Abstract
Second-order exponential (SOE) models have been proposed as item response models (e.g., Anderson et al., J. Educ. Behav. Stat. 35:422–452, 2010; Anderson, J. Classif. 30:276–303, 2013. doi: 10.1007/s00357-00357-013-9131-x; Hessen, Psychometrika 77:693–709, 2012. doi:10.1007/s11336-012-9277-1 Holland, Psychometrika 55:5–18, 1990); however, the philosophical and theoretical underpinnings of the SOE models differ from those of standard item response theory models. Although presented as reexpressions of item response theory models (Holland, Psychometrika 55:5–18, 1990), which are reflective models, the SOE models are formative measurement models. We extend Anderson and Yu (Psychometrika 72:5–23, 2007) who studied unidimensional models for dichotomous items to multidimensional models for dichotomous and polytomous items. The properties of the models for multiple latent variables are studied theoretically and empirically. Even though there are mathematical differences between the second-order exponential models and multidimensional item response theory (MIRT) models, the SOE models behave very much like standard MIRT models and in some cases better than MIRT models.
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Notes
- 1.
Files containing code and data that reproduce all analyses can be downloaded from http://faculty.education.illinois.edu/cja/homepage.
References
A. Agresti, Categorical Data Analysis, 3rd edn. (Wiley, New York, 2013)
C.J. Anderson, Multidimensional item response theory models with collateral information as Poisson regression models. J. Classif. 30, 276–303 (2013). doi: 10.1007/s00357-00357-013-9131-x
C.J. Anderson, J.K. Vermunt, Log-multiplicative association models as latent variable models for nominal and/or ordinal data. Sociol. Methodol. 30, 81–121 (2000)
C.J. Anderson, H.-T. Yu, Log-multiplicative association models as item response models. Psychometrika 72, 5–23 (2007)
C.J. Anderson, Z. Li, J.K. Vermunt, Estimation of models in a Rasch family for polytomous items and multiple latent variables. J. Stat. Softw. 20 (2007). http://www.jstatsoft.org/v20/i06/v20i06.pdf
C.J. Anderson, J.V. Verkuilen, B.L. Peyton, Modeling polytomous item responses using simultaneously estimated multinomial logistic regression models. J. Educ. Behav. Stat. 35, 422–452 (2010)
C.L.N. Azevedo, H. Bolfarine, D.F. Andrade, Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal. 55, 353–365 (2011)
K.A. Bollen, S. Bauldry, Three C’s in measurement models: causal indicators, composite indicators, and covariates. Psychol. Methods 16, 265–284 (2011)
J.M. Casabianca, B.W. Junker, Multivariate normal distribution, in Handbook of Item Response Theory, Volume Two: Statistical Tools, ed. by W.J. van der Linden (Talyor & Fransics/CRC Press, Boca Raton, 2016), pp. 35–46
D.J. Hessen, Fitting and testing conditional multinomial partial credit models. Psychometrika 77, 693–709 (2012). doi:10.1007/s11336-012-9277-1
P.H. Holland, The Dutch identity: a new tool for the study of item response models. Psychometrika 55, 5–18 (1990)
C.R. Houts, L. Cai, flexMIRT: Flexible Multilevel Multidimensional Item Analysis and Test Scoring (Vector Psychometric Group, LLC, Chapel Hill, 2013)
J. Lee, Multidimensional item response theory: an investigation of interaction effects between factors on item parameter recovery using Markov Chain Monte Carlo. Unpublished Doctoral Dissertation, Michigan State University (2002)
K.V. Mardia, J.M. Kent, J.M. Bibby, Multivariate Analysis (Academic, Orlando, 1979)
K.A. Markus, D. Borsboom, Frontiers of Test Validity Theory: Measurement, Causation and Meaning (Routlege, New York, 2013)
Y. Paek, Pseudo-likelihood estimation of multidimensional polytomous item response theory models. Unpublished Doctoral Dissertation, University of Illinois at Urbana-Champaign (2016)
Y. Paek, C.J. Anderson, Pseudo-likelihood estimation of multidimensional response models: polytomous and dichotomous items, in The 81st Annual Meeting of the Psychometric Society, Asheville, NC, ed. by L.A. van der Ark, S.A. Culpepper, J.A. Douglas, W.C. Wang, M. Wiberg (Springer, New York, 2017)
SAS Institute Inc., Statistical Analysis System, version 9.4 (SAS Institute, Cary, 2015)
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Anderson, C.J., Yu, HT. (2017). Properties of Second-Order Exponential Models as Multidimensional Response Models. In: van der Ark, L.A., Wiberg, M., Culpepper, S.A., Douglas, J.A., Wang, WC. (eds) Quantitative Psychology. IMPS 2016. Springer Proceedings in Mathematics & Statistics, vol 196. Springer, Cham. https://doi.org/10.1007/978-3-319-56294-0_2
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