Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-08T09:55:27.169Z Has data issue: false hasContentIssue false
  • English
  • Français

Copula Functions for Residual Dependency

Published online by Cambridge University Press:  01 January 2025

Johan Braeken ,
Francis Tuerlinckx  and
Paul De Boeck
Johan Braeken*
Affiliation:
University of Leuven
Francis Tuerlinckx
Affiliation:
University of Leuven
Paul De Boeck
Affiliation:
University of Leuven
*
Requests for reprints should be sent to Johan Braeken, Research Group Quantitative and Personality Psychology, Department of Psychology, University of Leuven, Tiensestraat 102, B-3000 Leuven, Belgium. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Most item response theory models are not robust to violations of conditional independence. However, several modeling approaches (e.g., conditioning on other responses, additional random effects) exist that try to incorporate local item dependencies, but they have some drawbacks such as the nonreproducibility of marginal probabilities and resulting interpretation problems. In this paper, a new class of models making use of copulas to deal with local item dependencies is introduced. These models belong to the bigger class of marginal models in which margins and association structure are modeled separately. It is shown how this approach overcomes some of the problems associated with other local item dependency models.

Keywords

item response theorylocal item dependencycopula
Type
Theory and Methods
Information
Psychometrika , Volume 72 , Issue 3 , September 2007 , pp. 393 - 411
Copyright
Copyright © 2007 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The authors wish to thank Yuri Goegebeur and Taoufik Bouezmarni for their helpful suggestions and comments. We are also indebted to the reviewers of this paper, their generous comments and remarks greatly improved the setup and clarity of the presented material. Preparation of this manuscript was supported in part by the Fund for Scientific Research Flanders (FWO) Grant G.0148.04 and by the K.U. Leuven Research Council Grant GOA/2005/04.

