Psychometrika

, Volume 45, Issue 2, pp 249–267 | Cite as

Inequalities among lower bounds to reliability: With applications to test construction and factor analysis

  • P. M. Bentler
  • J. Arthur Woodward
Article

Abstract

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.

Key words

reliability internal consistency test theory factor analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference notes

  1. Bentler, P. M. & Woodward, J. A.Inequalities among lower bounds to reliability. Paper presented at the annual meeting of the Psychometric Society, Chapel Hill, June, 1977.Google Scholar
  2. Della Riccia, G. & Shapiro, L.Minimum rank and minimum trace of correlation matrices. Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel, 1979.Google Scholar
  3. Rozeboom, W. W. Personal communication, March, 1979.Google Scholar
  4. Bentler, P. M.Fast, legitimate, rank-independent factor analytic procedures. Paper presented at the annual meeting of the Society of Multivariate Experimental Psychology, Fort Worth, October, 1972.Google Scholar

References

  1. Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency.Social Science Research, 1972,1, 343–357.Google Scholar
  2. Cronbach, L. J. Coefficient alpha and the internal structure of tests.Psychometrika, 1951,16, 297–334.Google Scholar
  3. Garey, M. R. & Johnson, D. S.Computers and intractability: A guide to the theory of NP-completeness. San Francisco: Freeman, 1979.Google Scholar
  4. Guttman, L. A basis for analyzing test-retest reliability.Psychometrika, 1945,10, 255–282.Google Scholar
  5. Harman, H.Modern factor analysis. Chicago: University of Chicago, 1976.Google Scholar
  6. Jackson, D. N. Reliability of the Jackson Personality Inventory.Psychological Reports, 1977,40, 613–614.Google Scholar
  7. Jackson, P. H. A note on the relation between coefficient alpha and Guttman's “split-half” lower bounds.Psychometrika, 1979,44, 251–252.Google Scholar
  8. 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, 1977,42, 567–578.Google Scholar
  9. Jöreskog, K. G. Analyzing psychological data by structural analysis of covariance matrices. In R. C. Atkinson, D. H. Krantz, R. D. Luce, & P. Suppes (Eds.),Contemporary developments in mathematical psychology, Vol. II. San Francisco: Freeman, 1974, pp. 1–56.Google Scholar
  10. Lord, F. M. A study of speed factors in tests and academic grades.Psychometrika, 1956,21, 31–50.Google Scholar
  11. Lord, F. M. & Novick, M. R.Statistical theories of mental test scores. Reading, Mass.: Addison-Wesley, 1968.Google Scholar
  12. Warner, W. L., Meeker, M. & Eels, K.Social class in America. New York: Harper & Row, 1960.Google Scholar
  13. Woodhouse, B. & Jackson, P. H. Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: II: A search procedure to locate the greatest lower bound.Psychometrika, 1977,42, 579–591.Google Scholar
  14. Woodward, J. A. & Bentler, P. M. A statistical lower bound to population reliability.Psychological Bulletin, 1978,85, 1323–1326.Google Scholar
  15. Woodward, J. A. & Bentler, P. M. Application of optimal sign vectors to reliability and cluster analysis.Psychometrika, 1979,44, 337–341.Google Scholar

Copyright information

© The Psychometric Society 1980

Authors and Affiliations

  • P. M. Bentler
    • 1
  • J. Arthur Woodward
    • 1
  1. 1.Department of PsychologyUniversity of CaliforniaLos Angeles

Personalised recommendations