, N. W., Anderson, T. W., (1968). Latent structure analysis. Boston, MA: Houghton Mifflin.Google Scholar
Lehmann, E. L., Casella, G.Theory of point estimation 2006 New York: Springer.Google Scholar
Leighton, J., Gierl, M.Cognitive diagnostic assessment for education: Theory and applications 2007 Cambridge: Cambridge University Pressdoi:10.1017/CBO9780511611186.CrossRefGoogle Scholar
Leighton, J. P., Gierl, M. J., Hunka, S. M.. (2004). The attribute hierarchy model for cognitive assessment: A variation on Tatsuoka’s rule-space approach. Journal of Educational Measurement, 41, 205–237. doi:10.1111/j.1745-3984.2004.tb01163.x.CrossRefGoogle Scholar
Li, X., Liu, J. & Ying, Z. (2016). Chernoff index for cox test of separate parametric families. arXiv:1606.08248.Google Scholar
Liu, J., Xu, G., & Ying, Z.. (2012). Data-driven learning of Q-matrix. Applied Psychological Measurement, 36, 548–564. doi:10.1177/0146621612456591 3733574.CrossRefGoogle ScholarPubMed
Liu, J., Xu, G., & Ying, Z.. (2013). Theory of self-learning Q-matrix. Bernoulli, 19, 1790–1817. doi:10.3150/12-BEJ430 4011940.CrossRefGoogle ScholarPubMed
Nishii, R. et al. (1984). Asymptotic properties of criteria for selection of variables in multiple regression. The Annals of Statistics, 12, 758–765. doi:10.1214/aos/1176346522.CrossRefGoogle Scholar
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 Perspective, 6, 219–262.Google Scholar
Rupp, A., Templin, J., & Henson, R. A., (2010). Diagnostic measurement: Theory, methods, and applications. New York: Guilford Press.Google Scholar
Stein, M. B., & Stein, D. J.. (2008). Social anxiety disorder. The Lancet, 371, 1115–1125. doi:10.1016/S0140-6736(08)60488-2.CrossRefGoogle ScholarPubMed
Tatsuoka, C.. (2002). Data analytic methods for latent partially ordered classification models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 51, 337–350. doi:10.1111/1467-9876.00272.Google Scholar
Tatsuoka, C., & Ferguson, T.. (2003). Sequential classification on partially ordered sets. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65, 143–157. doi:10.1111/1467-9868.00377.CrossRefGoogle Scholar
Tatsuoka, K.. (1985). A probabilistic model for diagnosing misconceptions in the pattern classification approach. Journal of Educational Statistics, 12, 55–73.CrossRefGoogle Scholar
Tatsuoka, K.Cognitive assessment: An introduction to the rule space method 2009 New York: Routledge.CrossRefGoogle Scholar
Templin, J., & Henson, R.. (2006). Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287–305. doi:10.1037/1082-989X.11.3.287.CrossRefGoogle ScholarPubMed
Tibshirani, R.. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 58, 267–288.CrossRefGoogle Scholar
von Davier, M. (2005) A general diagnosis model applied to language testing data. ETS Research Rep. No. RR-05-16. Princeton, NJ: ETS..CrossRefGoogle Scholar
von Davier, M.. (2008). A general diagnostic model applied to language testing data. British Journal of Mathematical and Statistical Psychology, 61, 287–307. doi:10.1348/000711007X193957.CrossRefGoogle ScholarPubMed
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. doi:10.1111/bmsp.12003.CrossRefGoogle Scholar
von Davier, M., & Haberman, S. J.. (2014). Hierarchical diagnostic classification models morphing into unidimensional ‘diagnostic’ classification models—a commentary. Psychometrika, 79, 340–346. doi:10.1007/s11336-013-9363-z.CrossRefGoogle ScholarPubMed
von Davier, M. & Yamamoto, K. (2004) A class of models for cognitive diagnosis. In 4th Spearman Conference, Philadelphia, PA..Google Scholar
Wang, H., Li, B., & Leng, C.. (2009). Shrinkage tuning parameter selection with a diverging number of parameters. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71, 671–683. doi:10.1111/j.1467-9868.2008.00693.x.CrossRefGoogle Scholar
Wang, H., Li, R., & Tsai, C-L. (2007). Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika, 94, 553–568. doi:10.1093/biomet/asm053 2663963.CrossRefGoogle ScholarPubMed
Wang, T., & Zhu, L.. (2011). Consistent tuning parameter selection in high dimensional sparse linear regression. Journal of Multivariate Analysis, 102, 1141–1151. doi:10.1016/j.jmva.2011.03.007.CrossRefGoogle Scholar
Xu, G. (2016). Identifiability of restricted latent class models with binary responses. arXiv:1603.04140.Google Scholar