Skip to main content
SpringerLink
Log in

A Geometrical Approach to Item Response Theory

  • Published:
Behaviormetrika Aims and scope Submit manuscript

Abstract

How critical is the concept of the latent trait to modern test theory? The appeal to some unobservable characteristic modulating response probability can lead to some confusion and misunderstanding among users of psychometric technology. This paper looks at a geometric formulation of item response theory that avoids the need to appeal to unobservables. It draws on concepts in differential geometry to represent the trait being measured as a differentiate manifold within the space of possible joint item response probabilities given conditional independence. The result is a manifest and in principle observable representation of the trait that is invariant under one to one transformations of trait scores. These concepts are illustrated by analyses of an actual test.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bock, R.D. and Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: An application of an EM algorithm, Psychometrika, 46, 443–459.

    Article  MathSciNet  Google Scholar 

  • Boothby, W.M. (1975). An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press.

    MATH  Google Scholar 

  • Douglas, J. (1995). Joint consistency of nonparmetric item characteristic curve and ability estimation, Psychometrika, in press.

    Google Scholar 

  • Ellis, J.L. and Junker, B.W. (1995). Tail-measurability in monotone latent variable models. Carnegie-Mellon University, unpublished manuscript.

    MATH  Google Scholar 

  • Fischer, G.H. (1995). Derivations of the Rasch model. In G.H. Fischer and I.W. Molenaar (Eds.) Rasch Models: Foundations, Recent Developments, and Applications. New York: Springer-Verlag.

    Chapter  Google Scholar 

  • Holland, P.W. and Rosenbaum, P.R. (1986). Conditional association and unidimensionality in monotone latent trait models. Annals of Statistics, 14, 1523–1543.

    Article  MathSciNet  Google Scholar 

  • Junker, B.W. (1993). Conditional association, essential independence and monotone unidimensional item response models, Annals of Statistics, 21, 1259–1378.

    Article  MathSciNet  Google Scholar 

  • Levine, M.V. (1984). An Introduction to Multilinear Formula Score Theory (Report 84-4). Champaign, IL: University of Illionois, Department of Educational Psychology, Model-Based Measurement Laboratory

    Google Scholar 

  • Ramsay, J.O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630.

    Article  Google Scholar 

  • Ramsay, J.O. (1995). A similarity-based smoothing approach to nondimensional item analysis. Psychometrika, 60, 323–339.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Psychology, McGill University, 1205 Dr. Penfield Ave., Montréal, Québec, Canada, H3A 1B1

    J. O. Ramsay

Authors
  1. J. O. Ramsay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. O. Ramsay.

About this article

Cite this article

Ramsay, J.O. A Geometrical Approach to Item Response Theory. Behaviormetrika 23, 3–16 (1996). https://doi.org/10.2333/bhmk.23.3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2333/bhmk.23.3

Key Words and Phrases

Search

Navigation