Skip to main content
SpringerLink
Log in

Non-Iterative Fitting of the Direct Product Model for Multitrait-Multimethod Matrices

  • Conference paper
Latent Variable Modeling and Applications to Causality

Part of the book series: Lecture Notes in Statistics ((LNS,volume 120))

Abstract

The Composite Direct Product Model is a multiplicative model that conveniently decomposes a multitrait-multimethod correlation matrix into trait correlations, method correlations and communality indices. An iterative procedure is required to fit the model. Since the logarithms of absolute values of elements of the correlation matrix satisfy an additive model, the model may be fitted non-iteratively by generalized least squares. Antilogs of parameter estimates for the additive model then yield parameter estimates for the multiplicative model. The choice of weight matrix is considered and asymptotic properties of the estimators and test statistic are examined.

An example is used to compare the proposed estimates with conventional maximum likelihood estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Bagozzi, R. P. (1993). Assessing construct validity in personality research: Applications to measures of self-esteem.Journal of Research in Personality27, 49–87.

    Article  Google Scholar 

  • Bagozzi, R. P., & Yi, Y. (1992). Testing hypotheses about methods, traits, and communalities in the direct product model.Applied Psychological Measurement 16373–380.

    Article  Google Scholar 

  • Bishop, Y. M. M., Fienberg, S. E. & Holland, P. W. (1975).Discrete multivariate analysis: Theory and Practice.Cambridge: MIT Press.

    MATH  Google Scholar 

  • Browne, M. W. (1977). The analysis of patterned correlation matrices by generalized least squares.British Journal of Mathematical and Statistical Psychology 30113–124.

    MathSciNet  MATH  Google Scholar 

  • Browne, M. W. (1984a). Asymptotically distribution-free methods in the analysis of covariance structures.British Journal of Mathematical and Statistical Psychology 3762–83.

    MathSciNet  MATH  Google Scholar 

  • Browne, M. W. (1984b). The decomposition of multitrait-multimethod matrices.British Journal of Mathematical and Statistical Psychology 371–21.

    MathSciNet  MATH  Google Scholar 

  • Browne, M. W. (1989). Relationships between an additive model and a multiplicative model for multitrait-multimethod matrices. In R. Coppi & S. Bolasco (Eds.)Multiway data analysis.(pp. 507–520). Amsterdam: Elsevier Science Publishers BV.

    Google Scholar 

  • Browne, M. W. (1992).MUTMUM: User’s Guide.Department of Psychology, The Ohio State University.

    Google Scholar 

  • Browne, M. W. & Cudeck, R. (1993). Alternative ways of assessing model fit. In: Bollen, K. A. & Long, J. S. (eds.)Testing Structural Equation Models.(pp. 136–162). Newbury Park: Sage.

    Google Scholar 

  • Browne, M. W. & Strydom, H. F. (1991).LINLOG: User’s Guide.Department of Psychology, The Ohio State University.

    Google Scholar 

  • Campbell, D. T. & Fiske, D. W. (1959). Convergent and discriminant validation by the multitrait-multimethod matrix.Psychological Bulletin 56, 81–105.

    Google Scholar 

  • Campbell, D. T. & O’Connell, E. S. (1967). Method factors in multitrait-multimethod matrices: Multiplicative rather than additive?Multivariate Behavioral Research2, 409–426.

    Article  Google Scholar 

  • Cudeck, R. (1988). Multiplicative models and MTMM matrices.Journal of Educational Statistics 13131–147.

    Article  Google Scholar 

  • Dijkstra, T. K. (1992). On statistical inference with parameter estimates on the boundary of the parameter space.British Journal of Mathematical and Statistical Psychology 45289–309.

    MathSciNet  MATH  Google Scholar 

  • Goffin, R. D. & Jackson, D. N. (1992). Analysis of multitrait-multirater performance appraisal data: Composite direct product method versus confirmatory factor analysis.Multivariate Behavioral Research27, 363–385.

