Skip to main content
Log in

Multidimensional Item Response Theory Models with Collateral Information as Poisson Regression Models

Journal of Classification Aims and scope Submit manuscript

Abstract

Multiple choice items on tests and Likert items on surveys are ubiquitous in educational, social and behavioral science research; however, methods for analyzing of such data can be problematic. Multidimensional item response theory models are proposed that yield structured Poisson regression models for the joint distribution of responses to items. The methodology presented here extends the approach described in Anderson, Verkuilen, and Peyton (2010) that used fully conditionally specified multinomial logistic regression models as item response functions. In this paper, covariates are added as predictors of the latent variables along with covariates as predictors of location parameters. Furthermore, the models presented here incorporate ordinal information of the response options thus allowing an empirical examination of assumptions regarding the ordering and the estimation of optimal scoring of the response options. To illustrate the methodology and flexibility of the models, data from a study on aggression in middle school (Espelage, Holt, and Henkel 2004) is analyzed. The models are fit to data using SAS.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  • AGRESTI, A., CHUANG, C., and KEZOUTH, A. (1987), “Order-Restricted Score Parameters in Association Models for Contingency Tables”, Journal of the American Statistical Assocation, 82, 619–623.

    Article  MATH  Google Scholar 

  • AGRESTI, A. (2002), Categorical Data Analysis (2nd ed.), New York: Wiley.

    Book  MATH  Google Scholar 

  • ANDERSEN, E.B. (1995), “The derivation of polytomous Rasch models”, in Rasch Models: Foundations, Recent Developments, and Applications, eds. G.H. Fischer and I.W. Molenaar, New York: Springer, pp. 271–291.

    Google Scholar 

  • ANDERSON, C.J. (2009), “Categorical Data Analysis with a Psychometric Twist”, in The Sage Handbook of Quantitative Methods in Psychology, eds. R.E. Millsap and A. Maydeu-Olivares, Thousand Oaks, CA: Sage, pp. 311–336.

    Chapter  Google Scholar 

  • ANDERSON, C.J., LI, Z., and VERMUNT, J.K. (2007), “Estimation of Models in a Rasch Family for Polytomous Items and Multiple Latent Variables”, Journal of Statistical Software, 20, published online at http://www.jstatsoft.org/v20/i06/paper.

  • ANDERSON, C.J., VERKUILEN, J.V., and PEYTON, B.L. (2010), “Modeling Polytomous Item Responses Using Simultaneously Estimated Multinomial Logistic Regression Models”, Journal of Educational and Behavioral Statistics, 35, 422–452.

    Article  Google Scholar 

  • ANDERSON, C.J., and VERMUNT, J.K. (2000), “Log-Multiplicative Association Models as Latent Variable Models for Nominal and/or Ordinal Data”, Sociological Methodology, 30, 81–121.

    Article  Google Scholar 

  • ANDERSON, C.J., and YU, H.T. (2007), “Log-Multiplicative Association Models as Item Response Models”, Psychometrika, 72, 5–23.

    Article  MathSciNet  MATH  Google Scholar 

  • ANRAKU, K. (1999), “An Information Criterion for Parameters Under a Simple Order Restriction”, Biometrika, 86, 141–152.

    Article  MathSciNet  MATH  Google Scholar 

  • BARTHOLOMEW, D.J., STEELE, F., MOUSTAKI, I., GALRAITH, J.I. (2008), Analysis of Multivariate Social Science Data, Boca Raton, FL: CRC Press.

    MATH  Google Scholar 

  • BARTOLUCCI, F., and FORCINA, A. (2002), “Extended RC Association Models Allowing for Order Restrictions and Modelling”, Journal of the American Statistical Association, 97, 1192–1199.

    Article  MathSciNet  MATH  Google Scholar 

  • BECKER, M.P. (1989), “Models for the Analysis of Association in Multivariate Contingency Tables”, Journal of the American Statistical Association, 84, 1014–1019.

    Article  Google Scholar 

  • BISHOP, Y.M.M., FIENBERG, S.E., and HOLLAND, P.W. (1975), Discrete Multivariate Analysis, Cambridge, MA: MIT Press.

    MATH  Google Scholar 

  • BOCK, D.R., GIBBONS, SCHILLING, S.G., MURAKI, E., WILSON, D.T., and WOOD, R. (2003), TESTFACT 4.0 Computer Software and Manual, Lincolnwood, IL: Scientific Software International.

    Google Scholar 

  • DE LA TORRE, J. (2009), “Improving the Quality of Ability Estimates Through Multidimensional Scoring and Incorporation of Ancillary Variables”, Applied Psychological Measurement, DOI:10.1177/0146621608329890, published online at http://apm.sagepub.com/content/33/6/465.short.

