Sage Journals: Discover world-class research

Abstract

This article introduces a class of restricted latent class model in statistics to handle the diagnosis problem of fuzzy knowledge structure, where the classification of the latent equivalence classes are based on the extent of proficiency in skills. The model includes skill magnitude and latent response probabilities, which consist of linear logistic constraints of the original parameters. Expectation-maximization algorithm is given for parameter estimation of the proposed fuzzy skill model. We also establish the identifiability results of the parametric model, which guarantee the validity of the parameter estimation. A simulation study was conducted to assess the recovery of model parameters. Additionally, the proposed model is illustrated through the use of real-world data to demonstrate its practical application. Finally, topics for further research are discussed.

Keywords

restricted latent class model polytomous fuzzy knowledge structure fuzzy skill maps knowledge space theory cognitive diagnosis models

Get full access to this article

View all access options for this article.

References

Agresti

(2013). Categorical data analysis (3rd ed.). Wiley.

Andersen

E. B.

(1982). Latent structure analysis: A survey. Scandinavian Journal of Statistics, 9, 1–12.

Anselmi

Stefanutti

de Chiusole

Robusto

(2017). The assessment of knowledge and learning in competence spaces: The gain-loss model for dependent skills. British Journal of Mathematical and Statistical Psychology, 70(3), 457–479.

Chen

J. S.

de la Torre

J. M.

(2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37(6), 419–437.

Chen

Liu

Ying

(2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82, 660–692.

Chen

Y. Y.

Culpepper

Liang

(2020). A sparse latent class model for cognitive diagnosis. Psychometrika, 85, 121–153.

Dayton

C. M.

(1998). Latent class scaling analysis. Sage.

DeCarlo

L. T.

(2011). On the analysis of fraction subtraction data: the DINA model, classification, class sizes, and the Q-matrix. Applied Psychological Measurement, 35, 8–26.

de Chiusole

Spoto

Stefanutti

. (2020). Extracting partially ordered clusters from ordinal polytomous data. Behavior Research Methods, 52, 503–520.

10.

de la Torre

J. M.

(2010). The partial-credit DINA model. Paper presented at the international meeting of the Psychometric Society, Athens, GA.

11.

de la Torre

J. M.

(2011). Development of a multiple-choice diagnostic classification model: MS-DINA. Journal of Educational Measurement, 48(3), 243–263.

12.

de la Torre

J. M.

Douglas

(2004). Higher order latent trait models for cognitive diagnosis. Psychometrika, 69, 333–353.

13.

Dempster

A. P.

Laird

N. M.

Rubin

D. B.

(1977). Maximum likelihood with incomplete data via the EM algorithm. Journal of the Royal Statistical Society (Series B), 39, 1–38.

14.

Doignon

J. P.

Falmagne

J. C.

(1985). Spaces for the assessment of knowledge. International Journal of Manmachine Studies, 23(2), 175–196.

15.

Doignon

J. P.

Falmagne

J. C.

(1999). Knowledge spaces. Springer.

16.

Doignon

J. P.

Falmagne

J. C.

(2011). Learning spaces: Interdisciplinary applied mathematics. Springer-Verlag Berlin Heidelberg.

17.

Formann

A. K.

(1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 86, 476–486.

18.

Goodman

L. A.

(1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61, 215–231.

19.

Hansen

Cai

(2013). Abstract: A hierarchical item response model for cognitive diagnosis. Multivariate Behavioral Research, 48(1), 158–158.

20.

Heller

(2017). Identifiability in probabilistic knowledge structures. Journal of Mathematical Psychology, 77, 46–57.

21.

Heller

(2021). Generalizing quasi-ordinal knowledge spaces to polytomous items. Journal of Mathematical Psychology, 101, Article 102515.

22.

Heller

Stefanutti

Anselmi

Robusto

(2015). On the link between cognitive diagnostic models and knowledge space theory. Psychometrika, 80(4), 995–1019.

23.

Huang

G. H.

Bandeen-Roche

(2004). Building an identifiable latent class model with covariate effects on underlying and measured variables. Psychometrika, 69(1), 5–32.

24.

Korossy

(1999). Modeling knowledge as competence and performance. In Albert

Lukas

(Eds.), Knowledge spaces: Theories, empirical research, and applications. (pp. 103–132). Lawrence Erlbaum Associates.

25.

Lim

Y. S.

(2024). Nonparametric cognitive diagnosis when attributes are polytomous. Journal of Classification, 41(1), 94–128.

26.

W. C.

(2021). A higher-order cognitive diagnosis model with ordinal attributes for dichotomous response data. Multivariate Behavioral Research, 57(2–3), 408–421.

27.

