A multilevel finite mixture item response model to cluster examinees and schools. (English) Zbl 1414.62491

Summary: Within the educational context, a key goal is to assess students’ acquired skills and to cluster students according to their ability level. In this regard, a relevant element to be accounted for is the possible effect of the school students come from. For this aim, we provide a methodological tool which takes into account the multilevel structure of the data (i.e., students in schools) and allows us to cluster both students and schools into homogeneous classes of ability and effectiveness, and to assess the effect of certain students’ and school characteristics on the probability to belong to such classes. The proposed approach relies on an extended class of multidimensional latent class IRT models characterised by: (i) latent traits defined at student and school level, (ii) latent traits represented through random vectors with a discrete distribution, (iii) the inclusion of covariates at student and school level, and (iv) a two-parameter logistic parametrisation for the conditional probability of a correct response given the ability. The approach is applied for the analysis of data collected by two national tests administered in Italy to middle school students in June 2009: the INVALSI Language Test and the Mathematics Test.


62P15 Applications of statistics to psychology
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62H12 Estimation in multivariate analysis
Full Text: DOI arXiv


[1] Bacci, S.; Bartolucci, F.; Gnaldi, M., A class of multidimensional latent class IRT models for ordinal polytomous item responses, Commu Stat Theory Methods, 43, 787-800, (2014) · Zbl 1462.62400
[2] Bartolucci, F., A class of multidimensional IRT models for testing unidimensionality and clustering items, Psychometrika, 72, 141-157, (2007) · Zbl 1286.62099
[3] Bartolucci, F.; Pennoni, F.; Vittadini, G., Assessment of school performance through a multilevel latent Markov Rasch model, J Educ Behav Stat, 36, 491-522, (2011)
[4] Bartolucci F, Bacci S, Gnaldi M (2014) MultiLCIRT: an R package for multidimensional latent class item response models. Comput Stat Data Anal 71:971-985 · Zbl 1471.62024
[5] Biernacki C, Govaert G (1999) Choosing models in model-based clustering and discriminant analysis. J Stat Comput Simul 64:49-71 · Zbl 1156.62335
[6] Birnbaum, A.; Lord, FM (ed.); Novick, MR (ed.), Some latent trait models and their use in inferring an examinee’s ability, 395-479, (1968), Reading
[7] Bolck, A.; Croon, M.; Hagenaars, J., Estimating latent structure models with categorical variables: one-step versus three-step estimators, Polit Anal, 12, 3-27, (2004)
[8] Bolt, D.; Cohen, A.; Wollack, J., Item parameter estimation under conditions of test speededness: application of a mixture Rasch model with ordinal constraints, J Educ Meas, 39, 331-348, (2002)
[9] Cho, SJ; Cohen, AS, A multilevel mixture IRT model with an application to DIF, J Educ Behav Stat, 35, 336-370, (2010)
[10] Christensen, K.; Bjorner, J.; Kreiner, S.; Petersen, J., Testing unidimensionality in polytomous Rasch models, Psychometrika, 67, 563-574, (2002) · Zbl 1297.62228
[11] Cizek G, Bunch M, Koons H (2004) Setting performance standards: contemporary methods. Educ Meas: Issues Pract 23:31-50
[12] Dayton, CM; Macready, GB, Concomitant-variable latent-class models, J Am Stat Assoc, 83, 173-178, (1988)
[13] Dempster, AP; Laird, NM; Rubin, DB, Maximum likelihood from incomplete data via the EM algorithm (with discussion), J R Stat Soc Ser B, 39, 1-38, (1977) · Zbl 0364.62022
[14] Formann, AK, Linear logistic latent class analysis for polytomous data, J Am Stat Assoc, 87, 476-486, (1992)
[15] Formann, AK; Fischer, G. (ed.); Molenaar, I. (ed.), Linear logistic latent class analysis and the Rasch model, 239-255, (1995), New York · Zbl 0831.62090
[16] Formann, AK; Davier, M. (ed.); Carstensen, C. (ed.), (Almost) equivalence between conditional and mixture maximum likelihood estimates for some models of the Rasch type, 177-189, (2007), New York
[17] Formann, AK, Mixture analysis of multivariate categorical data with covariates and missing entries, Computat Stat Data Anal, 51, 5236-5246, (2007) · Zbl 1445.62266
[18] Fox, JP, Multilevel IRT using dichotomous and polytomous response data, Br J Math Stat Psychol, 58, 145-172, (2005)
[19] Fraley, C.; Raftery, AE, Model-based clustering, discriminant analysis, and density estimation, J Am Stat Assoc, 97, 611-631, (2002) · Zbl 1073.62545
[20] Goldstein H (2011) Multilevel statistical models. Wiley, Hoboken · Zbl 1274.62006
[21] Goodman, LA, Exploratory latent structure analysis using both identifiable and unidentifiable models, Biometrika, 61, 215-231, (1974) · Zbl 0281.62057
[22] Grilli L, Rampichini C (2007) Multilevel factor models for ordinal variables. Struct Equ Model 14:1-25 · Zbl 1405.62256
[23] Heinen T (1996) Latent class and discrete latent traits models: similarities and differences. Sage, Thousand Oaks
