A Restricted Latent Class Hidden Markov Model for Polytomous Responses,Polytomous Attributes,and Covariates: Identifiability and Application

Abstract

We introduce a restricted latent class exploratory model for longitudinal data with ordinal attributes and respondent-specific covariates. Responses follow a time-inhomogeneous hidden Markov model where the probability of a respondent’s latent state at the current time point is conditional on the respondent’s latent state at the previous time point as well as the respondent’s covariates at the current time point. We prove that the model is identifiable, state a Bayesian formulation, and demonstrate its efficacy in a variety of scenarios through two simulation studies. We apply the model to response data from a mathematics examination, comparing the results to a previously published confirmatory analysis, and also apply it to emotional state response data, which was measured over a several-day period.

Keywords

Bayesian inference hidden Markov model longitudinal multivariate probit restricted latent class model

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References

Agresti

(2015). Foundations of linear and generalized linear models. Wiley.

Ashford

J. R.

Sowden

R. R.

(1970). Multi-variate probit analysis. Biometrics, 26(3), 535–546. https://doi.org/10.2307/2529107

Barnard

McCulloch

Meng

X.-L.

(2000). Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage. Statistica Sinica, 10(4), 1281–1311. https://www3.stat.sinica.edu.tw/statistica/oldpdf/A10n416.pdf

Bartolucci

Farcomeni

(2009). A multivariate extension of the dynamic logit model for longitudinal data based on a latent Markov heterogeneity structure. Journal of the American Statistical Association, 104(486), 816–831. https://doi.org/10.1198/jasa.2009.0107

Bartolucci

Farcomeni

Pennoni

(2012). Latent Markov models for longitudinal data. CRC Press. https://doi.org/10.1201/b13246

Baum

L. E.

Petrie

(1966). Statistical inference for probabilistic functions of finite state Markov chains. The Annals of Mathematical Statistics, 37(6), 1554–1563. https://doi.org/10.1214/aoms/1177699147

Chen

de la Torre

(2013). A general cognitive diagnosis model for expert-defined polytomous attributes. Applied Psychological Measurement, 37(6), 419–437. https://doi.org/10.1177/0146621613479818

Chen

Culpepper

S. A.

(2020). A multivariate probit model for learning trajectories: A fine-grained evaluation of an educational intervention. Applied Psychological Measurement, 44(7–8), 515–530. https://doi.org/10.1177/0146621620920928

Chen

Culpepper

S. A.

Wang

Douglas

(2018). A hidden Markov model for learning trajectories in cognitive diagnosis with application to spatial rotation skills. Applied Psychological Measurement, 42(1), 5–23. https://doi.org/10.1177/0146621617721250

10.

Chen

Culpepper

Liang

(2020). A sparse latent class model for cognitive diagnosis. Psychometrika, 85(1), 121–153. https://doi.org/10.1007/s11336-019-09693-2

11.

Chib

(2011). Introduction to simulation and MCMC methods. In Geweke

Koop

Van Dijk

(Eds.), The Oxford handbook of Bayesian econometrics (pp. 183–217). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199559084.013.0006

12.

Christoffersson

(1975). Factor analysis of dichotomized variables. Psychometrika, 40(1), 5–32. https://doi.org/10.1007/BF02291477

13.

Collins

L. M.

Wugalter

S. E.

(1992). Latent class models for stage-sequential dynamic latent variables. Multivariate Behavioral Research, 27(1), 131–157. https://doi.org/10.1207/s15327906mbr2701_8

14.

Culpepper

S. A.

(2019). An exploratory diagnostic model for ordinal responses with binary attributes: Identifiability and estimation. Psychometrika, 84(4), 921–940. https://doi.org/10.1007/s11336-019-09683-4

15.

Gelman

Hill

(2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press. https://doi.org/10.1017/CBO9780511790942

16.

Geweke

(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 4: Proceedings of the Fourth Valencia International Meeting, dedicated to the memory of Morris H. DeGroot, 1931–1989 (pp. 169–193). Oxford University Press. https://doi.org/10.1093/oso/9780198522669.003.0010

17.

Goodman

L. A.

(1974). Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika, 61(2), 215–231. https://doi.org/10.1093/biomet/61.2.215

18.

Haertel

(1984). An application of latent class models to assessment data. Applied Psychological Measurement, 8(3), 333–346. https://doi.org/10.1177/014662168400800311

19.

Haertel

E. H.

(1990). Continuous and discrete latent structure models for item response data. Psychometrika, 55(3), 477–494. https://doi.org/10.1007/BF02294762

20.

Hagenaars

J. A.

(1990). Categorical longitudinal data: Log-linear panel, trend, and cohort analysis. SAGE Publications.

21.

Culpepper

S. A.

Douglas

(2023). A sparse latent class model for polytomous attributes in cognitive diagnostic assessments. In van der Ark

L. A.

Emons

W. H. M.

Meijer

R. R.

(Eds.), Essays on contemporary psychometrics (pp. 413–442). Springer International Publishing. https://doi.org/10.1007/978-3-031-10370-4_21

22.

Kaya

Leite

W. L.

(2017). Assessing change in latent skills across time with longitudinal cognitive diagnosis modeling: An evaluation of model performance. Educational and Psychological Measurement, 77(3), 369–388. https://doi.org/10.1177/0013164416659314

23.

Kuo

Mallick

(1998). Variable selection for regression models. Sankhyā: The Indian Journal of Statistics, Series B, 60(1), 65–81.

24.

Lauritzen

S. L.

(1996). Graphical models. Clarendon Press. https://doi.org/10.1093/oso/9780198522195.001.0001

25.

Cohen

Bottge

Templin

(2016). A latent transition analysis model for assessing change in cognitive skills. Educational and Psychological Measurement, 76(2), 181–204. https://doi.org/10.1177/0013164415588946

26.

Little

R. J.

(2021). Missing data assumptions. Annual Review of Statistics and its Application, 8, 89–107. https://doi.org/10.1146/annurev-statistics-040720-031104

27.

Liu

(2008). Monte Carlo strategies in scientific computing. Springer Science+Business Media New York. https://doi.org/10.1007/978-0-387-76371-2

28.

Liu

J. S.

Y. N.

(1999). Parameter expansion for data augmentation. Journal of the American Statistical Association, 94(448), 1264–1274. https://doi.org/10.2307/2669940

29.

Liu

Ying

(2013). Theory of self-learning Q-matrix. Bernoulli, 19(5A), 1790–1817. https://doi.org/10.3150/12-BEJ430

30.

Liu

Culpepper

S. A.

Chen

(2023). Identifiability of hidden Markov models for learning trajectories in cognitive diagnosis. Psychometrika, 88(2), 361–386. https://doi.org/10.1007/s11336-023-09904-x

31.

J. G.

Liu

X. L.

Liao

Wang

(2020). Disentangling stereotypes from social reality: Astrological stereotypes and discrimination in China. Journal of Personality and Social Psychology, 119(6), 1359–1379. https://doi.org/10.1037/pspi0000237

32.

Madison

M. J.

Bradshaw

L. P.

(2018). Assessing growth in a diagnostic classification model framework. Psychometrika, 83(4), 963–990. https://doi.org/10.1007/s11336-018-9638-5

33.

Madison

M. J.

Jeon

Cotterell

Haab

Zor

(2025). TDCM: An R package for estimating longitudinal diagnostic classification models. Multivariate Behavioral Research, 60(3), 518–527. https://doi.org/10.1080/00273171.2025.2453454

34.

Marini

M. M.

Olsen

A. R.

Rubin

D. B.

(1980). Maximum-likelihood estimation in panel studies with missing data. Sociological Methodology, 11, 314–357. https://doi.org/10.2307/270868

35.

McDonald

R. P.

(1967). Nonlinear factor analysis. Psychometric Society.

36.

McGibbon

R. T.

(2015). pyhmc: Hamiltonian Monte Carlo in Python [Computer software]. https://github.com/rmcgibbo/pyhmc

37.

Meng

X.-L.

Van Dyk

D. A.

(1999). Seeking efficient data augmentation schemes via conditional and marginal augmentation. Biometrika, 86(2), 301–320. https://doi.org/10.1093/biomet/86.2.301

38.

Merkle

E. C.

Furr

Rabe-Hesketh

(2019). Bayesian comparison of latent variable models: Conditional versus marginal likelihoods. Psychometrika, 84(3), 802–829. https://doi.org/10.1007/s11336-019-09679-0

39.

Miller

J. W.

(2016). Hidden Markov models [Lecture notes]. https://jwmi.github.io/ASM/5-HMMs.pdf

40.

Murphy

K. P.

(2012). Machine learning: A probabilistic perspective. MIT Press.

41.

Muthén

(1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43(4), 551–560. https://doi.org/10.1007/BF02293813

42.

Plummer

Best

Cowles

Vines

(2006). CODA: Convergence diagnosis and output analysis for MCMC. R News, 6(1), 7–11. https://journal.r-project.org/articles/RN-2006-002/RN-2006-002.pdf

43.

Poulsen

C. S.

(1983). Latent structure analysis with choice modeling applications (Publication No. 1983.8316074) [Doctoral dissertation, University of Pennsylvania]. ProQuest Dissertations & Theses Global.

44.