References

Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Petrov, B.N. & Csáki, F. (Eds.), 2nd international symposium on information theory (pp. 267–281). Armenia, USSR: Tsahkadsov.Google Scholar
Ashford, J.R., Sowden, R.R. (1970). Multivariate probit analysis. Biometrics, 26, 535–546.CrossRefGoogle Scholar
Bahadur, R. (1961). A representation of the joint distribution of responses to n dichotomous items. In Solomon, H. (Ed.), Studies in item analysis and prediction (pp. 158–168). Palo Alto, CA: Standford University Press.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
Bradlow, E.T., Wainer, H., Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.CrossRefGoogle Scholar
Chen, W., Thissen, D. (1997). Local dependence indexes for item pairs using item response theory.. Journal of Educational and Behavioral Statistics, 22, 265–289.CrossRefGoogle Scholar
Clayton, D.G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151.CrossRefGoogle Scholar
Cook, R.D., Johnson, M.E. (1981). A family of distributions to modeling non-elliptically symmetric multivariate data. Journal of the Royal Statistical Society, Series B, 43, 210–218.CrossRefGoogle Scholar
Cox, D.R. (1972). The analysis of multivariate binary data. Applied Statistics, 21, 113–120.CrossRefGoogle Scholar
De Boeck, P., Wilson, M. (2004). Explanatory item response models: A generalized linear and nonlinear approach, New York: Springer.CrossRefGoogle Scholar
Ferrara, S., Huynh, H., Michaels, H. (1999). Contextual explanations of local dependence in item clusters in a large-scale hands-on science performance assessment. Journal of Educational Measurement, 36, 119–140.CrossRefGoogle Scholar
Fitzmaurice, G.M., Laird, N.M., Rotnitzky, A.G. (1993). Regression models for discrete longitudinal responses. Statistical Science, 8, 284–309.Google Scholar
Frank, M.J. (1979). On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Mathematica, 19, 194–226.CrossRefGoogle Scholar
Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Annales de l’Université Lyon, Série 3, 14, 53–77.Google Scholar
Frees, A.W., Valdez, E.A. (1998). Understanding relationships using copulas. Actuarial Research Clearing House, 1, 5–45.Google Scholar
Genest, C., MacKay, J. (1986). Copules archimédiennes et familles de lois bi-dimensionelles dont les marges sont donées. Canadian Journal of Statistics, 14, 145–159.CrossRefGoogle Scholar
Hoeffding, W. (1940). Masstabinvariante Korrelations-Theorie. Schriften des Matematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin, 5(3), 179–223. [Reprinted as Scale-invariant correlation theory in the Collected Works of Wassily Hoeffding, N.I. Fischer, & P.K. Sen (Eds.), New York: Springer]Google Scholar
Holland, P.W. (1990). The Dutch identity: A new tool for the study of item response models. Psychometrika, 55, 5–18.CrossRefGoogle Scholar
Holland, P.W., Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Hoskens, M., De Boeck, P. (1997). A parametric model for local item dependencies among test items. Psychological Methods, 2, 261–277.CrossRefGoogle Scholar
Ip, E. (2000). Adjusting for information inflation due to local dependence in moderately large item clusters. Psychometrika, 65, 73–91.CrossRefGoogle Scholar
Ip, E. (2001). Testing for local dependence in dichotomous and polutomous item response models. Psychometrika, 66, 109–132.CrossRefGoogle Scholar
Ip, E. (2002). Locally dependent latent trait model and the Dutch identity revisited. Psychometrika, 67, 367–386.CrossRefGoogle Scholar
Ip, E., Wang, Y.J., De Boeck, P., Meulders, M. (2004). Locally dependent latent trait models for polytomous responses. Psychometrika, 69, 191–216.CrossRefGoogle Scholar
Joe, H. (1993). Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis, 46, 262–282.CrossRefGoogle Scholar
Joe, H. (1997). Multivariate models and dependence concepts, London: Chapman & Hall.Google Scholar
Junker, B.W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.CrossRefGoogle Scholar
Kotz, S., Balakrishnan, N., Johnson, N. (2000). Continuous multivariate distributions (Vol. 1), New York: Wiley.CrossRefGoogle Scholar
Liang, K.-Y., Zeger, S.L., Qaqish, B. (1992). Multivariate regression analyses for categorical data. Journal of the Royal Statistical Society, Series B, 54, 3–40.CrossRefGoogle Scholar
Mantel, N., Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of National Cancer Institute, 22, 719–748.Google ScholarPubMed
Masters, G.N. (1988). Item discrimination: When more is worse. Journal of Educational Measurement, 25, 15–29.CrossRefGoogle Scholar
McCullagh, P. (1989). Models for discrete multivariate responses. Bulletin of the International Statistics Institute, 53, 407–418.Google Scholar
Meester, S.G. (1991). Methods for clustered categorical data. Unpublished doctoral dissertation. University of Waterloo, Canada.Google Scholar
Molenberghs, G., Verbeke, G. (2005). Models for discrete longitudinal data, New York: Springer.Google Scholar
Mood, A.M., Graybill, F.A., Boes, D.C. (1974). Introduction to the theory of statistics, New York: McGraw-Hill.Google Scholar
Nelsen, R.B. (1999). An introduction to copulas, New York: Springer.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and achievement tests, Copenhagen: Danish Institute for Educational Research.Google Scholar
Rosenbaum, P.R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 7.Google Scholar
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
Scott, S.L., Ip, E. (2002). Empirical Bayes and item clustering effects in a latent variable hierarchical model: A case study from the national assessment of educational progress. Journal of the American Statistical Association, 97, 409–419.CrossRefGoogle Scholar
Self, G.H., Liang, K.-Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605–610.CrossRefGoogle Scholar
Sireci, S.G., Thissen, D., Wainer, H. (1991). On the reliability of testlet-based tests. Journal of Educational Measurement, 28, 237–247.CrossRefGoogle Scholar
Sklar, A. (1959). Fonctions de répartition à n dimension et leurs marges. Publications Statistiques Université de Paris, 8, 229–231.Google Scholar
Tate, R. (2003). A comparison of selected empirical methods for assessing the structure of responses to test items. Applied Psychological Measurement, 27, 159–203.CrossRefGoogle Scholar
Tuerlinckx, F., De Boeck, P. (2001). The effect of ignoring item interactions on the estimated discrimination parameters in item response theory. Psychological Methods, 6, 181–195.CrossRefGoogle ScholarPubMed
Tuerlinckx, F., & De Boeck, P. (2001b). Non-modeled item interactions lead to distorted discrimination parameters: A case study. Methods of Psychological Research, 6. [Retrieved May 20, 2005 from http://www.mpr-online.de/issue14/art3/Tuerlinckx.pdf].CrossRefGoogle Scholar
Tuerlinckx, F., De Boeck, P. (2004). Models for residual dependencies. In De Boeck, P., Wilson, M. (Eds.), Explanatory item response models: A generalized linear and nonlinear approach (pp. 289–316). New York: Springer.CrossRefGoogle Scholar
Vansteelandt, K. (2000). Formal models for contextualized personality psychology. Unpublished doctoral dissertation. K.U. Leuven, Belgium.Google Scholar
Verhelst, N.D., Glas, C.A.W. (1993). A dynamic generalization of the Rasch model. Psychometrika, 58, 395–415.CrossRefGoogle Scholar
Yen, W.M. (1984). Effects of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8, 125–145.CrossRefGoogle Scholar
Yen, W.M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30, 187–213.CrossRefGoogle Scholar
Zhao, L.P., Prentice, R.L. (1990). Correlated binary regression using a generalized quadratic model. Biometrics, 77, 642–648.CrossRefGoogle Scholar