[24] Hoijtink, H.; Molenaar, I., A multidimensional item response model: constrained latent class analysis using the Gibbs sampler and posterior predictive checks, Psychometrika, 62, 171-190, (1997) · Zbl 1003.62548
[25] INVALSI (2009a) Esame di stato di primo ciclo. a.s. 2008/2009. In: INVALSI technical report
[26] INVALSI (2009b) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di Italiano
[27] INVALSI (2009c) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di Matematica
[28] Jiao, H.; Lissitz, R.; Macready, G.; Wang, S.; Liang, S., Exploring levels of performance using the mixture Rasch model for standard setting, Psychol Test Assess Model, 53, 499-522, (2012)
[29] Kamata, A., Item analysis by the hierarchical generalized linear model, J Educ Meas, 38, 79-93, (2001)
[30] Langheine R, Rost J (1988) Latent trait and latent class models. Plenum, New York
[31] Lazarsfeld PF, Henry NW (1968) Latent structure analysis. Houghton Mifflin, Boston · Zbl 0182.52201
[32] Lindsay, B.; Clogg, C.; Greco, J., Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis, J Am Stat Assoc, 86, 96-107, (1991) · Zbl 0735.62107
[33] Loomis, S.; Bourque, M.; Cizek, GJ (ed.), From tradition to innovation: standard setting on the national assessment of educational progress, (2001), Mahwah
[34] Maier, KS, A Rasch hierarchical measurement model, J Educ Behav Stat, 26, 307-330, (2001)
[35] Maij-de Meij, AM; Kelderman, H.; Flier, H., Fitting a mixture item response theory model to personality questionnaire data: characterizing latent classes and investigating possibilities for improving prediction, Appl Psychol Meas, 32, 611-631, (2008)
[36] Masters G (1985) A comparison of latent trait and latent class analyses of Likert-type data. Psychometrika 50:69-82
[37] McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York · Zbl 0963.62061
[38] Mislevy, RJ; Verhelst, N., Modeling item responses when different subjects employ different solution strategies, Psychometrika, 55, 195-215, (1990)
[39] Muthén L, Muthén B (2012) Mplus user’s guide. Muthén and Muthén edition, Los Angeles
[40] Nylund KL, Asparouhov T, Muthén BO (2007) Deciding on the number of classes in latent class analysis and growth mixture modeling: a Monte Carlo simulation study. Struct Equ Model 14:535-569
[41] Rasch G (1961) On general laws and the meaning of measurement in psychology. In: Proceedings of the IV Berkeley symposium on mathematical statistics and probability, The Regents of the University of California, pp 321-333
[42] Reckase MD (2009) Multidimensional item response theory. Springer, New York · Zbl 1291.62023
[43] Rost, J., Rasch models in latent classes: an integration of two approaches to item analysis, Appl Psychol Meas, 14, 271-282, (1990)
[44] Rost, J., A logistic mixture distribution model for polychotomous item responses, Br J Math Stat Psychol, 44, 75-92, (1991)
[45] Sani C, Grilli L (2011) Differential variability of test scores among schools: a multilevel analysis of the fifth grade INVALSI test using heteroscedastic random effects. J Appl Quant Methods 6:88-99
[46] Schwarz, G., Estimating the dimension of a model, Ann Stat, 6, 461-464, (1978) · Zbl 0379.62005
[47] Skrondal A, Rabe-Hesketh S (2004) Generalized latent variable modeling. Multilevel, longitudinal and structural equation models. Chapman and Hall/CRC, London · Zbl 1097.62001
[48] Smit A, Kelderman H, van der Flier H (1999) Collateral information and mixed Rasch models. Methods Psychol Res 4:19-32
[49] Smit A, Kelderman H, van der Flier H (2000) The mixed Birnbaum model: estimation using collateral information. Methods Psychol Res Online 5:31-43
[50] Smit A, Kelderman H, van der Flier H (2003) Latent trait latent class analysis of an Eysenck personality questionnaire. Methods Psychol Res Online 8:23-50
[51] Tay, L.; Vermunt, JK; Wang, C., Assessing the item response theory with covariate (IRT-C) procedure for ascertaining differential item functioning, Int J Test, 13, 201-222, (2013)
[52] Tay L, Newman DA, Vermunt JK (2011) Using mixed-measurement item response theory with covariates (MM-IRT-C) to ascertain observed and unobserved measurement equivalence. Organ Res Methods 14:147-176
[53] Vermunt, JK, The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models, Appl Psychol Meas, 25, 283-294, (2001)
[54] Vermunt, JK, Multilevel latent class models, Sociol Methodol, 33, 213-239, (2003)
[55] Vermunt JK, Magidson J (2005) Latent GOLD 4.0 user’s guide. Statistical Innovations Inc., Belmont
[56] Vermunt, JK, Multilevel latent variable modeling: an application in education testing, Austrian J Stat, 37, 285-299, (2008)
[57] Vermunt JK (2010) Latent class modeling with covariates: two improved three-step approaches. Polit Anal 18:450-469
[58] von Davier M (2005) mdltm [computer software]. ETS edn, Princeton
[59] von Davier M (2008) A general diagnostic model applied to language testing data. Br J MathStat Psychol 61:287-307
[60] von Davier M, Rost J (1995) Polytomous mixed Rasch models. In: Fischer G, Molenaar I (eds) Rasch models. Foundations, recent developments, and applications. Springer, New York, pp 371-379 · Zbl 0825.62928
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.