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Notes on Factorial Invariance

Published online by Cambridge University Press:  01 January 2025

William Meredith
William Meredith*
Affiliation:
University of California, Berkeley
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Abstract

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.

Type
Original Paper
Information
Psychometrika , Volume 29 , Issue 2 , June 1964 , pp. 177 - 185
Copyright
Copyright © 1964 Psychometric Society

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References

Ahmavaraa, Y. The mathematical theory of factorial invariance under selection. Psychometrika, 1954, 19, 27–38.CrossRefGoogle Scholar
Aitken, A. C. Note on selection from a multivariate normal population. Proc. Edinburgh Math. Soc., 1934, 4, 106–110.CrossRefGoogle Scholar
Barlow, J. A. and Burt, C. The identification of factors from different experiments. Brit. J. statist. Psychol., 1954, 7, 52–56.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Cattell, R. B. “Parallel proportional profiles” and other principles for determining the choice of factors by rotation. Psychometrika, 1944, 9, 267–283.CrossRefGoogle Scholar
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
Harman, H. Modern factor analysis, Chicago: Univ. Chicago Press, 1960.Google Scholar
Horst, P. Relations betweenm sets of variates. Psychometrika, 1961, 26, 129–150.CrossRefGoogle Scholar
Lawley, D. N. A note on Karl Pearson's selection formulae. Proc. roy. Soc. Edinburgh, 1943, 62, 28–30.Google Scholar
Thomson, G. H., Ledermann, W.. The influence of multivariate selection on the factorial analysis of ability. Brit. J. Psychol., 1939, 29, 288–305.Google Scholar
Thurstone, L. L.. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947.Google Scholar
Wrigley, C. and Neuhaus, J. The matching of two sets of factors. Contract Memorandum Report, A-32, Univ. Illinois, 1955.Google Scholar