Skip to main content
Log in

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds

  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

Let Σ x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of Σ x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ4 (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency.Social Science Research, 1972,1, 343–357.

    Article  Google Scholar 

  • Cronbach, L. J. Coefficient alpha and the internal structure of tests.Psychometrika, 1951,16, 279–334.

    Article  Google Scholar 

  • Gulliksen, H.Theory of mental tests. New York: Wiley, 1950.

    Google Scholar 

  • Guttman, L. A basis for analyzing test-retest reliability.Psychometrika, 1945,10, 255–282.

    Article  Google Scholar 

  • Lord, F. M. & Novick, M. R.Statistical theories of mental test scores. Reading, Mass: Addison-Wesley, 1968.

    Google Scholar 

  • Novick, M. R. & Lewis, C. Coefficient alpha and the reliability of composite measurements.Psychometrika, 1967,32, 1–13.

    Article  PubMed  Google Scholar 

  • Rozeboom, W. W.Foundations of the theory of prediction. Homewood, Illinois: Dorsey Press, 1966.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Present affiliation of the second author: Department of Statistics, University of Nigeria (Nsukka Campus). Work on this paper was carried out while on study leave in Aberystwyth.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jackson, P.H., Agunwamba, C.C. Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds. Psychometrika 42, 567–578 (1977). https://doi.org/10.1007/BF02295979

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02295979

Key words

Navigation