Skip to main content
Log in

Locally dependent latent trait model and the dutch identity revisited

  • Articles
  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

In this paper, we propose a class of locally dependent latent trait models for responses to psychological and educational tests. Typically, item response models treat an individual's multiple response to stimuli as conditional independent given the individual's latent trait. In this paper, instead the focus is on models based on a family of conditional distributions, or kernel, that describes joint multiple item responses as a function of student latent trait, not assuming conditional independence. Specifically, we examine a hybrid kernel which comprises a component for one-way item response functions and a component for conditional associations between items given latent traits. The class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research. An EM algorithm for marginal maximum likelihood of the hybrid kernel model is proposed. Furthermore, we delineate the relationship of the class of locally dependent models and the log-linear model by revisiting the Dutch identity (Holland, 1990).

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

Similar content being viewed by others

References

  • Andersen, E.B. (1980).Discrete statistical models with social science applications. Amsterdam: North-Holland.

    Google Scholar 

  • Bahadur, R. (1961). A representation of the joint distribution of responses ton dichotomous items. In H. Solomon (Ed.),Studies in item analysis and prediction (pp. 158–168). Palo Alto, CA: Stanford University Press.

    Google Scholar 

  • Baker, F.B. (1992).Item response theory. New York, NY: Marcel Dekker.

    Google Scholar 

  • Barndorff-Nielsen, O.E. (1978).Information and exponential families in statistical theory. New York, NY: John Wiley.

    Google Scholar 

  • Becker, R.A., Chambers, J.M., & Wilks, A.R. (1988).The new S language. New York, NY: Chapman & Hall.

    Google Scholar 

  • Bell, R.C., Pattison, P.E., & Withers, G.P. (1988). Conditional independence in a clustered item test.Applied Psychological Measurement,12, 15–26.

    Google Scholar 

  • Bishop, Y., Fienberg, S.E., & Holland, P. (1975).Discrete multivariate analysis. Cambridge, MA: MIT Press.

    Google Scholar 

  • Bock, R.D., & Aitkin, N, (1981). Marginal maximum likelihood estimation of item parameters: An application of an EM algorithm.Psychometrika, 46, 443–459.

    Article  Google Scholar 

  • Bock, R.D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrika, 35, 179–197.

    Google Scholar 

  • Bradlow, E., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets.Psychometrika, 64, 153–168.

    Article  Google Scholar 

  • Cox, D.R. (1972). The analysis of multivariate binary data.Applied Statistics, 21, 113–120.

    Google Scholar 

  • Deming, W.E., & Stephan, F.F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known.Annals of Mathematical Statistics, 11, 427–444.

    Google Scholar 

  • Dempster, A.P., Laird, N.M., & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion).Journal of the Royal Statistical Society, Series B,39, 1–38.

    Google Scholar 

  • Diggle, P.J., Liang, K., & Zeger, S.L. (1994).Analysis of longitudinal data. New York, NY: Oxford.

    Google Scholar 

  • Douglas, J., Kim, H.R., Habing, B., & Gao, F. (1998). Investigating local dependence with conditional covariance functions.Journal of Educational and Behavioral Statistics, 23, 129–151.

    Google Scholar 

  • Efron, B., & Hinkley, D. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expeted Fisher information.Biometrika, 65, 457–481.

    Google Scholar 

  • Ekholm, A., Smith, P.W.F., & McDonald, J.W. (1995). Marginal regression analysis of a multivariate binary response.Biometrika, 82, 847–854.

    Google Scholar 

  • Embretson (Whitely), S. (1984). A general latent trait model for response processes.Psychometrika, 49, 175–186.

    Article  Google Scholar 

  • Embretson (Whitely), S. (1985). Multicomponent latent trait models for test design. In S. Embretson (Ed.),Test design: Developments in psychology and psychometrics (pp. 195–218). Orlando, FL: Academic Press.

    Google Scholar 

  • Fischer, G., & Formann, A.K. (1982). Some applications of logistic latent trait models with linear constraints on the parameter.Applied Psychological Measurement, 6, 397–416.

    Google Scholar 

  • Fitzmaurice, G.M., Laird, N.M. (1993). A likelihood-based method for analyzing longitudinal binary responses.Biometrika, 80, 141–151.

    Google Scholar 

  • Fitzmaurice, G.M., Laird, N.M., & Rotnitzky, A.G. (1993). Regression models for discrete longitudinal responses (with discussion).Statistical Science, 8, 284–309.

    Google Scholar 

  • Glonek, G.F.V. (1996). A class of regression models for multivariate categorical responses.Biometrika, 83, 15–28.

    Article  Google Scholar 

  • Habing, B.T. (1998).Some issues in weak local dependence in item response theory. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign, Department of Statistics.

  • Harper, D. (1972). Local dependence latent structure models.Psychometrika, 37, 53–57.

    Article  Google Scholar 

  • Holland, P. (1990). The Dutch identity: A new tool for the study of item response models.Psychometrika, 55, 5–18.

    Google Scholar 

  • Hoskens, M., & De Boeck, P. (1997). A parametric model for local dependence among test items.Psychological Methods, 2, 261–277.

    Article  Google Scholar 

  • Ip, E.H. (2000). Adjusting for information inflation due to local dependency in moderately large item clusters.Psychometrika, 65, 73–91.

    Article  Google Scholar 

  • Ip, E.H. (2001). Testing for local dependency in dichotomous and polytomous item response models.Psychometrika, 66, 109–132.

    Google Scholar 

  • Jannarone, R.J. (1986). Conjunctive item response theory kernels.Psychometrika, 51, 357–373.

    Article  Google Scholar 

  • Jannarone, R.J. (1992). Conjunctive measurement theory: Cognitive research prospects. In M. Wilson (Ed.),Objective measurement: Theory and practice, Volume 1. (pp. 210–235). Norwood, NJ: Ablex Publishing.

    Google Scholar 

  • Kelderman, H., & Rijkes, C.P.M. (1994). Log-linear multidimensional IRT models for polytomously scored items.Psychometrika, 59, 149–176.

    Article  Google Scholar 

  • Laird, N.M. (1991). Topics in likelihood-based methods for longitudinal data analysis.Statistica Sinica, 1, 33–50.

    Google Scholar 

  • Lang, J.B., & Agresti, A. (1994). Simultaneously modeling joint and marginal distributions of multivariate categorical responses.Journal of American Statistical Association, 89, 625–632.

    Google Scholar 

  • Lehmann, E.L. (1983).Theory of point estimation. New York, NY: John Wiley & Sons.

    Google Scholar 

  • Liang, K.Y., Zeger, S.L., & Qaqish, B. (1992). Multivariate regression analysis for categorical data (with discussion).Journal of the Royal Statistical Society, Series B,54, 3–40.

    Google Scholar 

  • McCullagh, P., & Nelder, J.A. (1989).Generalized linear models (2nd ed.). London, U.K.: Chapman and Hall.

    Google Scholar 

  • Mislevy, R.J. (1988). Exploiting auxiliary information about items in the estimation of the Rasch item difficulty parameters.Applied Psychological Measurement, 12, 281–296.

    Google Scholar 

  • Mislevy, R.J. (1996). Test theory reconceived.Journal of Educational Measurement, 33, 379–416.

    Article  Google Scholar 

  • Mislevy, R.J., & Sheehan, K.M. (1989). The role of collateral information about examinees in item parameter estimation.Psychometrika, 54, 661–679.

    Google Scholar 

  • Patz, R., & Junker, B.W. (1999). A straightforward approach to Markov Chain Monte Carlo methods for item response models.Journal of Educational and Behavioral Statistics, 24, 146–178.

    Google Scholar 

  • Rosenbaum, P.R. (1988). Item bundles.Psychometrika, 53, 349–359.

    Article  Google Scholar 

  • Scott, S., & Ip, E.H. (2002). Empirical Bayes and item clustering effects in a latent variable hierarchical model: A case from the National Assessment of Educational Progress.Journal of the American Statistical Association, 97, 409–419.

    Google Scholar 

  • Stout, W. (1990). A new item response theory modeling approach with application to unidimensional assessment and ability estimation.Psychometrika, 55, 293–325.

    Google Scholar 

  • Ten Vergert, E., Gillespie, M., & Kingma, J. (1993). Testing the assumptions and interpreting the results of the Rasch model using log-linear procedures in SPSS.Behavioral Research Methods, Instruments, & Computers, 25, 350–359.

    Google Scholar 

  • Tuerlinckx, F., & De Boeck, P. (1999). Distinguishing constant and dimension-dependent interaction: a simulation study.Applied Psychological Measurement, 23, 299–307.

    Article  Google Scholar 

  • Verhelst, N.D., & Glas, C.A.W. (1993). A dynamic generalization of the Rasch model.Psychometrika, 58, 395–415.

    Article  Google Scholar 

  • Wainer, H., & Kiely, G.L. (1987). Item clusters and computerized adaptive testing: A case for testlets.Journal of Educational Measurement, 24, 185–201.

    Article  Google Scholar 

  • Zhang, J., & Stout, W. (1997). On Holland's Dutch identity conjecture.Psychometrika, 62, 375–392.

    Article  Google Scholar 

  • Zhao, L.P., & Prentice, R.L. (1990). Correlated binary regression using a quadratic exponential model.Biometrika, 77, 642–648.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Edward H. Ip.

Additional information

The work is supported by a research grant from the Marshall School of Business, University of Southern California. The author thanks the anonymous referees for their suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ip, E.H. Locally dependent latent trait model and the dutch identity revisited. Psychometrika 67, 367–386 (2002). https://doi.org/10.1007/BF02294990

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02294990

Key words

Navigation