IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v89y2024i2d10.1007_s11336-023-09940-7.html
   My bibliography  Save this article

Restricted Latent Class Models for Nominal Response Data: Identifiability and Estimation

Author

Listed:
  • Ying Liu

    (University of Illinois at Urbana-Champaign)

  • Steven Andrew Culpepper

    (University of Illinois at Urbana-Champaign)

Abstract

Restricted latent class models (RLCMs) provide an important framework for diagnosing and classifying respondents on a collection of multivariate binary responses. Recent research made significant advances in theory for establishing identifiability conditions for RLCMs with binary and polytomous response data. Multiclass data, which are unordered nominal response data, are also widely collected in the social sciences and psychometrics via forced-choice inventories and multiple choice tests. We establish new identifiability conditions for parameters of RLCMs for multiclass data and discuss the implications for substantive applications. The new identifiability conditions are applicable to a wealth of RLCMs for polytomous and nominal response data. We propose a Bayesian framework for inferring model parameters, assess parameter recovery in a Monte Carlo simulation study, and present an application of the model to a real dataset.

Suggested Citation

  • Ying Liu & Steven Andrew Culpepper, 2024. "Restricted Latent Class Models for Nominal Response Data: Identifiability and Estimation," Psychometrika, Springer;The Psychometric Society, vol. 89(2), pages 592-625, June.
    • Handle: RePEc:spr:psycho:v:89:y:2024:i:2:d:10.1007_s11336-023-09940-7
    DOI: 10.1007/s11336-023-09940-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-023-09940-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11336-023-09940-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Michela Gnaldi & Silvia Bacci & Thiemo Kunze & Samuel Greiff, 2020. "Students’ Complex Problem Solving Profiles," Psychometrika, Springer;The Psychometric Society, vol. 85(2), pages 469-501, June.
    2. Guan-Hua Huang & Karen Bandeen-Roche, 2004. "Building an identifiable latent class model with covariate effects on underlying and measured variables," Psychometrika, Springer;The Psychometric Society, vol. 69(1), pages 5-32, March.
    3. Chia-Yi Chiu & Jeffrey Douglas & Xiaodong Li, 2009. "Cluster Analysis for Cognitive Diagnosis: Theory and Applications," Psychometrika, Springer;The Psychometric Society, vol. 74(4), pages 633-665, December.
    4. Gongjun Xu & Zhuoran Shang, 2018. "Identifying Latent Structures in Restricted Latent Class Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1284-1295, July.
    5. Francesco Bartolucci, 2007. "A class of multidimensional IRT models for testing unidimensionality and clustering items," Psychometrika, Springer;The Psychometric Society, vol. 72(2), pages 141-157, June.
    6. Dunson, David B. & Xing, Chuanhua, 2009. "Nonparametric Bayes Modeling of Multivariate Categorical Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(487), pages 1042-1051.
    7. Joseph Kruskal, 1976. "More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling," Psychometrika, Springer;The Psychometric Society, vol. 41(3), pages 281-293, September.
    8. Yinghan Chen & Ying Liu & Steven Andrew Culpepper & Yuguo Chen, 2021. "Inferring the Number of Attributes for the Exploratory DINA Model," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 30-64, March.
    9. Steven Andrew Culpepper, 2019. "An Exploratory Diagnostic Model for Ordinal Responses with Binary Attributes: Identifiability and Estimation," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 921-940, December.
    10. Laine Bradshaw & Jonathan Templin, 2014. "Combining Item Response Theory and Diagnostic Classification Models: A Psychometric Model for Scaling Ability and Diagnosing Misconceptions," Psychometrika, Springer;The Psychometric Society, vol. 79(3), pages 403-425, July.
    11. Jared S. Murray & Jerome P. Reiter, 2016. "Multiple Imputation of Missing Categorical and Continuous Values via Bayesian Mixture Models With Local Dependence," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(516), pages 1466-1479, October.
    12. Nicholas G. Polson & James G. Scott & Jesse Windle, 2013. "Bayesian Inference for Logistic Models Using Pólya--Gamma Latent Variables," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(504), pages 1339-1349, December.
    13. Yunxiao Chen & Jingchen Liu & Gongjun Xu & Zhiliang Ying, 2015. "Statistical Analysis of Q -Matrix Based Diagnostic Classification Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 850-866, June.
    14. Jinsong Chen & Hui Zhou, 2017. "Test designs and modeling under the general nominal diagnosis model framework," PLOS ONE, Public Library of Science, vol. 12(6), pages 1-14, June.
    15. Jimmy de la Torre, 2011. "The Generalized DINA Model Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(2), pages 179-199, April.
    16. Chen, Yunxiao & Liu, Jingchen & Xu, Gongjun & Ying, Zhiliang, 2015. "Statistical analysis of Q-matrix based diagnostic classification models," LSE Research Online Documents on Economics 103183, London School of Economics and Political Science, LSE Library.
    17. James Joseph Balamuta & Steven Andrew Culpepper, 2022. "Exploratory Restricted Latent Class Models with Monotonicity Requirements under PÒLYA–GAMMA Data Augmentation," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 903-945, September.
