Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-07T18:15:57.641Z Has data issue: false hasContentIssue false
  • English
  • Français

A Nonparametric Approach for Assessing Latent Trait Unidimensionality

Published online by Cambridge University Press:  01 January 2025

William Stout
William Stout*
Affiliation:
Department of Statistics and Mathematics, University of Illinois at Urbana-Champaign
*
Request for reprints should be sent to William Stout, Department of Statistics, University of Illinois, 101 lllini Hall, 725 South Wright Street, Champaign, IL 61821.
Get access Rights & Permissions [Opens in a new window]

Abstract

Assuming a nonparametric family of item response theory models, a theory-based procedure for testing the hypothesis of unidimensionality of the latent space is proposed. The asymptotic distribution of the test statistic is derived assuming unidimensionality, thereby establishing an asymptotically valid statistical test of the unidimensionality of the latent trait. Based upon a new notion of dimensionality, the test is shown to have asymptotic power 1. A 6300 trial Monte Carlo study using published item parameter estimates of widely used standardized tests indicates conservative adherence to the nominal level of significance and statistical power averaging 81 out of 100 rejections for examinee sample sizes and psychological test lengths often incurred in practice.

Keywords

unidimensionalitylocal independencetest of unidimensionalitylatent trait theorylatent structure analysisitem response theorylarge sample theoryasymptotic distribution theorynonparametric test
Type
Original Paper
Information
Psychometrika , Volume 52 , Issue 4 , December 1987 , pp. 589 - 617
Copyright
Copyright © 1987 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 referees' comments were remarkably detailed and greatly enhanced the writeup and sensitized the author to certain pertinent issues. Discussions with Fritz Drasgow, Lloyd Humphreys, Dennis Jennings, Brian Junker, Robert Linn, Ratna Nandakumar, and Robin Shealy were also very useful.

This research was supported by the Office of Naval Research under grant N00014-84-K-0186; NR 150-533, and by the National Science Foundation under grant DMS 85-03321.

