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Locally Dependent Latent Trait Model and the Dutch Identity Revisited

Published online by Cambridge University Press:  01 January 2025

Edward H. Ip
Edward H. Ip*
Affiliation:
Marshall School of Business, University of Southern California
*
Requests for reprints should be sent to Edward Ip, Information and Operations Management, Marshall School of Business, University of Southern California, Los Angeles, CA 90089. E-Mail: eddie.ip@marshall.usc.edu
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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).

Keywords

hybrid kernelEM algorithmgeneralized log-linear modelreproducibility
Type
Articles
Information
Psychometrika , Volume 67 , Issue 3 , September 2002 , pp. 367 - 386
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

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.

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 Solomon, H. (Eds.), 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Bock, R.D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Bradlow, E., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.CrossRefGoogle Scholar
Cox, D.R. (1972). The analysis of multivariate binary data. Applied Statistics, 21, 113–120.CrossRefGoogle 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.CrossRefGoogle 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, 39, 1–38.CrossRefGoogle 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.CrossRefGoogle Scholar
Efron, B., & Hinkley, D. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expeted Fisher information. Biometrika, 65, 457–481.CrossRefGoogle Scholar
Ekholm, A., Smith, P.W.F., & McDonald, J.W. (1995). Marginal regression analysis of a multivariate binary response. Biometrika, 82, 847–854.CrossRefGoogle Scholar
Embretson (Whitely), S. (1984). A general latent trait model for response processes. Psychometrika, 49, 175–186.CrossRefGoogle Scholar
Embretson (Whitely), S. (1985). Multicomponent latent trait models for test design. In Embretson, S. (Eds.), Test design: Developments in psychology and psychometrics (pp. 195–218). Orlando, FL: Academic Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Fitzmaurice, G.M., & Laird, N.M. (1993). A likelihood-based method for analyzing longitudinal binary responses. Biometrika, 80, 141–151.CrossRefGoogle 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.CrossRefGoogle 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.Google Scholar
Harper, D. (1972). Local dependence latent structure models. Psychometrika, 37, 53–57.CrossRefGoogle Scholar
Holland, P. (1990). The Dutch identity: A new tool for the study of item response models. Psychometrika, 55, 5–18.CrossRefGoogle Scholar
Hoskens, M., & De Boeck, P. (1997). A parametric model for local dependence among test items. Psychological Methods, 2, 261–277.CrossRefGoogle Scholar
Ip, E.H. (2000). Adjusting for information inflation due to local dependency in moderately large item clusters. Psychometrika, 65, 73–91.CrossRefGoogle Scholar
Ip, E.H. (2001). Testing for local dependency in dichotomous and polytomous item response models. Psychometrika, 66, 109–132.CrossRefGoogle Scholar
Jannarone, R.J. (1986). Conjunctive item response theory kernels. Psychometrika, 51, 357–373.CrossRefGoogle Scholar
Jannarone, R.J. (1992). Conjunctive measurement theory: Cognitive research prospects. In Wilson, M. (Eds.), 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Lehmann, E.L. (1983). Theory of point estimation. New York, NY: John Wiley & Sons.CrossRefGoogle Scholar
Liang, K.Y., Zeger, S.L., & Qaqish, B. (1992). Multivariate regression analysis for categorical data (with discussion). Journal of the Royal Statistical Society, 54, 3–40.CrossRefGoogle Scholar
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models 2nd ed., London, U.K.: Chapman and Hall.CrossRefGoogle 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.CrossRefGoogle Scholar
Mislevy, R.J. (1996). Test theory reconceived. Journal of Educational Measurement, 33, 379–416.CrossRefGoogle Scholar
Mislevy, R.J., & Sheehan, K.M. (1989). The role of collateral information about examinees in item parameter estimation. Psychometrika, 54, 661–679.CrossRefGoogle 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.CrossRefGoogle Scholar
Rosenbaum, P.R. (1988). Item bundles. Psychometrika, 53, 349–359.CrossRefGoogle 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.CrossRefGoogle Scholar
Stout, W. (1990). A new item response theory modeling approach with application to unidimensional assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle 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.CrossRefGoogle Scholar
Tuerlinckx, F., & De Boeck, P. (1999). Distinguishing constant and dimension-dependent interaction: a simulation study. Applied Psychological Measurement, 23, 299–307.CrossRefGoogle Scholar
Verhelst, N.D., & Glas, C.A.W. (1993). A dynamic generalization of the Rasch model. Psychometrika, 58, 395–415.CrossRefGoogle Scholar
Wainer, H., & Kiely, G.L. (1987). Item clusters and computerized adaptive testing: A case for testlets. Journal of Educational Measurement, 24, 185–201.CrossRefGoogle Scholar
Zhang, J., & Stout, W. (1997). On Holland's Dutch identity conjecture. Psychometrika, 62, 375–392.CrossRefGoogle Scholar
Zhao, L.P., & Prentice, R.L. (1990). Correlated binary regression using a quadratic exponential model. Biometrika, 77, 642–648.CrossRefGoogle Scholar