    18. Zhehan Jiang & Jonathan Templin, 2019. "Gibbs Samplers for Logistic Item Response Models via the Pólya–Gamma Distribution: A Computationally Efficient Data-Augmentation Strategy," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 358-374, June.
    19. Yinyin Chen & Steven Culpepper & Feng Liang, 2020. "A Sparse Latent Class Model for Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 85(1), pages 121-153, March.
    20. Guanhua Fang & Jingchen Liu & Zhiliang Ying, 2019. "On the Identifiability of Diagnostic Classification Models," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 19-40, March.
    21. Richard McHugh, 1956. "Efficient estimation and local identification in latent class analysis," Psychometrika, Springer;The Psychometric Society, vol. 21(4), pages 331-347, December.
    22. Jimmy Torre, 2011. "Erratum to: The Generalized DINA Model Framework," Psychometrika, Springer;The Psychometric Society, vol. 76(3), pages 510-510, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. James Joseph Balamuta & Steven Andrew Culpepper, 2022. "Exploratory Restricted Latent Class Models with Monotonicity Requirements under PÒLYA–GAMMA Data Augmentation," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 903-945, September.
    2. Yuqi Gu, 2024. "Going Deep in Diagnostic Modeling: Deep Cognitive Diagnostic Models (DeepCDMs)," Psychometrika, Springer;The Psychometric Society, vol. 89(1), pages 118-150, March.
    3. Steven Andrew Culpepper, 2023. "A Note on Weaker Conditions for Identifying Restricted Latent Class Models for Binary Responses," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 158-174, March.
    4. Jing Ouyang & Gongjun Xu, 2022. "Identifiability of Latent Class Models with Covariates," Psychometrika, Springer;The Psychometric Society, vol. 87(4), pages 1343-1360, December.
    5. Motonori Oka & Kensuke Okada, 2023. "Scalable Bayesian Approach for the Dina Q-Matrix Estimation Combining Stochastic Optimization and Variational Inference," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 302-331, March.
    6. Mengqi Lin & Gongjun Xu, 2024. "Sufficient and Necessary Conditions for the Identifiability of DINA Models with Polytomous Responses," Psychometrika, Springer;The Psychometric Society, vol. 89(2), pages 717-740, June.
    7. Yuqi Gu, 2023. "Generic Identifiability of the DINA Model and Blessing of Latent Dependence," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 117-131, March.
    8. Yinghan Chen & Ying Liu & Steven Andrew Culpepper & Yuguo Chen, 2021. "Inferring the Number of Attributes for the Exploratory DINA Model," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 30-64, March.
    9. Seunghyun Lee & Yuqi Gu, 2024. "New Paradigm of Identifiable General-response Cognitive Diagnostic Models: Beyond Categorical Data," Psychometrika, Springer;The Psychometric Society, vol. 89(4), pages 1304-1336, December.
    10. Xin Xu & Guanhua Fang & Jinxin Guo & Zhiliang Ying & Susu Zhang, 2024. "Diagnostic Classification Models for Testlets: Methods and Theory," Psychometrika, Springer;The Psychometric Society, vol. 89(3), pages 851-876, September.
    11. Ying Liu & Steven Andrew Culpepper & Yuguo Chen, 2023. "Identifiability of Hidden Markov Models for Learning Trajectories in Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 361-386, June.
    12. Yinghan Chen & Steven Andrew Culpepper & Yuguo Chen, 2023. "Bayesian Inference for an Unknown Number of Attributes in Restricted Latent Class Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 613-635, June.
    13. Yinyin Chen & Steven Culpepper & Feng Liang, 2020. "A Sparse Latent Class Model for Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 85(1), pages 121-153, March.
    14. Chen-Wei Liu & Björn Andersson & Anders Skrondal, 2020. "A Constrained Metropolis–Hastings Robbins–Monro Algorithm for Q Matrix Estimation in DINA Models," Psychometrika, Springer;The Psychometric Society, vol. 85(2), pages 322-357, June.
    15. Steven Andrew Culpepper, 2019. "An Exploratory Diagnostic Model for Ordinal Responses with Binary Attributes: Identifiability and Estimation," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 921-940, December.
    16. Peida Zhan & Wen-Chung Wang & Xiaomin Li, 2020. "A Partial Mastery, Higher-Order Latent Structural Model for Polytomous Attributes in Cognitive Diagnostic Assessments," Journal of Classification, Springer;The Classification Society, vol. 37(2), pages 328-351, July.
    17. Chun Wang & Jing Lu, 2021. "Learning Attribute Hierarchies From Data: Two Exploratory Approaches," Journal of Educational and Behavioral Statistics, , vol. 46(1), pages 58-84, February.
    18. Chenchen Ma & Jimmy Torre & Gongjun Xu, 2023. "Bridging Parametric and Nonparametric Methods in Cognitive Diagnosis," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 51-75, March.
    19. Yuqi Gu & Gongjun Xu, 2019. "The Sufficient and Necessary Condition for the Identifiability and Estimability of the DINA Model," Psychometrika, Springer;The Psychometric Society, vol. 84(2), pages 468-483, June.
    20. Guanhua Fang & Jingchen Liu & Zhiliang Ying, 2019. "On the Identifiability of Diagnostic Classification Models," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 19-40, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:psycho:v:89:y:2024:i:2:d:10.1007_s11336-023-09940-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.