Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-08T08:18:51.834Z Has data issue: false hasContentIssue false
  • English
  • Français

Identification with Deficient Rank Loading Matrices in Confirmatory Factor Analysis: Multitrait-Multimethod Models

Published online by Cambridge University Press:  01 January 2025

David Grayson  and
Herbert W. Marsh
David Grayson*
Affiliation:
University of Sydney
Herbert W. Marsh
Affiliation:
University of Western Sydney
*
Requests for reprints should be sent to D. A. Grayson, Aged and Extended Care Department, C25, Concord Hospital, NSW 2139, AUSTRALIA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents some results on identification in multitrait-multimethod (MTMM) confirmatory factor analysis (CFA) models. Some MTMM models are not identified when the (factorial-patterned) loadings matrix is of deficient column rank. For at least one other MTMM model, identification does exist despite such deficiency. It is also shown that for some MTMM CFA models, Howe's (1955) conditions sufficient for rotational uniqueness can fail, yet the model may well be identified and rotationally unique. Implications of these results for CFA models in general are discussed.

Keywords

multitrait-multimethodidentificationrank conditionconfirmatory factor analysisrotational uniqueness

Information

Type
Article
Information
Psychometrika , Volume 59 , Issue 1 , March 1994 , pp. 121 - 134
Copyright
Copyright © 1994 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Algina, J. (1980). A note on identification in the oblique and orthogonal factor analysis models. Psychometrika, 45, 393–396.CrossRefGoogle Scholar
Anderson, T. W., Rubin, H. (1956). Statistical inference in factor analysis. In Neyman, Jerzy (Eds.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics of Probability (Vol. 5), Berkeley and Los Angeles: University of California Press.Google Scholar
Bekker, P. A. (1987). A note on the identification of restricted factor loading matrices. Psychometrika, 51, 607–611.CrossRefGoogle Scholar
Bollen, K. A., Jöreskog, K. G. (1985). Uniqueness does not imply identification. A note on confirmatory factor analysis. Sociological Methods and Research, 14, 155–163.CrossRefGoogle Scholar
Howe, W. G. (1955). Some contributions to factor analysis (Report No. ORNL-1919), Oak Ridge, TN: Oak Ridge National Laboratory.CrossRefGoogle Scholar
Jöreskog, K. G., Sörbom, D. (1989). LISREL 7: A guide to the program and applications, Chicago: SPSS.Google Scholar
Kenny, D. A., Kashy, D. A. (1992). The analysis of the multitrait-multimethod matrix by confirmatory factor analysis. Psychological Bulletin, 112, 165–172.CrossRefGoogle Scholar
Marsh, H. W. (1989). Confirmatory factor analysis of multitrait-multimethod data: Many problems and a few solutions. Applied Psychological Measurement, 13, 335–361.CrossRefGoogle Scholar
Marsh, H. W., Bailey, M. (1991). Confirmatory factor analysis of multitrait-multimethod data: A comparison of alternative modes. Applied Psychological Measurement, 15, 47–70.CrossRefGoogle Scholar
Shapiro, A. (1985). Identifiability of factor analysis: Some results and open problems. Linear Algebra and Its Applications, 70, 1–15.CrossRefGoogle Scholar
Wothke, W. (1984). The estimation of trait and method components in multitrait-multimethod measurement. Unpublished doctoral dissertation. University of Chicago.Google Scholar