{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T15:11:41Z","timestamp":1774278701303,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T00:00:00Z","timestamp":1739404800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T00:00:00Z","timestamp":1739404800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100013003","name":"Universit\u00e0 degli Studi di Cagliari","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100013003","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We present an extension of the multilevel latent class model for dealing with multilevel cross-classified categorical data. Cross-classified data structures arise when observations are simultaneously nested within two or more groups, for example, children nested within both schools and neighborhoods. More specifically, we propose extending the standard hierarchical latent class model, which contains mixture components at two levels, say for children and schools, by including a separate set of mixture components for each of the higher-level crossed classifications, say for schools and neighborhoods. Because of the complex dependency structure arising from the cross-classified nature of the data, it is no longer possible to obtain maximum likelihood estimates of the model parameters, for example, using the EM algorithm. As a solution to the estimation problem, we propose an approximate estimation approach using a stochastic version of the EM algorithm. The performance of this approach, which resembles Gibbs sampling, was investigated through a set of simulation studies. Moreover, the application of the new model is illustrated using an Italian dataset on the quality of university experience at degree programme level, with degree programmes nested in both universities and fields of study.\n<\/jats:p>","DOI":"10.1007\/s11222-025-10579-w","type":"journal-article","created":{"date-parts":[[2025,2,13]],"date-time":"2025-02-13T19:49:55Z","timestamp":1739476195000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multilevel latent class models for cross-classified categorical data: model definition and estimation through stochastic EM"],"prefix":"10.1007","volume":"35","author":[{"given":"S.","family":"Columbu","sequence":"first","affiliation":[]},{"given":"N.","family":"Piras","sequence":"additional","affiliation":[]},{"given":"J. K.","family":"Vermunt","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,13]]},"reference":[{"key":"10579_CR1","unstructured":"Almalaurea: 2018 Reports on Graduates\u2019 Profile and Occupational Condition. www.almalaurea.it"},{"issue":"3\u20134","key":"10579_CR2","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1016\/S0167-9473(02)00163-9","volume":"41","author":"C Biernacki","year":"2003","unstructured":"Biernacki, C., Celeux, G., Govaert, G.: Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput. Statist. Data Anal. 41(3\u20134), 561\u2013575 (2003)","journal-title":"Comput. Statist. Data Anal."},{"key":"10579_CR3","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s11634-016-0234-1","volume":"10","author":"M Bennink","year":"2016","unstructured":"Bennink, M., Croon, M.A., Kroon, B.E.A.: Micro-macro multilevel latent class models with multiple discrete individual-level variables. Adv. Data Anal. Classif. 10, 139\u2013154 (2016)","journal-title":"Adv. Data Anal. Classif."},{"key":"10579_CR4","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1007\/s00357-023-09441-3","volume":"40","author":"C Biernacki","year":"2023","unstructured":"Biernacki, C., Jacques, J., Keribin, C.: A survey on model-based co-clustering: high dimension and estimation challenges. J. Classif. 40, 332\u2013381 (2023)","journal-title":"J. Classif."},{"key":"10579_CR5","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1146\/annurev-statistics-040220-091910","volume":"5","author":"F Bartolucci","year":"2022","unstructured":"Bartolucci, F., Pandolfi, S., Pennoni, F.: Discrete latent variable models. Ann. Rev. Statist. 5, 425\u2013452 (2022)","journal-title":"Ann. Rev. Statist."},{"issue":"4","key":"10579_CR6","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1080\/00949659608811772","volume":"55","author":"G Celeux","year":"1996","unstructured":"Celeux, G., Chauveau, D., Diebolt, J.: Stochastic versions of the EM algorithm: an experimental study in the mixture case. J. Stat. Comput. Simul. 55(4), 287\u2013314 (1996)","journal-title":"J. Stat. Comput. Simul."},{"issue":"4","key":"10579_CR7","first-page":"35","volume":"34","author":"G Celeux","year":"1986","unstructured":"Celeux, G., Diebolt, J.: L\u2019algorithme sem: un algorithme d\u2019apprentissage probabiliste pour la reconnaissance de m\u00e9lange de densit\u00e9s. Revue de statistique appliqu\u00e9e 34(4), 35\u201352 (1986)","journal-title":"Revue de statistique appliqu\u00e9e"},{"key":"10579_CR8","unstructured":"Columbu, S., Piras, N., Vermunt, J.K.: Log-likelihood approximation in stochastic em for multilevel latent class models. In: Plaia, A., Egidi, L., Abbruzzo, A. (eds.) Proceedings of the Statistics and Data Science 2024 Conference - New Perspectives on Statistics and Data Science, Palermo (2024)"},{"issue":"1","key":"10579_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","volume":"39","author":"AP Dempster","year":"1977","unstructured":"Dempster, A.P., Laird, N.M., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. Ser. B (Methodol.) 39(1), 1\u201338 (1977)","journal-title":"J. Roy. Stat. Soc. Ser. B (Methodol.)"},{"issue":"4","key":"10579_CR10","doi-asserted-by":"publisher","first-page":"1144","DOI":"10.1007\/s11336-023-09929-2","volume":"88","author":"R Di Mari","year":"2023","unstructured":"Di Mari, R., Bakk, Z., Oser, J., Kuha, J.: A two-step estimator for multilevel latent class analysis with covariates. Psychometrika 88(4), 1144\u20131170 (2023)","journal-title":"Psychometrika"},{"issue":"3","key":"10579_CR11","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1016\/j.csda.2004.06.012","volume":"49","author":"A D\u2019Elia","year":"2005","unstructured":"D\u2019Elia, A., Piccolo, D.: A mixture model for preference data analysis. Comput. Stat. Data Anal. 49(3), 917\u2013934 (2005)","journal-title":"Comput. Stat. Data Anal."},{"key":"10579_CR12","unstructured":"Eddelbuettel, D., Francois, R., Allaire, J., Ushey, K., Kou, Q., Russell, N., Ucar, I., Bates, D., Chambers, J.: Rcpp: Seamless R and C++ Integration. (2024). R package version 1.0.13. https:\/\/CRAN.Rproject.org\/package=Rcpp"},{"key":"10579_CR13","doi-asserted-by":"publisher","DOI":"10.1002\/9780470973394","volume-title":"Multilevel Statistical Models","author":"H Goldstein","year":"2010","unstructured":"Goldstein, H.: Multilevel Statistical Models. Wiley Series in Probability and Statistics, London (2010)"},{"issue":"2","key":"10579_CR14","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","volume":"5","author":"PW Holland","year":"1983","unstructured":"Holland, P.W., Blackmond Laskey, K., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109\u2013137 (1983)","journal-title":"Soc. Netw."},{"key":"10579_CR15","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511499531","volume-title":"Applied Latent Class Analysis","author":"JA Hagenaars","year":"2002","unstructured":"Hagenaars, J.A., McCutcheon, A.L.: Applied Latent Class Analysis. Cambridge University Press, Cambridge (2002)"},{"key":"10579_CR16","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.csda.2018.01.014","volume":"123","author":"J Jacques","year":"2018","unstructured":"Jacques, J., Biernacki, C.: Model-based co-clustering for ordinal data. Comput. Stat. Data Anal. 123, 101\u2013115 (2018)","journal-title":"Comput. Stat. Data Anal."},{"issue":"6","key":"10579_CR17","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1007\/s11222-014-9472-2","volume":"25","author":"C Keribin","year":"2015","unstructured":"Keribin, C., Brault, V., Celeux, G., Govaert, G.: Estimation and selection for the latent block model on categorical data. Stat. Comput. 25(6), 1201\u20131216 (2015)","journal-title":"Stat. Comput."},{"key":"10579_CR18","unstructured":"Lazarsfeld, P.F.: The logical and mathematical foundation of latent structure analysis. Studies in Social Psychology in World War II Vol. IV : Measurement and Prediction, 362\u2013412 (1950)"},{"issue":"1","key":"10579_CR19","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1111\/j.1467-9531.2010.01231.x","volume":"40","author":"O Lukoc\u0306iene","year":"2010","unstructured":"Lukoc\u0306iene, O., Varriale, R., Vermunt, J.K.: The simultaneous decision(s) about the number of lower- and higher-level classes in multilevel latent class analysis. Sociol. Methodol. 40(1), 247\u2013283 (2010)","journal-title":"Sociol. Methodol."},{"key":"10579_CR20","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0049-089X(81)90003-X","volume":"10","author":"J Magidson","year":"1981","unstructured":"Magidson, J.: Qualitative variance, entropy, and correlation ratios for nominal dependent variables. Soc. Sci. Res. 10, 177\u2013194 (1981)","journal-title":"Soc. Sci. Res."},{"key":"10579_CR21","series-title":"Wiley Series in Probability and Statistics","doi-asserted-by":"publisher","DOI":"10.1002\/9780470191613","volume-title":"The EM Algorithm and Extensions","author":"GJ McLachlan","year":"2008","unstructured":"McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. Wiley Series in Probability and Statistics, Wiley, New Jersey (2008)"},{"key":"10579_CR22","first-page":"459","volume-title":"Handbook of Statistics","author":"JM Marin","year":"2005","unstructured":"Marin, J.M., Mengersen, K., Robert, C.P.: Bayesian modelling and inference on mixtures of distributions. In: Dey, D.K. (ed.) Handbook of Statistics, vol. 25, pp. 459\u2013507. Elsevier, Amsterdam (2005)"},{"key":"10579_CR23","series-title":"Wiley Series in Probability and Statistics","doi-asserted-by":"crossref","DOI":"10.1002\/047172842X","volume-title":"Finite Mixture Models","author":"GJ McLachlan","year":"2004","unstructured":"McLachlan, G.J., Peel, D.: Finite Mixture Models. Wiley Series in Probability and Statistics, Wiley, UK (2004)"},{"issue":"3","key":"10579_CR24","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1111\/insr.12436","volume":"89","author":"A Maruotti","year":"2021","unstructured":"Maruotti, A., Punzo, A.: Initialization of hidden Markov and semi-Markov models: a critical evaluation of several strategies. Int. Stat. Rev. 89(3), 447\u2013480 (2021)","journal-title":"Int. Stat. Rev."