Rupp

A. A.

Templin

Henson

R. A.

(2010). Diagnostic measurement: Theory, methods, and applications. The Guilford Press.

45.

Sanderson

Curtin

(2019). Practical sparse matrices in C++ with hybrid storage and template-based expression optimisation. Mathematical and Computational Applications, 24(3), Article 70. https://doi.org/10.3390/mca24030070

46.

Sanderson

Curtin

(2025). Armadillo: An efficient framework for numerical linear algebra. In 2025 17th International Conference on Computer and Automation Engineering (ICCAE) (pp. 303–307). https://doi.org/10.1109/ICCAE64891.2025.10980539

47.

Seneta

(2006). Non-negative matrices and Markov chains. Springer Science+Business Media. https://doi.org/10.1007/0-387-32792-4

48.

Shui

Zhang

Wang

Zhang

(2020). DAPPER dataset [Data set]. Synapse. https://doi.org/10.7303/syn22418021

49.

Shui

Zhang

Wang

Zhang

(2021). A dataset of daily ambulatory psychological and physiological recording for emotion research. Scientific Data, 8, Article 161. https://doi.org/10.1038/s41597-021-00945-4

50.

Steger

M. F.

Frazier

Oishi

Kaler

(2006). The Meaning in Life Questionnaire: Assessing the presence of and search for meaning in life. Journal of Counseling Psychology, 53(1), 80–93. https://doi.org/10.1037/0022-0167.53.1.80

51.

Sun

Xin

Zhang

de la Torre

(2013). A polytomous extension of the generalized distance discriminating method. Applied Psychological Measurement, 37(7), 503–521. https://doi.org/10.1177/0146621613487254

52.

Tang

Zhan

(2020). The development of an instrument for longitudinal learning diagnosis of rational number operations based on parallel tests. Frontiers in Psychology, 11, Article 2246. https://doi.org/10.3389/fpsyg.2020.02246

53.

Tang

Zhan

(2021). Does diagnostic feedback promote learning? Evidence from a longitudinal cognitive diagnostic assessment. AERA Open, 7(1), 1–15. https://doi.org/10.1177/23328584211060804

54.

van de Pol

Langeheine

(1990). Mixed Markov latent class models. Sociological Methodology, 20, 213–247. https://doi.org/10.2307/271087

55.

Vehtari

Gabry

Magnusson

Yao

Bürkner

P.-C.

Paananen

Gelman

(2024). loo: Efficient leave-one-out cross-validation and WAIC for Bayesian models [Computer software]. https://mc-stan.org/loo/

56.

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. https://doi.org/10.1177/01466210122032082

57.

Wang

Yang

Culpepper

S. A.

Douglas

J. A.

(2018). Tracking skill acquisition with cognitive diagnosis models: A higher-order, hidden Markov model with covariates. Journal of Educational and Behavioral Statistics, 43(1), 57–87. https://doi.org/10.3102/1076998617719727

58.

Watanabe

(2010). Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. Journal of Machine Learning Research, 11, 3571–3594. https://www.jmlr.org/papers/volume11/watanabe10a/watanabe10a.pdf

59.

Watson

Clark

L. A.

Tellegen

(1988). Development and validation of brief measures of positive and negative affect: The PANAS scales. Journal of Personality and Social Psychology, 54(6), 1063–1070. https://doi.org/10.1037/0022-3514.54.6.1063

60.

Wayman

E. A.

(2025). probitlcmlongit [Computer software]. https://github.com/ericwayman01/probitlcmlongit

61.

Wayman

E. A.

Culpepper

S. A.

Douglas

Bowers

(2025). A restricted latent class model with polytomous attributes and respondent-level covariates. Behaviormetrika. Advance online publication. https://doi.org/10.1007/s41237-025-00271-8

62.

Wiggins

L. M.

(1955). Mathematical models for the interpretation of attitude and behavior change: The analysis of multi-wave panel [Unpublished doctoral dissertation]. Columbia University.

63.

(2017). Identifiability of restricted latent class models with binary responses. The Annals of Statistics, 45(2), 675–707. https://doi.org/10.1214/16-AOS1464

64.

Shang

(2018). Identifying latent structures in restricted latent class models. Journal of the American Statistical Association, 113(523), 1284–1295. https://doi.org/10.1080/01621459.2017.1340889

65.

Zhan

(2021). Data for Does diagnostic feedback promote learning? Evidence from a longitudinal cognitive diagnostic assessment [Data set]. Inter-university Consortium for Political and Social Research. https://doi.org/10.3886/E153061V1

66.

Zhang

Chang

H.-H.

(2020). A multilevel logistic hidden Markov model for learning under cognitive diagnosis. Behavior Research Methods, 52, 408–421. https://doi.org/10.3758/s13428-019-01238-w

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