Abstract
Mathematics proficiency involves several content domains and processes at different levels. This essentially means that mathematics ability is a complex latent variable. In standardised testing, the complex, and unobserved, latent constructs underlying a test are traditionally appraised by expert panels through subjective measures. In the present research, we deal with the issue of dimensionality of the latent structure behind a test measuring the mathematics ability of Italian students from a statistical and objective point of view, within an IRT framework. The data refer to a national standardised test developed and collected by the Italian National Institute for the Evaluation of the Education System (INVALSI), and administered to lower secondary school students (grade 8). The model we apply is based on a class of multidimensional latent class IRT models, which allows us to ascertain the test dimensionality based on an explorative approach, and by concurrently accounting for non-constant item discrimination and a discrete latent variable formulation. Our results show that the latent abilities underlying the INVALSI test mirror the assessment objectives defined at the national level for the mathematics curriculum. We recommend the use of the proposed extended IRT models in the practice of test construction, primarily—but not exclusively—in the educational field, to support the meaningfulness of the inferences made from test scores about students’ abilities.
Similar content being viewed by others
References
Bacci, S., Bartolucci, F., Gnaldi, M.: A class of multidimensional latent class IRT models for ordinal polytomous item responses. Commun. Stat Theory Methods 43, 787–800 (2014)
Bartolini Bussi, M.G., Boni, M., Ferri, F., Garuti, R.: Early approach to theoretical thinking: gears in primary school. Educ. Stud. Math. 39, 67–87 (1999)
Bartolucci, F.: A class of multidimensional IRT models for testing unidimensionality and clustering items. Psychometrika 72, 141–157 (2007)
Bartolucci, F., Bacci, S., Gnaldi, M.: MultiLCIRT: an R package for multidimensional latent class item response models. Comput. Stat. Data Anal. 71, 971–985 (2014)
Bartolucci, F., Bacci, S., Gnaldi, M.: Statistical Analysis of Questionnaires: A Unified Approach Based on R and Stata. Chapman & Hall/CRCHall, New York (2015)
Birnbaum, A.: Some latent trait models and their use in inferring an examinee’s ability. In: Lord, F.M., Novick, M.R. (eds.) Statistical Theories of Mental Test Scores, pp. 395–479. Addison-Wesley, Reading, MA (1968)
Briggs, D., Wilson, M.: An introduction to multidimensional measurement using Rasch models. J. Appl. Meas. 4, 87–100 (2003)
Cai L, Yang JS, Hansen M.: Generalized full-information item bifactor analysis. Psychol. Methods 16, 221–248 (2011)
Camilli, G.: A conceptual analysis of differential item functioning in terms of a multidimensional item response model. Appl. Psychol. Meas. 16, 129–147 (1992)
Cizek, G., Bunch, M., Koons, H.: Setting performance standards: contemporary methods. Educ. Meas. 23(4), 31–50 (2004)
Douek, N.: Some remarks about argumentation and proof. In: Boero, P. (ed.) Theorems in School: From History, Epistemology and Cognition to Classroom Practice. Sense Publishers, Rotterdam (2006)
Embretson, S.E.: A multidimensional latent trait model for measuring learning and change. Psychometrika 56, 495–515 (1991)
Formann, A.K.: Linear logistic latent class analysis and the Rasch model. In: Fischer, G., Molenaar, I. (eds.) Rasch Models: Foundations, Recent Developments, and Applications, pp. 239–255. Springer, New York (1995)
Glas, C.A.W., Verhelst, N.D.: Testing the rasch model. In: Fischer, G.H., Molenaar, I. (eds.) Rasch Models. Their Foundations, Recent Developments and Applications, pp. 69–95. Springer, New York (1995)
Gnaldi, M., Bartolucci, F., Bacci, S.: A multilevel finite mixture item response model to cluster examinees and schools. Adv. Data Anal. Classif. (2015). doi:10.1007/s11634-014-0196-0
Golay, P., Lecerf, T.: On higher order structure and confirmatory factor analysis of the French Wechsler Adult Intelligence Scale (WAIS-III). Psychol. Assess. 23, 143–152 (2011)
Holzinger, K., Swineford, S.: The bi-factor method. Psychometrika 47, 41–54 (1937)
INVALSI: Quadro di riferimento per il primo ciclo di istruzione. Technical report, INVALSI (2012a)
INVALSI: Quadro di riferimento per il secondo ciclo di istruzione. Technical report INVALSI (2012b)
Jennrich, R., Bentler, P.: Exploratory bi-factor analysis. Psychometrika 76, 537–549 (2011)
Jennrich, R., Bentler, P.: Exploratory bi-factor analysis: the Oblique case. Psychometrika 77, 442–454 (2012)
Kane, M.: Content-related validity evidence in test development. In: Downing, S.M., Haladyna, T.M. (eds.) Handbook of Test Development. Lawrence Erlbaum Associates, Mahwah, New Jersey (2006)
Lazarsfeld, P.F., Henry, N.W.: Latent Structure Analysis. Houghton Mifflin, Boston (1968)
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)
Loomis, S., Bourque, M.: From tradition to innovation: Standard setting on the national assessment of educational progress. In: Cizek, G.J. (ed.) Setting Performance Standards: Concepts Methods and Perspectives. Lawrence Erlbaum Associates, Mahwah, NJ (2001)
Luecht, R.M., Miller, R.: Unidimensional calibrations and interpretations of composite traits for multidimensional tests. Appl. Psychol. Meas. 16, 279–293 (1992)
Martin-Löf, P.: Statistiska Modeller. Institütet för Försäkringsmatemetik och Matematisk Statistisk vid Stockholms Universitet, Stockholm (1973)
Matteucci M, Mignani S (2015) Multidimensional irt models to analyze learning outcomes of italian students at the end of lower secondary school. In: Millsap R, Bolt D, van der Ark L, Wang W (eds) Quantitative Psychology Research, Springer Proceedings in Mathematics & Statistics, Springer International Publishing Switzerland, vol 89, pp. 91–111
Messick, S.: Validity. In: Linn, R.L. (ed.) Educational Measurement. American Council on Education and Macmillan, New York (1989)
Mokken, R.: A Theory and Procedure of Scale Analysis. De Gruyter, Berlin, Germany (1971)
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, University of California Press, pp. 321–333
Raykov, T., Marcoulides, G.A.: Introduction to Psychometric Theory. Routledge, Taylor & Francis Group, New York (2011)
Reckase, M.: Multidimensional Item Response Theory. Springer, NewYork (2009)
Schoenfeld, A.: Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In: Grows, D. (ed.) Handbook for Research on Mathematics Teaching and Learning. Macmillan, New York (1992)
Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978)
Sijtsma, K., Molenaar, I.: Introduction to Nonparametric Item Response Theory. Sage, Thousand Oaks (2002)
Stout, W.: A non parametric approach for assessing latent trait unidimensionality. Psychometrika 52(4), 589–617 (1987)
Tout, D., Spithill, J.: The challenges and complexities of writing items to test mathematical literacy. In: Turner, R., Stacey, K. (eds.) Assessing Mathematical Literacy, The PISA Experience. Springer, New York (2014)
Vermunt, J.: 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)
Webb, N.L.: Identifying content for student achievement tests. In: Downing, S.M., Haladyna, T.M. (eds.) Handbook of Test Development. Lawrence Erlbaum Associates, Mahwah, New Jersey (2006)
Wirth, R., Edwards, M.: Item factor analysis: current approaches and future directions. Psychol. Methods 12(1), 58–79 (2007)
Zhang, J., Stout, W.: Conditional covariance structure of generalized compensatory multidimensional item. Psychometrika 64(2), 129–152 (1999a)
Zhang, J., Stout, W.: The theoretical detect index of dimensionality and its application to approximate simple structure. Psychometrika 64(2), 213–249 (1999b)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gnaldi, M. A multidimensional IRT approach for dimensionality assessment of standardised students’ tests in mathematics. Qual Quant 51, 1167–1182 (2017). https://doi.org/10.1007/s11135-016-0323-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11135-016-0323-4