},{"key":"10579_CR25","doi-asserted-by":"publisher","first-page":"106866","DOI":"10.1016\/j.csda.2019.106866","volume":"144","author":"M Selosse","year":"2020","unstructured":"Selosse, M., Jacques, J., Biernacki, C.: Model-based co-clustering for mixed type data. Comput. Stat. Data Anal. 144, 106866 (2020)","journal-title":"Comput. Stat. Data Anal."},{"key":"10579_CR26","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s11222-024-10444-2","volume":"34","author":"A Sportisse","year":"2024","unstructured":"Sportisse, A., Marbac, M., Laporte, F., Celeux, G., Boyer, C., Josse, J., Biernacki, C.: Model-based clustering with missing not at random data. Statist. Comput. 34, 135 (2024)","journal-title":"Statist. Comput."},{"key":"10579_CR27","doi-asserted-by":"publisher","DOI":"10.1201\/9780203489437","volume-title":"Generalized Latent Variable Modelling: Multilevel Longitudinal and Structural Equation Models","author":"A Skrondal","year":"2004","unstructured":"Skrondal, A., Rabe-Hesketh, S.: Generalized Latent Variable Modelling: Multilevel Longitudinal and Structural Equation Models. Chapman and Hall\/CRC, Boca Raton (2004)"},{"key":"10579_CR28","unstructured":"UNESCO: Isced fields of education and international standard classification of education 2011. Technical report, UNESCO Institute for Statistics, Montr\u00e9al (2014)"},{"key":"10579_CR29","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1111\/j.0081-1750.2003.t01-1-00131.x","volume":"33","author":"JK Vermunt","year":"2003","unstructured":"Vermunt, J.K.: Multilevel latent class models. Sociol. Methodol. 33, 213\u2013239 (2003)","journal-title":"Sociol. Methodol."},{"key":"10579_CR30","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1046\/j.0039-0402.2003.00257.x","volume":"58","author":"JK Vermunt","year":"2004","unstructured":"Vermunt, J.K.: An EM algorithm for the estimation of parametric and nonparametric hierarchical nonlinear models. Stat. Neerl. 58, 220\u2013233 (2004)","journal-title":"Stat. Neerl."},{"issue":"3","key":"10579_CR31","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1207\/s15327906mbr4003_1","volume":"40","author":"JK Vermunt","year":"2005","unstructured":"Vermunt, J.K.: Mixed-effects logistic regression models for indirectly observed discrete outcome variables. Multivar. Behav. Res. 40(3), 281\u2013301 (2005)","journal-title":"Multivar. Behav. Res."},{"key":"10579_CR32","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1177\/0962280207081238","volume":"17","author":"JK Vermunt","year":"2008","unstructured":"Vermunt, J.K.: Latent class and finite mixture models for multilevel data sets. Stat. Methods Med. Res. 17, 33\u201351 (2008)","journal-title":"Stat. Methods Med. Res."},{"key":"10579_CR33","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1093\/pan\/mpq025","volume":"18","author":"JK Vermunt","year":"2010","unstructured":"Vermunt, J.K.: Latent class modeling with covariates: Two improved three-step approaches. Polit. Anal. 18, 450\u2013469 (2010)","journal-title":"Polit. Anal."},{"key":"10579_CR34","volume-title":"Upgrade Manual for Latent GOLD 5.1: Basic, Advanced, and Syntax","author":"JK Vermunt","year":"2016","unstructured":"Vermunt, J.K., Magidson, J.: Upgrade Manual for Latent GOLD 5.1: Basic, Advanced, and Syntax. Statistical Innovations Inc., Belmont Massachusetts (2016)"},{"key":"10579_CR35","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1214\/aos\/1176346060","volume":"11","author":"CJ Wu","year":"1983","unstructured":"Wu, C.J.: On the convergence properties of the EM algorithm. Ann. Stat. 11, 95\u2013103 (1983)","journal-title":"Ann. Stat."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10579-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-025-10579-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-025-10579-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T16:15:20Z","timestamp":1740586520000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-025-10579-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,13]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["10579"],"URL":"https:\/\/doi.org\/10.1007\/s11222-025-10579-w","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,13]]},"assertion":[{"value":"26 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"50"}}