Hostname: page-component-5b777bbd6c-5mwv9 Total loading time: 0 Render date: 2025-06-19T07:30:36.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing the Conditional Independence and Monotonicity Assumptions of Item Response Theory

Published online by Cambridge University Press:  01 January 2025

Paul R. Rosenbaum
Paul R. Rosenbaum*
Affiliation:
Research Statistics Group, Educational Testing Service
*
Requests for reprints should be sent to Paul R. Rosenbaum, Research Statistics Group, 2 I-T, Educational Testing Service, Princeton, N.J. 08541
Get access Rights & Permissions [Opens in a new window]

Abstract

When item characteristic curves are nondecreasing functions of a latent variable, the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses. This general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves. The proposed tests are simple, have known asymptotic null distributions, and possess certain optimal properties. In an example, the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves.

Keywords

Latent trait modelsAssociation of random variablesDiscrete dataContingency tables
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984 , pp. 425 - 435
Copyright
Copyright © 1984 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.)

Article purchase

Temporarily unavailable

Footnotes

The author acknowledges Paul W. Holland for valuable conversations on the subject of this paper; Henry Braun and Fred Lord for comments at a presentation on this subject which led to improvements in the paper; Carl H. Haag for permission to use the data in §4; Bruce Kaplan for assistance with computing; and two referees for helpful suggestions.

References

Andersen, E. B. (1980). Discrete Statistical Models with Social Science Applications, Amsterdam: North Holland.Google Scholar
Birch, M. W. (1964). The detection of partial association, I: the 2 × 2 case. Journal of the Royal Statistical Society, 26, 313–324.CrossRefGoogle Scholar
Birch, M. W. (1965). The detection of partial association, II: the general case. Journal of the Royal Statistical Society, 27, 111–124.CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability (Part 5). In Lord, F. and Novick, M. (Eds.), Statistical Theories of Mental Test Scores, Reading, MA: Addison-Wesley.Google Scholar
Bishop, Y., Fienberg, S., and Holland, P. (1975). Discrete Multivariate Analysis, Cambridge, MA: MIT Press.Google Scholar
Bock, D., and Lieberman, M. (1970). Fitting a response model forn dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Breslow, N. (1981). Odds ratio estimators when the data are sparse. Biometrika, 68, 73–84.CrossRefGoogle Scholar
Clayton, D. G. (1974). Some odds ratio statistics for the analysis of ordered categorical data. Biometrika, 61, 525–531.CrossRefGoogle Scholar
Cox, D. R. (1966). A simple example of a comparison involving quantal data. Biometrika, 53, 215–220.CrossRefGoogle ScholarPubMed
Cressie, N. and Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–141.CrossRefGoogle Scholar
Esary, J. D., Proschan, F., and Walkup, D. W. (1967). Association of random variables, with applications. Annals of Mathematical Statistics, 38, 1466–1474.CrossRefGoogle Scholar
Ferguson, T. (1967). Mathematical Statistics: A Decision Theoretic Approach, New York: Academic Press.Google Scholar
Goldstein, H. (1980). Dimensionality, bias, independence and measurement scale problems in latent test score models. British Journal of Mathematical and Statistical Psychology, 33, 234–246.CrossRefGoogle Scholar
Goodman, L. and Kruskal, W. (1979). Measures of Association for Cross Classifications, New York: Springer-Verlag.CrossRefGoogle Scholar
Holland, P. W. (1981). When are item response models consistent with observed data?. Psychometrika, 46, 79–92.CrossRefGoogle Scholar
Lehmann, E. L. (1955). Ordered families of distributions. Annals of Mathematical Statistics, 26, 399–419.CrossRefGoogle Scholar
Lehmann, E. L. (1966). Some concepts of dependence. Annals of Mathematical Statistics, 37, 1137–1153.CrossRefGoogle Scholar
Lord, F. (1952). A Theory of Test Scores. Psychometric Monograph # 7. Psychometric Society.Google Scholar
Lord, F. (1980). Applications of Item Response Theory to Practical Testing Problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Mantel, N. and Haenszel, W. (1959). Statistical aspects of the retrospective study of disease. Journal of the National Cancer Institute, 22, 719–748.Google ScholarPubMed
Miller, R. G. (1981). Simultaneous Statistical Inference, New York: Springer-Verlag.CrossRefGoogle Scholar
Molenaar, I. W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear Statistical Inference and Its Applications (pp. 389–391). New York: Wiley.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic Models for Some Intelligence and Attainment Tests, Copenhagen: Neilson and Lydiche.Google Scholar
Robertson, T. and Wright, F. T. (1981). Likelihood ratio tests for and against stochastic ordering between multinomial populations. Annals of Statistics, 9, 1248–1257.CrossRefGoogle Scholar
Schweder, T. (1970). Composable Markov processes. Journal of Applied Probability, 7, 400–410.CrossRefGoogle Scholar
Schweder, T. and Spjotvoll, E. (1982). Plots ofp-values to evaluate many tests simultaneously. Biometrika, 69, 493–502.CrossRefGoogle Scholar
Tjur, T. (1982). A connection between Rasch's item analysis model and a multiplicative Poisson model. Scand. J. Statist., 9, 23–30.Google Scholar
Traub, R. E. (1983). A priori considerations in choosing an item response model. In Hambleton, R. K. (Eds.), Applications of Item Response Theory (pp. 57–70). Vancouver: Educational Research Institute of British Columbia.Google Scholar
Traub, R. E. and Wolfe, R. G. (1981). Latent trait theories and the assessment of educational achievement. Review of Research in Education, 9, 377–435.Google Scholar
Tukey, J. W. (1977). Exploratory Data Analysis, Reading Massachusetts: Addison-Wesley.Google Scholar
Van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar