Models for residual dependencies

  • Francis Tuerlinckx
  • Paul De Boeck
Chapter
Part of the Statistics for Social Science and Public Policy book series (SSBS)

Abstract

The models discussed in the previous chapters recognize the clustered structure of data one is confronted with most often in psychometrics (i.e., items within persons). The within-person dependencies arising from this clustering are handled through a random effect or latent variable for person p, denoted as θ p . In some cases, there are several major sources of individual differences, and they have to be accounted for by more than one random effect (see Chapter 8 on multidimensionality). Conditional on these random effects, the responses to the different items in the data set should be independent — this requirement is called conditional independence or local (stochastic) independence. However, it appears that in many applications, not all dependence between the responses can be explained by the random effects one assumed to underly the responses. In those cases, it is said that there remain some residual dependencies not accounted for by the model, a phenomenon also denoted as local item dependencies (LIDs). Situations in which residual dependencies may occur are ample. Consider for instance the case where items of a reading test can be subdivided into groups of items each sharing the same reading passage. Data from a test with reading passages may show more dependencies than can be accounted for alone by a single reading ability dimension.

Keywords

Probit Model Conditional Independence Item Parameter Recursive Model Polytomous Item 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. Arnold, B.C., Castillo, E., & Sarabia, J.M. (1999). Conditional Specification of Statistical Models. New York: Springer.MATHGoogle Scholar
  2. Ashford, J.R., & Sowden, R.R. (1970). Multivariate probit analysis. Biometrics, 26, 535–546.CrossRefGoogle Scholar
  3. 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: Stanford University Press.Google Scholar
  4. Bishop, Y.M., Fienberg, S.E., & Holland, P. (1975). Discrete Multivariate Analysis. Cambridge, MA: MIT Press.MATHGoogle Scholar
  5. Bollen, K.A. (1989). Structural Equations with Latent Variables. New York: Wiley.MATHGoogle Scholar
  6. Bonney, G.E. (1987). Logistic regression for dependent binary observations. Biometrics, 43, 951–973.MATHCrossRefGoogle Scholar
  7. Bradlow, E.T., Wainer, H., & Wang, X. (1999). A Bayesian random effects model for testlets. Psychometrika, 64, 153–168.CrossRefGoogle Scholar
  8. Chen, W.H., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265–289.Google Scholar
  9. Chib, S., & Greenberg, E. (1998). Analysis of multivariate probit models. Biometrika, 85, 347–361.MATHCrossRefGoogle Scholar
  10. Connolly, M.A., & Liang, K.-Y. (1988). Conditional logistic regression results for correlated binary data. Biometrika, 75, 501–506.MathSciNetMATHCrossRefGoogle Scholar
  11. Cox, D.R. (1972). The analysis of multivariate binary data. Applied Statistics, 21, 113–120.CrossRefGoogle Scholar
  12. Diggle, P.J., Heagerty, P.J., Liang, K.-Y., & Zeger, S.L. (2002). Analysis of Longitudinal Data (2nd ed.). Oxford: Oxford University Press.Google Scholar
  13. 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
  14. Efron, B., & Tibshirani, R. (1993). An Introduction to the Bootstrap. London: Chapman & Hall.MATHGoogle Scholar
  15. Fahrmeir, L., & Tutz, G. (2001). Multivariate Statistical Modeling Based on Generalized Linear Models (2nd ed.). New York: Springer.CrossRefGoogle Scholar
  16. Fitzmaurice, G.M., Laird, N.M., & Rotnitzky, A.G. (1993). Regression models for discrete longitudinal responses. Statistical Science, 8, 284–309.MathSciNetMATHCrossRefGoogle Scholar
  17. Gelman, A. (2002). Exploratory data analysis for complex models. Technical report, Department of Statistics, Columbia University.Google Scholar
  18. Gelman, A., Goegebeur, Y., Tuerlinckx, F., & Van Mechelen, I. (2000). Diagnostic checks for discrete-data regression models using posterior predictive simulations. Applied Statistics, 42, 247–268.Google Scholar
  19. Haaijer, M.E., Vriens, M., Wansbeek, T.J., & Wedel, M. (1998). Utility covariances and context effects in conjoint MNP models. Marketing Science, 17, 236–252.CrossRefGoogle Scholar
  20. Hoskens M., & De Boeck, P. (1997). A parametric model for local item dependencies among test items. Psychological Methods, 2, 261–277.CrossRefGoogle Scholar
  21. Hoskens M., & De Boeck, P. (2001). Multidimensional componential IRT models. Applied Psychological Measurement, 25, 19–37.MathSciNetCrossRefGoogle Scholar
  22. Ip, E. (2000). Adjusting for information inflation due to local dependence in moderately large item clusters. Psychometrika, 65, 73–91.MathSciNetCrossRefGoogle Scholar
  23. Ip, E. (2001). Testing for local dependence in dichotomous and polytomous item response models. Psychometrika, 66, 109–132.MathSciNetCrossRefGoogle Scholar
  24. Ip, E. (2002). Locally dependent latent trait model and the Dutch identity revisited. Psychometrika, 67, 367–386.MathSciNetCrossRefGoogle Scholar
