Skip to main content
SpringerLink
Log in

Lower Bounds to the Reliabilities of Factor Score Estimators

  • Published:
Psychometrika Aims and scope Submit manuscript

Abstract

Under the general common factor model, the reliabilities of factor score estimators might be of more interest than the reliability of the total score (the unweighted sum of item scores). In this paper, lower bounds to the reliabilities of Thurstone’s factor score estimators, Bartlett’s factor score estimators, and McDonald’s factor score estimators are derived and conditions are given under which these lower bounds are equal. The relative performance of the derived lower bounds is studied using classic example data sets. The results show that estimates of the lower bounds to the reliabilities of Thurstone’s factor score estimators are greater than or equal to the estimates of the lower bounds to the reliabilities of Bartlett’s and McDonald’s factor score estimators.

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

  • Anderson, R. D., & Rubin, H. (1956). Statistical inference in factor analysis. In Proceedings of the third Berkeley symposium of mathematical statistics and probability (Vol. 5, pp. 111–150).

  • Bartlett, M. S. (1937). The statistical conception of mental factors. British Journal of Psychology, 28, 97–104.

    Google Scholar 

  • Bentler, P. M. (2009). Alpha, dimension-free, and model-based internal consistency reliability. Psychometrika, 74, 137–143.

    Article  PubMed  PubMed Central  Google Scholar 

  • Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.

    Article  Google Scholar 

  • De Leeuw, J. (1983). Models and methods for the analysis of correlation coefficients. Journal of Econometrics, 22(1–2), 113–137.

    Article  Google Scholar 

  • Grice, J. W. (2001). Computing and evaluating factor scores. Psychological Methods, 6, 430–450.

    Article  PubMed  Google Scholar 

  • Guttman, L. (1945). A basis for analyzing test-retest reliability. Psychometrika, 10(4), 255–282.

    Article  PubMed  Google Scholar 

  • Heise, D. R., & Bohrnstedt, G. W. (1970). Validity, invalidity, and reliability. In E. F. Borgatta (Ed.), Sociological methodology 1970 (pp. 104–129). San Francisco: Jossey-Bass.

    Google Scholar 

  • Henderson, H. V., & Searle, S. R. (1981). On deriving the inverse of a sum of matrices. SIAM Review, 23, 53–60.

    Article  Google Scholar 

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

    Google Scholar 

  • McDonald, R. P. (1978). Generalizability in factorable domains: “Domain validity and generalizability”: 1. Educational and Psychological Measurement, 38, 75–79.

    Article  Google Scholar 

  • McDonald, R. P. (1981). Constrained least squares estimators of oblique common factors. Psychometrika, 46, 337–341.

    Article  Google Scholar 

  • McDonald, R. P. (1999). Test theory: A unified treatment. Hillsdale, MI: Erlbaum.

    Google Scholar 

  • R Core Team (2015). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.R-project.org/.

  • Revelle, W. (2014). psych: Procedures for personality and psychological research (R package version 1.4.3).

  • Revelle, W., & Zinbarg, R. E. (2009). Coefficients alpha, beta, omega, and the glb: Comments on Sijtsma. Psychometrika, 74, 145–154.

    Article  Google Scholar 

  • Schmid, J. J., & Leiman, J. M. (1957). The development of hierarchical factor solutions. Psychometrika, 22(1), 83–90.

    Article  Google Scholar 

  • Sijtsma, K. (2009). On the use, the misuse, and the very limited usefulness of Cronbach’s alpha. Psychometrika, 74, 107–120.

    Article  PubMed  Google Scholar 

  • ten Berge, J. M. F., Krijnen, W., Wansbeek, T., & Shapiro, A. (1999). Some new results on correlation-preserving factor scores prediction methods. Linear Algebra and Its Applications, 289, 311–318.

    Article  Google Scholar 

  • ten Berge, J. M. F., Snijders, T. A. B., & Zegers, F. E. (1981). Computational aspects of the greatest lower bound to reliability and constrained minimum trace factor analysis. Psychometrika, 46, 201–213.

    Article  Google Scholar 

  • ten Berge, J. M. F., & Sočan, G. (2004). The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality. Psychometrika, 69, 613–625.

    Article  Google Scholar 

  • ten Berge, J. M. F., & Zegers, F. E. (1978). A series of lower bounds to the reliability of a test. Psychometrika, 34(4), 575–579.

    Article  Google Scholar 

  • Thompson, B. (1993). Calculation of standardized, noncentered factor scores: An alternative to conventional factor scores. Perceptual and Motor Skills, 77(3), 1128–1130.

    Article  Google Scholar 

  • Thomson, G. H. (1946). The factorial analysis of human ability (2nd ed.). New York: Houghton Mifflin.

    Google Scholar 

  • Thurstone, L. L. (1935). The vectors of mind. Chicago: University of Chicago Press.

    Google Scholar 

  • Warner, W. L., Meeker, & Eels, (1960). Social class in America: A manual of procedure for the measurement of social status. New York: Harper.

    Google Scholar 

  • Woodhouse, B., & Jackson, P. H. (1977). 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, 42, 579591.

    Article  Google Scholar 

  • Zinbarg, R. E., Revelle, W., Yovel, I., & Li, W. (2005). Cronbach’s \(\alpha \), Revelles \(\beta \), and McDonalds \(\omega _{H}\): their relations with each other and two alternative conceptualizations of reliability. Psychometrika, 70, 123–133.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Methodology and Statistics, Utrecht University, Padualaan 14, PO Box 80.140, 3508 TC, Utrecht, The Netherlands

    David J. Hessen

Authors
  1. David J. Hessen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David J. Hessen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hessen, D.J. Lower Bounds to the Reliabilities of Factor Score Estimators. Psychometrika 82, 648–659 (2017). https://doi.org/10.1007/s11336-016-9538-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11336-016-9538-5

Keywords

Search

Navigation