Hostname: page-component-5db58dd55d-ggg9q Total loading time: 0 Render date: 2026-06-01T23:07:56.896Z Has data issue: false hasContentIssue false
  • English
  • Français

Direct Estimation of Diagnostic Classification Model Attribute Mastery Profiles via a Collapsed Gibbs Sampling Algorithm

Published online by Cambridge University Press:  01 January 2025

Kazuhiro Yamaguchi Open the ORCID record for Kazuhiro Yamaguchi [Opens in a new window]  and
Jonathan Templin Open the ORCID record for Jonathan Templin [Opens in a new window]
Kazuhiro Yamaguchi*
Affiliation:
University Of Tsukuba
Jonathan Templin
Affiliation:
The University Of Iowa
*
Correspondence should be made to Kazuhiro Yamaguchi, Division of Psychology, Faculty of Human Sciences, University of Tsukuba, Institutes of Human Sciences A314, 1-1-1 Tennodai, Tsukuba, Ibaraki 3050006, Japan. Email: yamaguchi.kazuhir.ft@u.tsukuba.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes a novel collapsed Gibbs sampling algorithm that marginalizes model parameters and directly samples latent attribute mastery patterns in diagnostic classification models. This estimation method makes it possible to avoid boundary problems in the estimation of model item parameters by eliminating the need to estimate such parameters. A simulation study showed the collapsed Gibbs sampling algorithm can accurately recover the true attribute mastery status in various conditions. A second simulation showed the collapsed Gibbs sampling algorithm was computationally more efficient than another MCMC sampling algorithm, implemented by JAGS. In an analysis of real data, the collapsed Gibbs sampling algorithm indicated good classification agreement with results from a previous study.

Keywords

diagnostic classification modelsMarkov chain Monte Carlo methodGibbs samplingBayesian inference

Information

Type
Theory and Methods
Information
Psychometrika , Volume 87 , Issue 4 , December 2022 , pp. 1390 - 1421
Copyright
Copyright © 2022 The Author(s) under exclusive licence to The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/S0033312300005512a.

