, Volume 31, Issue 4, pp 545–561 | Cite as

Alpha factor analysis of infallible variables

  • Gene V Glass


The relationship between the factor pattern,F, derived from fallible (containing measurement error) observations on variables and the factor pattern,F*, derived from infallible observations on variables is investigated. A widely believed relationship betweenF andF*, viz.,F*=AF whereA is a diagonal matrix containing the inverses of the square roots of the reliabilities of the variables, is shown to be false for several factor analytic techniques. Under suitable assumptions, it is shown that for Kaiser and Caffrey's “alpha factor analysis”F* andF are related byF*=AF. Empirical examples for which the corresponding elements ofF* andAF are equal to two decimal places are presented. The implications of the equality ofF* andAF for alpha factor analysis are discussed.


Measurement Error Public Policy Diagonal Matrix Statistical Theory Alpha Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Cattell, R. B. A note on factor invariance and the identification of factors.Brit. J. Psychol. (Statist. Sec.), 1949,2, 134–139.Google Scholar
  2. [2]
    Cattell, R. B.Personality and motivation structure and measurement. New York: World Book, 1957.Google Scholar
  3. [3]
    Cronbach, L. J. and Hartmann, W. A note on negative reliabilities.Educ. psychol. Meast, 1954,14, 342–346.Google Scholar
  4. [4]
    Cronbach, L. J., Rajaratnam, N., and Gleser, G. Theory of generalizability: a liberalization of reliability theory.Brit. J. statist. Psychol., 1963,16, 137–163.Google Scholar
  5. [5]
    Cureton, E. E.,et al. Verbal abilities experiment: analysis of new word meaning and verbal analogies tests. PRS Report No. 548, Personal Research Section, The Adjutant General's Office, War Department, 1944 (mimeographed).Google Scholar
  6. [6]
    Davis, F. B. Fundamental factors of comprehension in reading.Psychometrika, 1944,9, 185–197.Google Scholar
  7. [7]
    Glass, G. V. The resolution of complexes of infallible variables into common factors and principal components. Unpublished doctoral dissertation, University of Wisconsin, 1965.Google Scholar
  8. [8]
    Guilford, J. P., Merrifield, P. R., Cristensen, P. R., and Frick, J. W. An investigation of symbolic factors of cognition and convergent production. Psychology Laboratory Report No. 23, University of Southern California, April, 1960.Google Scholar
  9. [9]
    Gulliksen, H.Theory of mental tests.New York: Wiley, 1950.Google Scholar
  10. [10]
    Guttman, L. Some necessary conditions for common-factor analysis.Psychometrika, 1954,19, 149–162.Google Scholar
  11. [11]
    Harris, C. W. Some Rao-Guttman relationships.Psychometrika, 1962,27, 247–263.Google Scholar
  12. [12]
    Harris, C. W. and Liba, M. University of Wisconsin studies in factor analysis. Paper presented at the annual convention of the American Educational Research Association, Chicago, 1965.Google Scholar
  13. [13]
    Holzinger, K. J. and Swineford, F. A study in factor analysis: the stability of a bifactor solution.Supplementary Educational Monographs, No. 48. Chicago: Dept. Educ., Univ. Chicago, 1939. (Data reprinted in Harman, H. H.Modern factor analysis, Chicago: Univ. Chicago Press, 1960.)Google Scholar
  14. [14]
    Hotelling, H. Analysis of a complex of statistical variables into principal components.J. Educ. Psychol., 1933,24, 417–441, 498–520.Google Scholar
  15. [15]
    Kaiser, H. F. The varimax criterion for analytic rotation in factor analysis.Psychometrika, 1958,23, 187–200.Google Scholar
  16. [16]
    Kaiser, H. F. and Caffrey, J. Alpha factor analysis.Psychometrika, 1965,30, 1–14.Google Scholar
  17. [17]
    Meredith, W. Canonical correlations with fallible data.Psychometrika, 1964,29, 55–65.Google Scholar
  18. [18]
    Michael, W. B. and Hunka, S. Research tools: statistical methods.Rev. Educ. Res., 1960,30, 440–486.Google Scholar
  19. [19]
    Rao, C. R. Estimation and tests of significance in factor analysis.Psychometrika, 1955,20, 93–111.Google Scholar
  20. [20]
    Roff, M. The relation between results obtainable with raw and corrected correlation coefficients in multiple factor analysis.Psychometrika, 1937,2, 35–39.Google Scholar
  21. [21]
    Saunders, D. R. Factor analysis I: Some effects of chance error.Psychometrika, 1948,13, 251–257.Google Scholar
  22. [22]
    Spearman, C.The abilities of man. New York: Macmillan, 1927.Google Scholar
  23. [23]
    Thurstone, L. L.Multiple-factor analysis. Chicago: Univ. Chicago Press, 1947.Google Scholar
  24. [24]
    Wechsler, D.Wechsler intelligence scale for children. New York: Psychological Corp., 1948.Google Scholar

Copyright information

© Psychometric Society 1966

Authors and Affiliations

  • Gene V Glass
    • 1
  1. 1.University of IllinoisUSA

Personalised recommendations