  • DE BOECK, P, and WILSON, M. (2004), Explanatory Item Response Models: A Generalized Linear and Nonlinear Approach, New York: Springer.

    Book  Google Scholar 

  • ESPELAGE, D.L., and HOLT, M.K. (2001), “Bullying and Victimization During Early Adolescence”, Journal of Emotional Abuse, 2, 123–142.

    Article  Google Scholar 

  • ESPELAGE, D.L., HOLT, M.K., and HENKEL, R.R. (2004), “Examination of Peer-Group Contextual Effects on Aggression During Early Adolescence”, Child Development, 74, 205–220.

    Article  Google Scholar 

  • FAHRMEIR, L., and TUTZ, G. (2001), Multivariate Statistical Modelling Based on Generalized Linear Models, New York: Springer.

    Book  MATH  Google Scholar 

  • FISHER, G.H. (1997), “Unidimensional Linear Logistic Rasch Models”, in Handbook of Modern Item Response Theory, eds. W.J. van der Linden and R.K. Hambletion, New York: Springer, pp. 225–243.

    Chapter  Google Scholar 

  • GALINDO-GARRE, F., and VERMUNT, J.K. (2004), “The Order-Restricted Association Model: Two Estimation Algorithms and Issues in Testing”, Psychometrika, 69, 641–654.

    Article  MathSciNet  Google Scholar 

  • GALINDO-GARRE, F., and VERMUNT, J.K. (2005), “Testing Log-Linear Models with Ordinal Constraints: A Comparison of Asymptotic, Bootstrap, and Posterior Predictive p-Values”, Statistica Neerlandica, 59, 82–94.

    Article  MathSciNet  MATH  Google Scholar 

  • GLAS, C.A.E. (2005), “Review of de Bock, P. and Wilson, M. Explanatory Item Response Theory Models: A Generalized Linear and Nonlinear Approach”, Journal of Educational Measurement, 42, 303–307.

    Article  Google Scholar 

  • GOODMAN, L.A. (1979), “Simple Models for the Analysis of Association in Cross-Classifications Having Ordered Categories”, Journal of the American Statistical Association, 74, 537–552.

    Article  MathSciNet  Google Scholar 

  • GOODMAN, L.A. (1985), “The Analysis of Cross-Classified Data Having Ordered and/or Unordered Categories: Association Models, Correlation Models, and Asymmetry Models for Contingency Tables With or Without Missing Entries”, The Annals of Statistics, 13, 10–69.

    Article  MathSciNet  MATH  Google Scholar 

  • HABERMAN, S.J. (1995), “Computation of Maximum Likelihood Estimates in Association Models”, Journal of the American Statistical Association 90, 1438–1446.

    Article  MathSciNet  MATH  Google Scholar 

  • HEINEN, T. (1993), Discrete Latent Variable Models, The Netherlands: Tilburg University Press.

    Google Scholar 

  • HEINEN, T. (1996), Latent Class and Discrete Latent Trait Models: Similarities and Differences, Thousand Oaks: Sage Publications, Inc.

    Google Scholar 

  • HOLLAND, P.H. (1990), “The Dutch Identity: A New Tool for the Study of Item Response Models”, Psychometrika, 55, 5–18.

    Article  MathSciNet  MATH  Google Scholar 

  • ILIOPOULOS, G., KATERI, M., and NTZOUFRAS, I. (2007), “Bayesian Estimation of Unrestricted and Order-Restricted Association Models for Two-Way Contingency Tables”, Computational Statistics and Data Analysis, 51, 4643–4655.

    Article  MathSciNet  MATH  Google Scholar 

  • ILIOPOULOS, G., and KATERI, M., and NTZOUFRAS, I. (2009), “Bayesian Comparison for the Order Restricted RC Association Model”, Psychometrika, 74, 561–587.

    Article  MathSciNet  MATH  Google Scholar 

  • JOE, H., and LIU, Y. (1996), “A Model for Multivariate Binary Response with Covariates Based on Conditionally Specified Logistic Regressions”, Statistics and Probability Letters, 31, 113–120.

    Article  MathSciNet  MATH  Google Scholar 

  • JUNKER, B.W. (1993), “Conditional Association, Essential Independence and Monotone Unidimensional Item Response Models”, Annals of Statistics, 21, 1359–1378.

    Article  MathSciNet  MATH  Google Scholar 

  • JUNKER, B.W., and SIJTSMA, K. (2000), “Latent and Manifest Monotonicity in Item Response Models”, Applied Psychological Measurement, 24, 65–81.