  25. Ip, E., Wang, J.W., De Boeck, P., & Meulders, M. (2003). Locally dependent latent trait models for polytomous responses. Psychometrika. Manuscript accepted for publication.Google Scholar
  26. Jannerone, R.J. (1986). Conjunctive item response theory kernels. Psychometrika, 51, 357–373.CrossRefGoogle Scholar
  27. Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika, 49, 223–245.MATHCrossRefGoogle Scholar
  28. Keller, L.A., Swaminathan, H., & Sireci, S.G. (2003). Evaluating scoring procedures for context-dependent item sets. Applied Measurement in Education, 16, 207–222.CrossRefGoogle Scholar
  29. Lee, L.-F. (1981). Fully recursive probability models and multivariate log-linear probability models for the analysis of qualitative data. Journal of Econometrics, 16, 51–69.MATHCrossRefGoogle Scholar
  30. Lesaffre, E., & Molenberghs, G. (1991). Multivariate probit analysis: A neglected procedure in medical statistics. Statistics in Medicine, 10, 1391–1403.CrossRefGoogle Scholar
  31. Liang, K.-Y., & Zeger, S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13–22.MathSciNetMATHCrossRefGoogle Scholar
  32. McCulloch, C.E., & Searle, S.R. (2001). Generalized, Linear, and Mixed Models. New York: Wiley.MATHGoogle Scholar
  33. Nerlove, M., & Press, S.J. (1973). Univariate and Multivariate Loglinear and Logistic Models (Report R-1306.) Santa Barbara, CA: Rand Corporation.Google Scholar
  34. Prentice, R.L. (1988). Correlated binary regression with covariates specific to each binary observation. Biometrics, 44, 1033–1048.MathSciNetMATHCrossRefGoogle Scholar
  35. Qu, Y., Williams, G.W., Beck, G.J., & Goormastic, M. (1987) A generalized model of logistic regression for clustered data. Communications in Statistics: Theory and Methodology, 16, 3447–3476.MathSciNetMATHCrossRefGoogle Scholar
  36. Rosenbaum, P. (1984). Testing the conditional independence and monotonic-ity assumptions of item response theory. Psychometrika, 49, 425–435.MathSciNetMATHCrossRefGoogle Scholar
  37. Rosenbaum, P. (1988). Item bundles. Psychometrika, 53, 349–359.MathSciNetMATHCrossRefGoogle Scholar
  38. Schmidt, P., & Strauss, R. (1975). The prediction of occupation using multiple logit models. International Economic Review, 16, 471–486.CrossRefGoogle Scholar
  39. Scott, S., & Ip, E. (2002). Empirical Bayes and item clustering effects in a latent variables hierarchical model: A case study from the National Assessment of Educational Progress. Journal of the American Statistical Association, 97, 409–419.MathSciNetMATHCrossRefGoogle Scholar
  40. Smits, D.J.M., & De Boeck, P. (2003). Random local item dependencies. Paper presented at the 13th International Meeting and the 68th Annual American Meeting of the Psychometric Society, Cagliari, Italy.Google Scholar
  41. Smits, D.J.M., De Boeck, P., & Hoskens, M. (2003). Examining the structure of concepts: Using interaction between items. Applied Psychological Measurement, 27, 415–439.MathSciNetGoogle Scholar
  42. Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models, Psychometrika, 51, 567–577.MATHCrossRefGoogle Scholar
  43. Tsai, R.-C., & Böckenholt, U. (2001). Maximum likelihood estimation of factor and ideal point models for paired comparison data. Journal of Mathematical Psychology, 45, 795–811.MathSciNetMATHCrossRefGoogle Scholar
  44. Tuerlinckx, F., & De Boeck, P. (2001). The effect of ignoring item interaction on the estimated discrimination parameter of the 2PLM. Psychological Methods, 6, 181–195.CrossRefGoogle Scholar
  45. Tuerlinckx, F., De Boeck, P., & Lens, W. (2002). Measuring needs with the Thematic Apperception Test: A psychometric study. Journal of Personality and Social Psychology, 82, 448–461.CrossRefGoogle Scholar
  46. van den Wollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.MATHCrossRefGoogle Scholar
  47. Verbeke, G., & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. New York: Springer.MATHGoogle Scholar
  48. Verguts, T., & De Boeck, P. (2000). A Rasch model for learning while solving an intelligence test. Applied Psychological Measurement, 24, 151–162.CrossRefGoogle Scholar
  49. Verhelst, N.D., & Glas, C.A.W. (1993). A dynamic generalization of the Rasch model. Psychometrika, 58, 395–415.MATHCrossRefGoogle Scholar
  50. Verhelst, N.D., & Glas, C.A.W. (1995). Dynamic generalizations of the Rasch model. In G.H. Fischer & I.W. Molenaar (Eds), Rasch Models: Foundations, Recent Developments, and Applications, (pp. 181–201). New York: Springer.Google Scholar
  51. 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
  52. Wilson, M., & Adams, R.J. (1995). Rasch models for item bundles. Psy-chometrika, 60, 181–198.MATHGoogle Scholar
  53. Yen, W.M. (1984). Effect of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8, 125–145.CrossRefGoogle Scholar
  54. Yen, W.M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, SO, 187–213.Google Scholar
  55. Zeger, S.L., & Liang, K.-Y. (1986). Longitudinal data analysis for discrete and continuous outcomes. Biometrics, 42, 121–130.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Francis Tuerlinckx
  • Paul De Boeck

There are no affiliations available

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