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.
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Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B.N. Petrov & F. Csáki (Eds.), 2nd international symposium on information theory (pp. 267–281). Armenia, USSR: Tsahkadsov.
Ashford, J.R., & Sowden, R.R. (1970). Multivariate probit analysis. Biometrics, 26, 535–546.
Bahadur, R. (1961). A representation of the joint distribution of responses to n dichotomous items. In H. Solomon (Ed.), Studies in item analysis and prediction (pp. 158–168). Palo Alto, CA: Standford University Press.
Bell, R.C., Pattison, P.E., & Withers, G.P. (1988). Conditional independence in a clustered item test. Applied Psychological Measurement, 12, 15–16.
Bradlow, E.T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.
Chen, W., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265–289.
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.
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.
Cox, D.R. (1972). The analysis of multivariate binary data. Applied Statistics, 21, 113–120.
De Boeck, P., & Wilson, M. (2004). Explanatory item response models: A generalized linear and nonlinear approach. New York: Springer.
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.
Fitzmaurice, G.M., Laird, N.M., & Rotnitzky, A.G. (1993). Regression models for discrete longitudinal responses. Statistical Science, 8, 284–309.
Frank, M.J. (1979). On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Mathematica, 19, 194–226.
Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Annales de l’Université Lyon, Série 3, 14, 53–57.
Frees, A.W., & Valdez, E.A. (1998). Understanding relationships using copulas. Actuarial Research Clearing House, 1, 5–15.
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.
Hoeffding, W. (1940). Masstabinvariante Korrelations-Theorie. Schriften des Matematischen Instituts und des Instituts für angewandte Mathematik der Universität Berlin (5:3, pp. 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].
Holland, P.W. (1990). The Dutch identity: A new tool for the study of item response models. Psychometrika, 55, 5–18.
Holland, P.W., & Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent variable models. Annals of Statistics, 14, 1523–1543.
Hoskens, M., & De Boeck, P. (1997). A parametric model for local item dependencies among test items. Psychological Methods, 2, 261–277.
Ip, E. (2000). Adjusting for information inflation due to local dependence in moderately large item clusters. Psychometrika, 65, 73–81.
Ip, E. (2001). Testing for local dependence in dichotomous and polutomous item response models. Psychometrika, 66, 109–132.
Ip, E. (2002). Locally dependent latent trait model and the Dutch identity revisited. Psychometrika, 67, 367–386.
Ip, E., Wang, Y.J., De Boeck, P., & Meulders, M. (2004). Locally dependent latent trait models for polytomous responses. Psychometrika, 69, 191–216.
Joe, H. (1993). Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis, 46, 262–282.
Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.
Junker, B.W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.
Kotz, S., Balakrishnan, N., & Johnson, N. (2000). Continuous multivariate distributions (Vol. 1). New York: Wiley.
Liang, K.-Y., Zeger, S.L., & Qaqish, B. (1992). Multivariate regression analyses for categorical data. Journal of the Royal Statistical Society, Series B, 54, 3–10.
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.
Masters, G.N. (1988). Item discrimination: When more is worse. Journal of Educational Measurement, 25, 15–19.
McCullagh, P. (1989). Models for discrete multivariate responses. Bulletin of the International Statistics Institute, 53, 407–418.
Meester, S.G. (1991). Methods for clustered categorical data. Unpublished doctoral dissertation. University of Waterloo, Canada.
Molenberghs, G., & Verbeke, G. (2005). Models for discrete longitudinal data. New York: Springer.
Mood, A.M., Graybill, F.A., & Boes, D.C. (1974). Introduction to the theory of statistics. New York: McGraw-Hill.
Nelsen, R.B. (1999). An introduction to copulas. New York: Springer.
Rasch, G. (1960). Probabilistic models for some intelligence and achievement tests. Copenhagen: Danish Institute for Educational Research.
Rosenbaum, P.R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 7.
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.
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.
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.
Sireci, S.G., Thissen, D., & Wainer, H. (1991). On the reliability of testlet-based tests. Journal of Educational Measurement, 28, 237–247.
Sklar, A. (1959). Fonctions de répartition à n dimension et leurs marges. Publications Statistiques Université de Paris, 8, 229–231.
Tate, R. (2003). A comparison of selected empirical methods for assessing the structure of responses to test items. Applied Psychological Measurement, 27, 159–203.
Tuerlinckx, F., & De Boeck, P. (2001a). The effect of ignoring item interactions on the estimated discrimination parameters in item response theory. Psychological Methods, 6, 181–195.
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].
Tuerlinckx, F., & De Boeck, P. (2004). Models for residual dependencies. In P. De Boeck & M. Wilson (Eds.), Explanatory item response models: A generalized linear and nonlinear approach (pp. 289–316). New York: Springer.
Vansteelandt, K. (2000). Formal models for contextualized personality psychology. Unpublished doctoral dissertation. K.U. Leuven, Belgium.
Verhelst, N.D., & Glas, C.A.W. (1993). A dynamic generalization of the Rasch model. Psychometrika, 58, 395–415.
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.
Yen, W.M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30, 187–213.
Zhao, L.P., & Prentice, R.L. (1990). Correlated binary regression using a generalized quadratic model. Biometrics, 77, 642–648.
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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.
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Braeken, J., Tuerlinckx, F. & De Boeck, P. Copula Functions for Residual Dependency. Psychometrika 72, 393–411 (2007). https://doi.org/10.1007/s11336-007-9005-4
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DOI: https://doi.org/10.1007/s11336-007-9005-4