    Article  Google Scholar 

  • LI, Z. (2010), Loglinear Models as Item Response Models, unpublished doctoral dissertation, University of Illinois at Urbana-Champaign.

  • LINTING, M., MEULMEAN, J.J., GROENEN, P.J.F., and VAN DER KOOIJ, A.J. (2007), “Nonlinear Principle Components Analysis: Introduction and Applications”, Psychological Methods, 12, 336–358.

    Article  Google Scholar 

  • MCDONALD, R.P. (1997), “Normal-ogive multidimensional model” in Handbook of Modern Item Response Theory, eds. W.J. van der Linden and R.K. Hambleton, New York: Springer.

    Google Scholar 

  • RECKASE, M.D. (2009), Multidimensional Item Response Theory, New York: Springer.

    Book  Google Scholar 

  • RIJMEN, F., TUERLINCKX, F., DE BOECK, F., and KUPPENS, P. (2003), “A Nonlinear Mixed Model Framework for Item Response Theory”, Psychological Methods, 8, 185–205.

    Article  Google Scholar 

  • RITOV, Y., and GILULA, Z. (1991), “The Order-Restricted RC Model for Ordered Contingency Tables: Estimation and Testing for Fit”, Annals of Statistics, 19, 2090–2101.

    Article  MathSciNet  MATH  Google Scholar 

  • RUTKOWSKI, L., VASTERLING, J.V., PROCTER, S.P., and ANDERSON, C.J. (2010), “Posttraumatic Stress Disorder and Standardized Test-Taking Ability”, Journal of Educational Psychology, 102, 223–233.

    Article  Google Scholar 

  • SHIEU, C.F., CHEN, C.T., SU, Y.H., and WANG, W.C. (2005), “Using SAS PROC NLMIXED to Fit Item Response Theory Models”, Behavior Research Methods, 37, 202–218.

    Article  Google Scholar 

  • TETTEGAH, S., and ANDERSON, C.J. (2007), “Pre-service Teachers’ Empathy and Cognitions: Statistical Analysis of Text Data by Graphical Models”, Contemporary Educational Psychology, 32, 48–82, published online at http://dx.doi.org/10.1016/j.cedpsych.2006.10.010.

    Article  Google Scholar 

  • THOMAS, N. (2002), “The Role of Secondary Covariates When Estimating Latent Trait Population Distributions”, Psychometrika, 67, 33–48.

    Article  MathSciNet  Google Scholar 

  • VERMUNT, J.K. (1997), ℓEM: A General Program for the Analysis of Categorical Data (Computer Software Manual), Tilburg, The Netherlands. Published online at http://members.home.nl/jeroenvermunt/.

  • VERMUNT, J.K. (1999), “A General Class of Nonparametric Models for Ordinal Categorical Data”, Sociological Methodology, 29, 187–223.

    Article  Google Scholar 

  • WANG, W.C., CHEN, P.H., and CHEN, Y.Y. (2004), “Improving Measurement Precision of Test Batteries Using Multidimensional Item Response Models”, Psychological Methods, 9, 116–136.

    Article  Google Scholar 

  • WONG, R.S. (1995), “Extensions in the Use of Log-Multiplicative Scaled Association Models in Multiway Contingency Tables”, Sociological Methods and Research, 23, 507–538.

    Article  Google Scholar 

  • WONG, R.S. (2001), “Multidimensional Association Models: A Multilinear Approach”, Sociological Methods and Research, 30, 197–240

    Article  MathSciNet  Google Scholar 

  • YEE, T.W., and HASTIE, T. J. (2003), “Reduced-Rank Vector Generalized Linear Models”, Statistical Modelling, 3, 1541.

    Article  MathSciNet  Google Scholar 

  • YEE, T.W. (2010), “The VGAM Package for Categorical Data Analysis”, Journal of Statistical Software, 32, 1–34, published online at http://www.jstatsoft.org/v32/i10/.

    MathSciNet  Google Scholar 

  • ZWINDERMAN, A.H. (1991), “A Generalized Rasch Model for Manifest Predictions”, Psychometrika, 56, 589–600.

    Article  MATH  Google Scholar 

  • ZWINDERMAN, A.H. (1997), “Response Models with Mainfest Predictors”, in Handbook of Modern Item Response Theory, eds. W.J. van der Linden and R.K. Hambletion, New York: Springer, pp. 245–256.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carolyn J. Anderson.

Additional information

A special thanks goes to Dorothy Espelage for the use of her data.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anderson, C.J. Multidimensional Item Response Theory Models with Collateral Information as Poisson Regression Models. J Classif 30, 276–303 (2013). https://doi.org/10.1007/s00357-013-9131-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00357-013-9131-x

Keywords

Navigation