A multilevel finite mixture item response model to cluster examinees and schools

  • Michela GnaldiEmail author
  • Silvia Bacci
  • Francesco Bartolucci
Regular Article


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.


EM algorithm INVALSI Tests Latent class model Multilevel multidimensional item response models Two-parameter logistic model 

Mathematics Subject Classification

62-07 62H30 62H12 62F99 


  1. Bacci S, Bartolucci F, Gnaldi M (2014) A class of multidimensional latent class IRT models for ordinal polytomous item responses. Commu Stat Theory Methods 43:787–800CrossRefMathSciNetzbMATHGoogle Scholar
  2. Bartolucci F (2007) A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika 72:141–157CrossRefMathSciNetzbMATHGoogle Scholar
  3. Bartolucci F, Pennoni F, Vittadini G (2011) Assessment of school performance through a multilevel latent Markov Rasch model. J Educ Behav Stat 36:491–522CrossRefGoogle Scholar
  4. Bartolucci F, Bacci S, Gnaldi M (2014) MultiLCIRT: an R package for multidimensional latent class item response models. Comput Stat Data Anal 71:971–985Google Scholar
  5. Biernacki C, Govaert G (1999) Choosing models in model-based clustering and discriminant analysis. J Stat Comput Simul 64:49–71Google Scholar
  6. Birnbaum A (1968) Some latent trait models and their use in inferring an examinee’s ability. In: Lord FM, Novick MR (eds) Statistical theories of mental test scores. Addison-Wesley, Reading, pp 395–479Google Scholar
  7. Bolck A, Croon M, Hagenaars J (2004) Estimating latent structure models with categorical variables: one-step versus three-step estimators. Polit Anal 12:3–27CrossRefGoogle Scholar
  8. Bolt D, Cohen A, Wollack J (2002) Item parameter estimation under conditions of test speededness: application of a mixture Rasch model with ordinal constraints. J Educ Meas 39:331–348CrossRefGoogle Scholar
  9. Cho SJ, Cohen AS (2010) A multilevel mixture IRT model with an application to DIF. J Educ Behav Stat 35:336–370CrossRefGoogle Scholar
  10. Christensen K, Bjorner J, Kreiner S, Petersen J (2002) Testing unidimensionality in polytomous Rasch models. Psychometrika 67:563–574CrossRefMathSciNetzbMATHGoogle Scholar
  11. Cizek G, Bunch M, Koons H (2004) Setting performance standards: contemporary methods. Educ Meas: Issues Pract 23:31–50Google Scholar
  12. Dayton CM, Macready GB (1988) Concomitant-variable latent-class models. J Am Stat Assoc 83:173–178CrossRefMathSciNetGoogle Scholar
  13. Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 39:1–38MathSciNetzbMATHGoogle Scholar
  14. Formann AK (1992) Linear logistic latent class analysis for polytomous data. J Am Stat Assoc 87:476–486CrossRefGoogle Scholar
  15. Formann AK (1995) Linear logistic latent class analysis and the Rasch model. In: Fischer G, Molenaar I (eds) Rasch models: foundations, recent developments, and applications. Springer, New York, pp 239–255CrossRefGoogle Scholar
  16. Formann AK (2007a) (Almost) equivalence between conditional and mixture maximum likelihood estimates for some models of the Rasch type. In: von Davier M, Carstensen C (eds) Multivariate and mixture distribution Rasch models. Springer, New York, pp 177–189CrossRefGoogle Scholar
  17. Formann AK (2007b) Mixture analysis of multivariate categorical data with covariates and missing entries. Computat Stat Data Anal 51:5236–5246CrossRefMathSciNetzbMATHGoogle Scholar
  18. Fox JP (2005) Multilevel IRT using dichotomous and polytomous response data. Br J Math Stat Psychol 58:145–172CrossRefGoogle Scholar
  19. Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–631CrossRefMathSciNetzbMATHGoogle Scholar
  20. Goldstein H (2011) Multilevel statistical models. Wiley, HobokenzbMATHGoogle Scholar
  21. Goodman LA (1974) Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61:215–231CrossRefMathSciNetzbMATHGoogle Scholar
  22. Grilli L, Rampichini C (2007) Multilevel factor models for ordinal variables. Struct Equ Model 14:1–25Google Scholar
  23. Heinen T (1996) Latent class and discrete latent traits models: similarities and differences. Sage, Thousand OaksGoogle Scholar
  24. Hoijtink H, Molenaar I (1997) A multidimensional item response model: constrained latent class analysis using the Gibbs sampler and posterior predictive checks. Psychometrika 62:171–190CrossRefzbMATHGoogle Scholar
  25. INVALSI (2009a) Esame di stato di primo ciclo. a.s. 2008/2009. In: INVALSI technical reportGoogle Scholar
  26. INVALSI (2009b) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di ItalianoGoogle Scholar
  27. INVALSI (2009c) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di MatematicaGoogle Scholar
  28. Jiao H, Lissitz R, Macready G, Wang S, Liang S (2012) Exploring levels of performance using the mixture Rasch model for standard setting. Psychol Test Assess Model 53:499–522Google Scholar
  29. Kamata A (2001) Item analysis by the hierarchical generalized linear model. J Educ Meas 38:79–93CrossRefGoogle Scholar
  30. Langheine R, Rost J (1988) Latent trait and latent class models. Plenum, New YorkCrossRefGoogle Scholar
  31. Lazarsfeld PF, Henry NW (1968) Latent structure analysis. Houghton Mifflin, BostonzbMATHGoogle Scholar
  32. Lindsay B, Clogg C, Greco J (1991) Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. J Am Stat Assoc 86:96–107CrossRefzbMATHGoogle Scholar
  33. Loomis S, Bourque M (2001) From tradition to innovation: standard setting on the national assessment of educational progress. In: Cizek GJ (ed) Setting performance standards: concepts methods and perspectives. Lawrence Erlbaum Associates, MahwahGoogle Scholar
  34. Maier KS (2001) A Rasch hierarchical measurement model. J Educ Behav Stat 26:307–330CrossRefGoogle Scholar
  35. Maij-de Meij AM, Kelderman H, van der Flier H (2008) Fitting a mixture item response theory model to personality questionnaire data: characterizing latent classes and investigating possibilities for improving prediction. Appl Psychol Meas 32:611–631CrossRefMathSciNetGoogle Scholar
  36. Masters G (1985) A comparison of latent trait and latent class analyses of Likert-type data. Psychometrika 50:69–82Google Scholar
  37. McLachlan G, Peel D (2000) Finite mixture models. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  38. Mislevy RJ, Verhelst N (1990) Modeling item responses when different subjects employ different solution strategies. Psychometrika 55:195–215CrossRefGoogle Scholar
  39. Muthén L, Muthén B (2012) Mplus user’s guide. Muthén and Muthén edition, Los AngelesGoogle Scholar
  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 14:535–569Google Scholar
  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 321–333Google Scholar
  42. Reckase MD (2009) Multidimensional item response theory. Springer, New YorkCrossRefGoogle Scholar
  43. Rost J (1990) Rasch models in latent classes: an integration of two approaches to item analysis. Appl Psychol Meas 14(3):271–282CrossRefMathSciNetGoogle Scholar
  44. Rost J (1991) A logistic mixture distribution model for polychotomous item responses. Br J Math Stat Psychol 44:75–92CrossRefGoogle Scholar
  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 6:88–99Google Scholar
  46. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464CrossRefzbMATHGoogle Scholar
  47. Skrondal A, Rabe-Hesketh S (2004) Generalized latent variable modeling. Multilevel, longitudinal and structural equation models. Chapman and Hall/CRC, LondonCrossRefzbMATHGoogle Scholar
  48. Smit A, Kelderman H, van der Flier H (1999) Collateral information and mixed Rasch models. Methods Psychol Res 4:19–32Google Scholar
  49. Smit A, Kelderman H, van der Flier H (2000) The mixed Birnbaum model: estimation using collateral information. Methods Psychol Res Online 5:31–43Google Scholar
  50. Smit A, Kelderman H, van der Flier H (2003) Latent trait latent class analysis of an Eysenck personality questionnaire. Methods Psychol Res Online 8:23–50Google Scholar
  51. Tay L, Vermunt JK, Wang C (2013) Assessing the item response theory with covariate (IRT-C) procedure for ascertaining differential item functioning. Int J Test 13:201–222CrossRefGoogle Scholar
  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 14:147–176Google Scholar
  53. Vermunt JK (2001) The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Appl Psychol Meas 25:283–294CrossRefMathSciNetGoogle Scholar
  54. Vermunt JK (2003) Multilevel latent class models. Sociol Methodol 33:213–239CrossRefGoogle Scholar
  55. Vermunt JK, Magidson J (2005) Latent GOLD 4.0 user’s guide. Statistical Innovations Inc., BelmontGoogle Scholar
  56. Vermunt JK (2008) Multilevel latent variable modeling: an application in education testing. Austrian J Stat 37:285–299Google Scholar
  57. Vermunt JK (2010) Latent class modeling with covariates: two improved three-step approaches. Polit Anal 18:450–469Google Scholar
  58. von Davier M (2005) mdltm [computer software]. ETS edn, PrincetonGoogle Scholar
  59. von Davier M (2008) A general diagnostic model applied to language testing data. Br J MathStat Psychol 61:287–307Google Scholar
  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 371–379Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Michela Gnaldi
    • 1
    Email author
  • Silvia Bacci
    • 2
  • Francesco Bartolucci
    • 2
  1. 1.Department of Political SciencesUniversity of PerugiaPerugiaItaly
  2. 2.Department of EconomicsUniversity of PerugiaPerugiaItaly

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