References

Bartholomew, D. J. (1980). Factor analysis for categorical data. Journal of the Royal Statistical Society, Series B, 42, 293–321.CrossRefGoogle Scholar
Bejar, I. I. (1980). A procedure of investigating the unidimensionality of achievement tests based on item parameter estimates. Journal of Educational Measurement, 17, 283–296.CrossRefGoogle Scholar
Bock, R. D. (1984, September). Contributions of empirical Bayes and marginal maximum likelihood methods to the measurement of individual differences. Proceedings of the 23rd International Conference of Psychology, Acapulco, Mexico.Google Scholar
Bock, R. D., Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., Gibbons, R., Murake, E. (1985). Full information factor analysis, Washington, DC: Office of Naval Research.Google Scholar
Christoffersson, A. (1975). Factor analysis of dichotomous variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
Chung, K. L. (1974). A course in probability theory 2nd ed.,, New York: Academic Press.Google Scholar
Divgi, D. R. (1981, April). Potential pitfalls of item response theory. Paper presented at annual meeting of National Council of Measurement in Education, Los Angeles.Google Scholar
Drasgow, F. (1987). A study of the measurement bias of two standardized psychological tests. Journal of Applied Psychology, 72, 19–30.CrossRefGoogle Scholar
Drasgow, F., Lissak, R. (1983). Modified parallel analysis: A procedure for examining the latent dimensionality of dichotomously scored item responses. Journal of Applied Psychology, 68, 363–373.CrossRefGoogle Scholar
Goldstein, H. (1980). Dimensionality, bias, independence, and measurement scale problems in latent trait score models. British Journal of Mathematical and Statistical Psychology, 33, 234–246.CrossRefGoogle Scholar
Hambleton, R. K., Swaminathan, (1985). Item Response Theory: Principles and Applications, Boston: Kluwer-Nijhoff.CrossRefGoogle Scholar
Hambleton, R. K., Traub, R. E. (1973). Analysis of empirical data using two logistic latent trait models. British Journal of Mathematical and Statistical Psychology, 26, 195–211.CrossRefGoogle Scholar
Hattie, J. (1985). Methodology Review: Assessing unidimensionality of tests and items. Applied Psychology Measurement, 9, 139–164.CrossRefGoogle Scholar
Holland, P. W. (1981). When are item response models consistent with observed data?. Psychometrika, 46, 79–92.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
Hulin, C. L., Drasgow, F., Parsons, L. K. (1983). Item Response Theory, Homewood, IL: Dow Jones-Irwin.Google Scholar
Humphreys, L. C. (1985). General intelligence: An integration of factor, test, and simplex theory. In Wolman, B. J. (Eds.), Handbook of intelligence: Theories, measurements, and applications (pp. 201–224). New York: John Wiley and Sons.Google Scholar
Lord, F. M. (1957). A significance test for the hypothesis that two variables measure the same trait except for errors of measurement. Psychometrika, 22, 207–220.CrossRefGoogle Scholar
Lord, F. M. (1968). An analysis of the verbal scholastic aptitude test using Birnbaum's three-parameter logistic model. Educational and Psychological Measurement, 28, 989–1020.CrossRefGoogle Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Lumsden, J. (1976). Test theory. In Rosenzwieg, M. R., Porter, L. W. (Eds.), Annual Review of Psychology, Palo Alto, CA: Annual Reviews.Google Scholar
McDonald, R. P. (1967a). Nonlinear factor analysis. Psychometrika Monograph No. 15, 32(4, Pt. 2).Google Scholar
McDonald, R. P. (1967). Numerical methods for polynomial models in nonlinear factor analysis. Psychometrika, 32, 77–112.CrossRefGoogle Scholar
McDonald, R. P. (1981). The dimensionality of test and items. British Journal of Mathematical and Statistical Psychology, 34, 100–117.CrossRefGoogle Scholar
McDonald, R. P. (1983). Exploratory and confirmatory factor analysis. In Warner, H., Messick, S. (Eds.), Principles of modern psychological measurement, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
McDonald, R. P., Ahlawat, K. S. (1974). Difficulty factors in binary data. British Journal of Mathematical and Statistical Psychology, 27, 82–99.CrossRefGoogle Scholar
McNemar, Q. (1946). Opinion-attitude methodology. Psychological Bulletin, 43, 289–374.CrossRefGoogle ScholarPubMed
Meredith, W. (1965). Some results on a general stochastic model for mental tests. Psychometrika, 30, 419–440.CrossRefGoogle ScholarPubMed
Mislevy, R. J., Bock, R. D. (1984). Item operating characteristics of the Armed Services Aptitute Battery (ASVAB), Washington, DC: Office of Naval Research.Google Scholar
Molenaar, I. W. (1983). Some improved diagnostics for failure of the Rasch model. Psychometrika, 48, 49–72.CrossRefGoogle Scholar
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
Nandakumar, R. (1987). Refinement of Stout's procedure for assessing latent trait unidimensionality. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign.Google Scholar
Reckase, M. D. (1979). Unifactor latent trait models applied to multifactor tests: Results and implications. Journal of Educational Statistics, 4, 207–230.CrossRefGoogle Scholar
Rosenbaum, P. R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–436.CrossRefGoogle Scholar
Sarrazin, G. (1983, July). The detection of item bias for different cultural groups using latent trait and chi-square methods. Paper presented at the Joint Meeting of the Psychometric Society and the Classification Society, Jouy-en-Joses, France.Google Scholar
Serfling, R. J. (1980). Approximation theories of mathematical statistics, New York: John Wiley.CrossRefGoogle Scholar
Stout, W. F. (1974). Almost sure convergence, New York: Academic Press.Google Scholar
Stout, W. F. (1984). A statistical test of unidimensionality for binary data with applications, Washington, DC: Office of Naval Research.Google Scholar
Stout, W. F. (in press). A nonparametric multidimensional IRT approach with applications to ability estimation and test bias. Psychometrika.Google Scholar
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar