Skip to main content
SpringerLink
Log in

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

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Downloadable from http://www.CRAN.R-project.org/package=MultiLCIRT

References

  • Bacci S, Bartolucci F, Gnaldi M (2014) A class of multidimensional latent class IRT models for ordinal polytomous item responses. Commu Stat Theory Methods 43:787–800

    Article  MathSciNet  MATH  Google Scholar 

  • Bartolucci F (2007) A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika 72:141–157

    Article  MathSciNet  MATH  Google Scholar 

  • Bartolucci F, Pennoni F, Vittadini G (2011) Assessment of school performance through a multilevel latent Markov Rasch model. J Educ Behav Stat 36:491–522

    Article  Google Scholar 

  • 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

  • Biernacki C, Govaert G (1999) Choosing models in model-based clustering and discriminant analysis. J Stat Comput Simul 64:49–71

  • 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 395–479

    Google Scholar 

  • Bolck A, Croon M, Hagenaars J (2004) Estimating latent structure models with categorical variables: one-step versus three-step estimators. Polit Anal 12:3–27

    Article  Google Scholar 

  • 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 39:331–348

    Article  Google Scholar 

  • Cho SJ, Cohen AS (2010) A multilevel mixture IRT model with an application to DIF. J Educ Behav Stat 35:336–370

    Article  Google Scholar 

  • Christensen K, Bjorner J, Kreiner S, Petersen J (2002) Testing unidimensionality in polytomous Rasch models. Psychometrika 67:563–574

    Article  MathSciNet  MATH  Google Scholar 

  • Cizek G, Bunch M, Koons H (2004) Setting performance standards: contemporary methods. Educ Meas: Issues Pract 23:31–50

  • Dayton CM, Macready GB (1988) Concomitant-variable latent-class models. J Am Stat Assoc 83:173–178

    Article  MathSciNet  Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 39:1–38

    MathSciNet  MATH  Google Scholar 

  • Formann AK (1992) Linear logistic latent class analysis for polytomous data. J Am Stat Assoc 87:476–486

    Article  Google Scholar 

  • 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 239–255

    Chapter  Google Scholar 

  • 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 177–189

    Chapter  Google Scholar 

  • Formann AK (2007b) Mixture analysis of multivariate categorical data with covariates and missing entries. Computat Stat Data Anal 51:5236–5246

    Article  MathSciNet  MATH  Google Scholar 

  • Fox JP (2005) Multilevel IRT using dichotomous and polytomous response data. Br J Math Stat Psychol 58:145–172

    Article  Google Scholar 

  • Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–631

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein H (2011) Multilevel statistical models. Wiley, Hoboken

    MATH  Google Scholar 

  • Goodman LA (1974) Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61:215–231

    Article  MathSciNet  MATH  Google Scholar 

  • Grilli L, Rampichini C (2007) Multilevel factor models for ordinal variables. Struct Equ Model 14:1–25

  • Heinen T (1996) Latent class and discrete latent traits models: similarities and differences. Sage, Thousand Oaks

    Google Scholar 

  • Hoijtink H, Molenaar I (1997) A multidimensional item response model: constrained latent class analysis using the Gibbs sampler and posterior predictive checks. Psychometrika 62:171–190

    Article  MATH  Google Scholar 

  • INVALSI (2009a) Esame di stato di primo ciclo. a.s. 2008/2009. In: INVALSI technical report

  • INVALSI (2009b) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di Italiano

  • INVALSI (2009c) Prove invalsi 2009. In: Report IT (ed) Quadro di riferimento di Matematica

  • 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 53:499–522

    Google Scholar 

  • Kamata A (2001) Item analysis by the hierarchical generalized linear model. J Educ Meas 38:79–93

    Article  Google Scholar 

  • Langheine R, Rost J (1988) Latent trait and latent class models. Plenum, New York

    Book  Google Scholar 

  • Lazarsfeld PF, Henry NW (1968) Latent structure analysis. Houghton Mifflin, Boston

    MATH  Google Scholar 

  • 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 86:96–107

    Article  MATH  Google Scholar 

  • 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, Mahwah

    Google Scholar 

  • Maier KS (2001) A Rasch hierarchical measurement model. J Educ Behav Stat 26:307–330

    Article  Google Scholar 

  • 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 32:611–631

    Article  MathSciNet  Google Scholar 

  • Masters G (1985) A comparison of latent trait and latent class analyses of Likert-type data. Psychometrika 50:69–82

  • McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York

    Book  MATH  Google Scholar 

  • Mislevy RJ, Verhelst N (1990) Modeling item responses when different subjects employ different solution strategies. Psychometrika 55:195–215

    Article  Google Scholar 

  • Muthén L, Muthén B (2012) Mplus user’s guide. Muthén and Muthén edition, Los Angeles

    Google Scholar 

  • 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

  • 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

  • Reckase MD (2009) Multidimensional item response theory. Springer, New York

    Book  Google Scholar 

  • Rost J (1990) Rasch models in latent classes: an integration of two approaches to item analysis. Appl Psychol Meas 14(3):271–282

    Article  MathSciNet  Google Scholar 

  • Rost J (1991) A logistic mixture distribution model for polychotomous item responses. Br J Math Stat Psychol 44:75–92

    Article  Google Scholar 

  • 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

  • Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464

    Article  MATH  Google Scholar 

  • Skrondal A, Rabe-Hesketh S (2004) Generalized latent variable modeling. Multilevel, longitudinal and structural equation models. Chapman and Hall/CRC, London

    Book  MATH  Google Scholar 

  • Smit A, Kelderman H, van der Flier H (1999) Collateral information and mixed Rasch models. Methods Psychol Res 4:19–32

  • Smit A, Kelderman H, van der Flier H (2000) The mixed Birnbaum model: estimation using collateral information. Methods Psychol Res Online 5:31–43

  • 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

  • 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 13:201–222

    Article  Google Scholar 

  • 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

  • Vermunt JK (2001) The use of restricted latent class models for defining and testing nonparametric and parametric item response theory models. Appl Psychol Meas 25:283–294

    Article  MathSciNet  Google Scholar 

  • Vermunt JK (2003) Multilevel latent class models. Sociol Methodol 33:213–239

    Article  Google Scholar 

  • Vermunt JK, Magidson J (2005) Latent GOLD 4.0 user’s guide. Statistical Innovations Inc., Belmont

    Google Scholar 

  • Vermunt JK (2008) Multilevel latent variable modeling: an application in education testing. Austrian J Stat 37:285–299

    Google Scholar 

  • Vermunt JK (2010) Latent class modeling with covariates: two improved three-step approaches. Polit Anal 18:450–469

  • von Davier M (2005) mdltm [computer software]. ETS edn, Princeton

  • von Davier M (2008) A general diagnostic model applied to language testing data. Br J MathStat Psychol 61:287–307

  • 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

Download references

Author information

Authors and Affiliations

  1. Department of Political Sciences, University of Perugia, via A. Pascoli, 20, 06123, Perugia, Italy

    Michela Gnaldi

  2. Department of Economics, University of Perugia, via A. Pascoli, 20, 06123, Perugia, Italy

    Silvia Bacci & Francesco Bartolucci

Authors
  1. Michela Gnaldi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Silvia Bacci
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Francesco Bartolucci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michela Gnaldi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gnaldi, M., Bacci, S. & Bartolucci, F. A multilevel finite mixture item response model to cluster examinees and schools. Adv Data Anal Classif 10, 53–70 (2016). https://doi.org/10.1007/s11634-014-0196-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-014-0196-0

Keywords

Mathematics Subject Classification

Search

Navigation