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The Direct Product Model for the Mtmm Matrix Parameterized as a Second Order Factor Analysis Model

Published online by Cambridge University Press:  01 January 2025

Werner Wothke  and
Michael W. Browne
Werner Wothke*
Affiliation:
Scientific Software, Chicago, IL
Michael W. Browne
Affiliation:
University of South Africa
*
Requests for reprints or LISREL sample input files should be sent to Werner Wothke, Scientific Software, Inc., 1525 E. 53rd Street, Suite 830, Chicago, IL 60615.
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Abstract

The composite direct product model for the multitrait-multimethod matrix is reparameterized as a second-order factor analysis model. This facilitates the use of widely available computer programs such as LISREL and LISCOMP for fitting the model.

Keywords

multitrait-multimenthod matrixmultiplicative modelsecond order factor analysisequivalent multivariate models
Type
Original Paper
Information
Psychometrika , Volume 55 , Issue 2 , June 1990 , pp. 255 - 262
Copyright
Copyright © 1990 The Psychometric Society

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Footnotes

Bruce Bloxom. Paul Horst and Karl Jöreskog contributed helpful comments to an earlier version of this paper. Their suggestions are gratefully acknowledged.

References

Bentler, P. M., & Lee, S. Y. (1978). Statistical aspect of a three-mode factor analysis model. Psychometrika, 43, 343–352.CrossRefGoogle Scholar
Bentler, P. M., Poon, W. Y., & Lee, S. Y. (1988). Generalized multimode latent variable models: Implementation by standard programs. Computational Statistics and Data Analysis, 6, 107–118.CrossRefGoogle Scholar
Bloxom, B. (1968). A note on invariance in three-mode factor analysis. Psychometrika, 33, 347–350.CrossRefGoogle ScholarPubMed
Bloxom, B. (1984). Tucker's three-mode factor analysis model. In Law, H. G., Snyder, C. W. Jr., Hattie, J. A., & McDonald, R. P. (Eds.), Research methods for multiomde data analysis, New York: Praeger.Google Scholar
Boruch, R. F., & Wolins, L. (1970). A procedure for estimation of trait, method, and error variance attributable to a measure. Educational and Psychological Measurement, 30, 547–574.CrossRefGoogle Scholar
Brown, M. W. (1984). the decomposition of multitrait-multimethod matrices. The British Journal of Mathematical and Statistical Psychology, 37, 1–21.CrossRefGoogle Scholar
Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. The British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Browne, M. W. (1985). MUTMUM: Decomposition of multitrait-multimethod matrices, minimum option version. Unpublished manuscript, University of South Africa.Google Scholar
Browne, M. W. (in press). Relationships between an additive model and a multiplicative model for multitraitmultimethod matrices. In Coppi, R. & Bolasco, S. (Eds.), Analysis of multiway data matrices. Amsterdam: Elsevier.Google Scholar
Campbell, D. T., & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix. Psychological Bulletin, 56, 81–105.CrossRefGoogle ScholarPubMed
Campbell, D. T., & O'Connell, E. J. (1967). Method factors in multitrait-multimethod matrices: Multiplicative rather than additive?. Multivariate Behavioral Research, 2, 409–426.CrossRefGoogle ScholarPubMed
Campbell, D. T., & O'Connell, E. J. (1982). Methods as diluting trait relationships rather than adding irrelevant systematic variance. In Brinberg, D. & Kidder, L. H. (Eds.), Forms of validity in research. New directions for methodology of social and behavioral sience (Vol. 12), San Francisco: Jossey-Bass.Google Scholar
Cudeck, R. (1988). Multiplicative models and MTMM matrices. Journal of Educational Statistics, 13(2), 131–147.Google Scholar
Jackson, D. N. (1975). Multimethod factor analysis: A reformulation. Multivariate Behavioral Research, 10, 259–275.CrossRefGoogle ScholarPubMed
Jöreskog, K. G. (1966). Testing a simple structure hypothesis in factor analysis. Psychometrika, 31, 165–178.CrossRefGoogle ScholarPubMed
Jöreskog, K. G. (1971). Statistical analysis of congeneric tests. Psychometrika, 36, 109–133.CrossRefGoogle Scholar
Jöreskog, K. G. (1978). Structural analysis of covariance and correlation matrices. Psychometrika, 43, 443–477.CrossRefGoogle Scholar
Jöreskog, K. G., & Sörbom, D. (1988). LISREL 7. A guide to the program and applications, Chicago, IL: SPSS Publications.Google Scholar
Lawler, E. E. III (1967). The multitrait-multimethod approach to measuring managerial job performance. Journal of Applied Psychology, 51, 369–381.CrossRefGoogle Scholar
Muthén, B. O. (1987). LISCOMP, Analysis of linear structural equations with a comprehensive measurement model, Mooresville, IN: Scientific Software.Google Scholar
Rindskopf, D. (1983). Parameterizing inequality constraints on unique variances in linear structural models. Psychometrika, 48, 73–83.CrossRefGoogle Scholar
Schmitt, N., & Stults, D. M. (1986). Methodology review: Analysis of multitrait-multimethod matrices. Applied Psychological Measurement, 10(1), 1–22.CrossRefGoogle Scholar
Searle, S. R. (1982). Matrix algebra useful for statistics, New York: Wiely.Google Scholar
Swain, A. J. (1975). Analysis of parametric structures for variance matrices. Unpublished doctoral dissertation, University of Adelaide.Google Scholar
Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279–311.CrossRefGoogle ScholarPubMed
Werts, C. E. & Linn, R. L. (1970). Path analysis. Psychological examples. Psychological Bulletin, 74(3), 193–212.CrossRefGoogle Scholar
Widaman, K. F. (1985). Hierarchically nested covariance structure models for multitrait-multimethod data. Applied Psychological Measurement, 9(1), 1–26.CrossRefGoogle Scholar
Wiley, D., Schmidt, W. H. & Bramble, W. J. (1973). Studies of a class of covariance structure models. Journal of the American Statistical Association, 68(342), 317–323.CrossRefGoogle Scholar
wothke, W. (1987, April). Multivariate linear models of the multitrait-multimethod matrix. paper presented at the meeting of the American Educational Research Association, Washington, DC. (Educational Resources Information Center: ED 283 850, TM 870 369)Google Scholar