, Volume 29, Issue 2, pp 177–185 | Cite as

Notes on factorial invariance

  • William Meredith


Lawley's selection theorem is applied to subpopulations derived from a parent in which the classical factor model holds for a specified set of variables. The results show that there exists an invariant factor pattern matrix that describes the regression of observed on factor variables in every subpopulation derivable by selection from the parent, given that selection does not occur directly on the observable variables and does not reduce the rank of the system. However, such a factor pattern matrix is not unique, which in turn implies that if a simple structure factor pattern matrix can be satisfactorily determined in one such subpopulation the same simple structure can be found in any subpopulation derivable by selection. The implications of these results for “parallel proportional profiles” and “factor matching” techniques are discussed.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ahmavaraa, Y. The mathematical theory of factorial invariance under selection.Psychometrika, 1954,19, 27–38.CrossRefGoogle Scholar
  2. [2]
    Aitken, A. C. Note on selection from a multivariate normal population.Proc. Edinburgh Math. Soc., 1934,4, 106–110.CrossRefGoogle Scholar
  3. [3]
    Barlow, J. A. and Burt, C. The identification of factors from different experiments.Brit. J. statist. Psychol., 1954,7, 52–56.CrossRefGoogle Scholar
  4. [4]
    Birnbaum, Z. W., Paulson, E., and Andrews, F. C. On the effects of selection performed on some coordinates of a multi-dimensional population.Psychometrika, 1950,15, 191–204.PubMedCrossRefGoogle Scholar
  5. [5]
    Cattell, R. B. “Parallel proportional profiles” and other principles for determining the choice of factors by rotation.Psychometrika, 1944,9, 267–283.CrossRefGoogle Scholar
  6. [6]
    Cattell, R. B. and Cattell, A. K. S. Factor solutions for proportional profiles: Analytical solution and an example.Brit. J. statist. Psychol., 1955,8, 83–92.CrossRefGoogle Scholar
  7. [7]
    Harman, H.Modern factor analysis. Chicago: Univ. Chicago Press, 1960.Google Scholar
  8. [8]
    Horst, P. Relations betweenm sets of variates.Psychometrika, 1961,26, 129–150.CrossRefGoogle Scholar
  9. [9]
    Lawley, D. N. A note on Karl Pearson's selection formulae.Proc. roy. Soc. Edinburgh (Section A), 1943–44,62, 28–30.Google Scholar
  10. [10]
    Thomson, G. H. and Ledermann, W. The influence of multivariate selection on the factorial analysis of ability.Brit. J. Psychol., 1939,29, 288–305.Google Scholar
  11. [11]
    Thurstone, L. L.Multiple-factor analysis. Chicago: Univ. Chicago Press, 1947.Google Scholar
  12. [12]
    Wrigley, C. and Neuhaus, J. The matching of two sets of factors. Contract Memorandum Report, A-32, Univ. Illinois, 1955.Google Scholar

Copyright information

© Psychometric Society 1964

Authors and Affiliations

  • William Meredith
    • 1
  1. 1.University of CaliforniaBerkeley

Personalised recommendations