References

Bishop, M. (2006). Pattern recognition and machine learning. Springer. https://10.1641/B580519CrossRefGoogle Scholar
Blei, D. M., Jordan, M. I., (2006). Variational inference for Dirichlet process mixtures Bayesian Analysis 1 1A 121–144 10.1214/06-BA104CrossRefGoogle Scholar
Brooks, S. P., Gelman, A., (1998). General methods for monitoring convergence of iterative simulations Journal of Computational and Graphical Statistics 7 434–455 10.2307/1390675CrossRefGoogle Scholar
Chen, Y., Culpepper, S. A., Chen, Y., Douglas, J., (2018). Baysean estimation of the the DINA Q matrix Psychometrika 83 89–108 10.1007/s11336-017-9579-4 28861685CrossRefGoogle Scholar
Chen, Y., Culpepper, S., Liang, F., (2020). A sparse latent class model for cognitive diagnosis Psychometrika 85 (1) 121–153 10.1007/s11336-019-09693-2 31927684CrossRefGoogle ScholarPubMed
Chiu, C. Y., Douglas, J., (2013). A nonparametric approach to cognitive diagnosis by proximity to ideal response patterns Journal of Classification 30 225–250 10.1007/s00357-013-9132-9CrossRefGoogle Scholar
Chung, M., (2019). A Gibbs sampling algorithm that estimates the Q-matrix for the DINA model Journal of Mathematical Psychology 93 10.1016/j.jmp.2019.07.002CrossRefGoogle Scholar
Cohen, J., (1960). A coefficient of agreement for nominal scales Educational and Psychological Measurement 20 37–46 10.1177/001316446002000104CrossRefGoogle Scholar
Cowles, M. K., Carlin, B. P., (1996). Markov chain Monte Carlo convergence diagnostics: A comparative review Journal of the American Statistical Association 91 (434) 883–904 10.1080/01621459.1996.10476956CrossRefGoogle Scholar
Culpepper, S. A., (2015). Bayesian estimation of the DINA model with Gibbs sampling Journal of Educational and Behavioral Statistics 40 454–476 10.3102/1076998615595403CrossRefGoogle Scholar
Culpepper, S. A., (2019). An exploratory diagnostic model for ordinal responses with binary attributes: Identifiability and estimation Psychometrika 84 (3) 921–940 10.1007/s11336-019-09683-4 31432312CrossRefGoogle ScholarPubMed
Culpepper, S. A., (2019). Estimating the cognitive diagnosis Q matrix with expert knowledge: Application to the fraction-subtraction dataset Psychometrika 84 (2) 333–357 10.1007/s11336-018-9643-8 30456748CrossRefGoogle Scholar
Culpepper, S. A., Hudson, A., (2018). An improved strategy for Bayesian estimation of the reduced reparameterized unified model Applied Psychological Measurement 42 99–115 10.1177/0146621617707511 29881115CrossRefGoogle ScholarPubMed
de la Torre, J., (2011). The generalized DINA model framework Psychometrika 76 179–199 10.1007/S11336-011-9207-7CrossRefGoogle Scholar
de la Torre, J., Douglas, J. A., (2004). Higher-order latent trait models for cognitive diagnosis Psychometrika 69 333–353 10.1007/BF02295640CrossRefGoogle Scholar
DeCarlo, L. T., (2011). On the analysis of fraction subtraction data: The DINA model, classification, latent class sizes, and the Q-matrix Applied Psychological Measurement 35 8–26 10.1177/0146621610377081CrossRefGoogle Scholar
DeCarlo, L. T., (2012). Recognizing uncertainty in the Q-Matrix via a Bayesian extension of the DINA Model Applied Psychological Measurement 36 447–468 10.1177/0146621612449069CrossRefGoogle Scholar
Galindo-Garre, F., Vermunt, J. K., (2006). Avoiding boundary estimates in latent class analysis by Bayesian posterior mode estimation Behaviormetrika 33 (1) 43–59 10.2333/bhmk.33.43CrossRefGoogle Scholar
George, A. C., Robitzsch, A., Kiefer, T., Groß, J., & Ünlü, A. (2016). The R package CDM for cognitive diagnosis models. Journal of Statistical Software. https://10.18637/jss.v074.i02CrossRefGoogle Scholar
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In Bernardo, J. M. Berger, J. O. Dawid, A. P. & Smith, A. F. M. (Eds.), Bayesian statistics (4th ed., pp. 169–193). Oxford University Press.CrossRefGoogle Scholar
Gu, Y., Xu, G., (2020). Partial identifiability of restricted latent class models Annals of Statistics 48 (3) 2082–2107 10.1214/19-AOS1878CrossRefGoogle 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 10.1007/S11336-008CrossRefGoogle Scholar
Jiang, Z., Carter, R., (2019). Using Hamiltonian Monte Carlo to estimate the log-linear cognitive diagnosis model via Stan Behavior Research Methods 51 651–662 10.3758/s13428-018-1069-9 29949073CrossRefGoogle ScholarPubMed
Lee, Y-S Park, Y. S., Taylan, D., (2011). A cognitive diagnostic modeling of attribute mastery in Massachusetts, Minnesota, and the U.S. national sample using the TIMSS 2007 International Journal of Testing 11 144–177 10.1080/15305058.2010.534571CrossRefGoogle Scholar
Li, F., Cohen, A., Bottge, B., Templin, J., (2016). A latent transition analysis model for assessing change in cognitive skills Educational and Psychological Measurement 76 181–204 10.1177/0013164415588946 29795862CrossRefGoogle ScholarPubMed
Liu, J. S., (1994). The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem Journal of the American Statistical Association 89 (427) 958–966 10.1080/01621459.1994.10476829CrossRefGoogle Scholar
Ma, W., & de la Torre, J. (2020). GDINA: An R package for cognitive diagnosis modeling. Journal of Statistical Software, 93(12), 1–26. https://10.18637/jss.v093.i14CrossRefGoogle Scholar
Ma, W., Jiang, Z., (2021). Estimating cognitive diagnosis models in small samples: Bayes modal estimation and monotonic constraints Applied Psychological Measurement 45 (2) 95–111 10.1177/0146621620977681 33627916CrossRefGoogle ScholarPubMed
Maris, E., (1999). Estimating multiple classification latent class models Psychometrika 64 187–212 10.1007/BF02294535CrossRefGoogle Scholar
McLachlan, G. J., Lee, S. X., Rathnayake, S. I., (2019). Finite mixture models Annual Review of Statistics and Its Application 6 355–375 10.1146/annurev-statistics-031017-100325CrossRefGoogle Scholar
McLachlan, G., & Peel, D. (2000). Finite mixture models. Wiley.CrossRefGoogle Scholar
Muthén, L. K., & Muthén, B. O. (1998–2017). Mplus user’s guide (8th ed.). Muthén & Muthén.Google Scholar
Nakajima, S., Watanabe, K., & Sugiyama, M. (2019). Variational Bayesian learning theory. Cambridge University Press. https://10.1017/9781139879354CrossRefGoogle Scholar
Papastamoulis, P. (2015). label.switching: An R package for dealing with the label switching problem in MCMC outputs. VV(February). https://10.18637/jss.v069.c01CrossRefGoogle Scholar
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 88–115 10.3102/1076998617719728CrossRefGoogle Scholar
Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. The 3rd international workshop on distributed statistical computing (Vol. 124, pp. 1–8). http://www.ci.tuwien.ac.at/Conferences/DSC-2003/Google Scholar
Plummer, M., Best, N., Cowles, K., & Vines, K. (2006). CODA: Convergence diagnosis and output analysis for MCMC. R News 6, 7–11. https://journal.r-project.org/archive/Google Scholar
Porteous, I., Newman, D., Ihler, A., Asuncion, A., Smyth, P., Welling, M., (2008). Fast collapsed Gibbs sampling for latent Dirichlet allocation IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E98A (1) 569–577 10.1587/transfun.E98.A.144Google Scholar
R Core Team. (2019). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/Google Scholar
Revelle, W. (2020). psych: Procedures for personality and psychological pesearch (Version 2.1.3) [Computer software]. CRAN. https://CRAN.R-project.org/package=psychGoogle Scholar
Rupp, A. A., Templin, J., (2008). Unique characteristics of diagnostic classification models: A comprehensive review of the current state-of-the-art Measurement: Interdisciplinary Research & Perspective 6 219–262 10.1080/15366360802490866Google Scholar
Rupp, A. A., Templin, J. L., & Henson, R. A. (2010). Diagnostic measurement: Theory, methods and applications. Guilford Press.Google Scholar
Sato, I. (2016). Bayesian Nonparametrics. Kodansha.Google Scholar
Stephens, M., (2000). Dealing with label switching in mixture models Journal of the Royal Statistical Society: Series B (Statistical Methodology) 62 (3) 795–809 10.1111/1467-9868.00265CrossRefGoogle Scholar
Suyama, A., & Sugiyama, M. (2017). Introduction to machine learning by Bayesian inference. Kodansha.Google Scholar
Templin, J., Bradshaw, L., (2014). Hierarchical diagnostic classification models: A family of models for estimating and testing attribute hierarchies Psychometrika 79 317–339 10.1007/s11336-013-9362-0 24478021CrossRefGoogle ScholarPubMed
Templin, J. L., Henson, R. A., (2006). Measurement of psychological disorders using cognitive diagnosis models Psychological Methods 11 (2) 287 10.1037/1082-989X.11.3.287 16953706CrossRefGoogle ScholarPubMed
Templin, J., Hoffman, L., (2013). Obtaining diagnostic classification model estimates using Mplus Educational Measurement: Issues and Practice 32 37–50 10.1111/emip.12010CrossRefGoogle Scholar
von Davier, M., (2008). A general diagnostic model applied to language testing data The British Journal of Mathematical and Statistical Psychology 61 Pt 2 287–307 10.1348/000711007X193957CrossRefGoogle ScholarPubMed
Wang, S., Douglas, J., (2015). Consistency of nonparametric classification in cognitive diagnosis Psychometrika 80 (1) 85–100 10.1007/s11336-013-9372-y 24297434CrossRefGoogle ScholarPubMed
Xu, G., Shang, Z., (2018). Identifying latent structures in restricted latent class models Journal of the American Statistical Association 113 (523) 1284–1295 10.1080/01621459.2017.1340889CrossRefGoogle Scholar
Yamaguchi, K., (2020). Variational Bayesian inference for the multiple-choice DINA model Behaviormetrika 47 159–187 10.1007/s41237-020-00104-wCrossRefGoogle Scholar
Yamaguchi, K., Okada, K., (2020). Variational Bayes inference for the DINA model Journal of Educational and Behavioral Statistics 45 (4) 569–597 10.3102/1076998620911934CrossRefGoogle Scholar
Yamaguchi, K., Okada, J., (2021). Variational Bayes inference algorithm for the saturated diagnostic classification model Psychometrika 85 (3) 973–995 10.1007/s11336-020-09739-wCrossRefGoogle Scholar
Yamaguchi, K., & Templin, J. (2022). A Gibbs sampling algorithm with monotonicity constraints for diagnostic classification models. Journal of Classification, 39, 24–54. https://10.1007/s00357-021-09392-7CrossRefGoogle Scholar
Zhan, P., Jiao, H., Man, K., Wang, L., (2019). Using JAGS for Bayesian cognitive diagnosis modeling: A tutorial Journal of Educational and Behavioral Statistics 44 (3) 473–503 10.3102/1076998619826040CrossRefGoogle Scholar
Zheng, Y., Chiu, C.-Y., & Douglas, A. J. (2019). Package ‘NPCD’ (1,0-11). https://cran.r-project.org/web/packages/NPCD/index.htmlGoogle Scholar
Supplementary material: File

Yamaguchi and Templin supplementary material

Yamaguchi and Templin supplementary material
Download Yamaguchi and Templin supplementary material(File)
File 129 KB