W. C.

de la Torre

J. M.

(2016). A sequential cognitive diagnosis model for polytomous responses. British Journal of Mathematical and Statistical Psychology, 69(3), 253–275.

28.

W. C.

de la Torre

J. M.

(2019). Category-level model selection for the sequential G-DINA model. Journal of Educational and Behavorial Statistics, 44(1), 45–77.

29.

Maris

Bechger

T. M.

(2009). Equivalent diagnostic classification models. Measurement, 7, 41–46.

30.

Nicotra

E. F.

Spoto

(2019). Connections and dissimilarities among formal concept analysis, knowledge space theory and cognitive diagnostic models in a systemic perspective. In Minati

Abram

Pessa

(Eds.), Systemics of incompleteness and quasi-systems, contemporary systems thinking (pp. 235–241). Springer.

31.

Noventa

Spoto

Heller

Kelava

(2019). On a generalization of local independence in item response theory based on knowledge space theory. Psychometrika, 84, 395–421.

32.

Robitzsch

(2020). Regularized latent class analysis for polytomous item responses: An application to SPM-LS data. Journal of Intelligence, 8(3), 30–54.

33.

Rusch

Wille

(1996). Knowledge spaces and formal concept analysis. In Data analysis and information systems: Statistical and conceptual approaches (pp. 427–436). Proceedings of the 19th Annual Conference of the Gesellschaft für Klassifikation eV University of Basel, Springer Berlin Heidelberg.

34.

Singh

P. K.

Cherukuri

A. K.

(2017). Concepts reduction in formal concept analysis with fuzzy setting using Shannon entropy. International Journal of Machine Learning & Cybernetics, 8, 179–189.

35.

Spoto

Stefanutti

(2020). On the necessary and sufficient conditions for delineating forward- and backward-graded knowledge structures from skill maps. Journal of Mathematical Psychology, 94, Article 102451.

36.

Spoto

Stefanutti

Vidotto

(2010). Knowledge space theory, formal concept analysis, and computerized psychological assessment. Behavior Research Methods, 42(1), 342–250.

37.

Stefanutti

de Chiusole

Anselmi

Spoto

(2020). Extending the basic local independence model to polytomous data. Psychometrika, 85(3), 684–715.

38.

Sun

J. J.

Lin

Y. D.

(2021). Knowledge structures delineated by fuzzy skill maps. Fuzzy Sets and Systems, 407, 50–66.

39.

Sun

J. J.

Lin

F. C.

Zh. R.

(2023). Constructing polytomous knowledge structures from fuzzy skills. Fuzzy Sets and Systems, 461, Article 108395.

40.

Tatsuoka

K. K.

(1984). Analysis of errors in fraction addition and subtraction problems (Final Report for Grant No. nie-g-81-0002). University of Illinois, Computer-Based Education Research Laboratory.

41.

Vermunt

J. K.

Magidson

J. A.

(2004). Latent class analysis. Sage.

42.

Vermunt

J. K.

Magidson

J. A.

(2017). Latent class analysis. Journal of Mathematical Psychology, 77, 46–57.

43.

von Davier

(2008) A general diagnostic model applied to language testing data, British Journal of Mathematical and Statistical Psychology, 61, 287–307.

44.

von Davier

(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.

45.

Wei

Zhang

W. X.

J. J.

Cao

(2020). Concept reduction and concept characteristics in formal concept analysis (in Chinese). Scientia Sinica Informationis, 50, 1817–1833.

46.

R. Z.

Liu

Y. P.

Chen

E. H.

Chen

Zh. G.

G.P.

(2015). Cognitive modelling for predicting examinee performance. Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence. AAAI Press.

47.

B. C.

J. J.

(2023). The inclusion degrees of fuzzy skill maps and knowledge structures. Fuzzy Sets and Systems, 465, Article 108540.

48.

G. J.

(2017). Identifiability of restricted latent class models with binary responses. The Annals of Statistics, 45(2), 675–707.

49.

G. J.

Zhang

(2016). Identifiability of diagnostic classification models. Psychometrika, 81, 625–649.

50.

G. J.

Zhang

(2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113(523), 1284–1295.

51.

W. H.

W. T.

(2016). Granular computing approach to two-way learning based on formal concept analysis in fuzzy datasets. IEEE Transactions on Cybernetics, 46(2), 366–379.

52.

Zhou

Y. F.

J. J.

Wang

H. K.

Sun

(2022). Skills and fuzzy knowledge structures. Journal of Intelligent & Fuzzy Systems, 42(3), 2629–2645.

53.

Zhou

Y. F.

J. J.

Yang

H. L.

Q. Y.

Zhou

Y. L.

(2024). Knowledge structure construction and skill reduction methods based on multi-scale context. Journal of Experimental & Theoretical Artificial Intelligence, 8, 36.