    Article  Google Scholar 

  • Henly, S. J., Klebe, K. J., McBride, J. R. & Cudeck, R. (1989). Adaptive and conventional versions of the DAT: The first complete test battery comparison.Applied Psychological Measurement 13363–371.

    Article  Google Scholar 

  • Hsu, P. L. (1949). The limiting distribution of functions of sample means and application to testing hypotheses.Proceedings of the First Berkeley Symposium on mathematical statistics and probability359–402.

    Google Scholar 

  • Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression.Biometrika 11185–190.

    Article  Google Scholar 

  • Jennrich, R. I (1970). An asymptotic chi-square test for the equality of two correlation matrices.Journal of the American Statistical Association 65904–912.

    Article  MathSciNet  MATH  Google Scholar 

  • Joreskog, K. G. (1977). Structural equation models in the social sciences: Specification, estimation and testing. In P. R Krishnaiah (Ed.)Applications of Statistics(pp. 265–287). Amsterdam: North Holland.

    Google Scholar 

  • Lastovicka, J. L., Murry, J. P. & Joachimsthaler, E. A. (1990). Evaluating the measurement validity of lifestyle typologies with qualitative measures and multiplicative factoring.Journal of Marketing Research2, 11–23.

    Article  Google Scholar 

  • Meier, S. T. (1984). The construct validity of burnout.Journal of Occupational Psychology 57211–219.

    Google Scholar 

  • Mooijaart, A. (1985). A note on computational efficiency in asymptotically distribution-free correlational models.British Journal of Mathematical and Statistical Psychology 38112–115.

    MathSciNet  Google Scholar 

  • Muthén, B. & Kaplan, D. (1992). A comparison of some methodologies for the factor analysis of non-normal Liken variables: A note on the size of the model.British Journal of Mathematical and Statistical Psychology 4519–30.

    Google Scholar 

  • Pearson, K. & Filon, L. N. G. (1898). Mathematical contributions to the theory of evolution: IV. On the probable error of frequency constants and on the influence of random selection of variation and correlation.Philosophical Transactions of the Royal Society of London Series A 191229–311.

    Article  Google Scholar 

  • Rao, C. R. (1973).Linear Statistical Inference and its Applications Second Edition.New York: Wiley.

    Book  MATH  Google Scholar 

  • Shapiro, A. (1985). Asymptotic equivalence of minimum discrepancy estimators to GLS estimatorsSouth African Statistical Journal 1973–81.

    MathSciNet  MATH  Google Scholar 

  • Steiger, J. H. (1989).EzPATH: A supplementary module for SYSTAT and SYGRAPH. Evanston, IL: SYSTAT Inc.

    Google Scholar 

  • Steiger, J. H. & Hakstian, A. R. (1982). The asymptotic distribution of a correlation matrix: Theory and application.British Journal of Mathematical and Statistical Psychology 35208–215.

    MathSciNet  MATH  Google Scholar 

  • Steiger, J. H. & Lind, J. C. (1980). Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa CityIA.

    Google Scholar 

  • Strydom, H. F. (1989). Non-iterative estimation of parameters in the composite direct product model. Unpublished MSc thesis. University of South Africa.

    Google Scholar 

  • Swain, A. J. (1975). A class of factor analysis estimation procedures with common asymptotic properties.Psychometrika 40315–335.

    Article  MathSciNet  MATH  Google Scholar 

  • Verhees, J. & Wansbeek, T. J. (1989). A multimode direct product model for covariance structure analysis.British Journal of Mathematical and Statistical Psychology 43231240.

    Google Scholar 

  • Wothke, W. & Browne, M. W. (1990). The direct product model for the MTMM matrix parameterized as a second order factor analysis model.Psychometrika 55255–262.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Psychology and Department of Statistics, The Ohio State University, Columbus, Ohio, USA

    M. W. Browner

  2. Department of Statistics, Vista University, Pretoria, South Africa

    H. F. Strydom

Authors
  1. M. W. Browner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. F. Strydom
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Maia Berkane

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag New York, Inc.

About this paper

Cite this paper

Browner, M.W., Strydom, H.F. (1997). Non-Iterative Fitting of the Direct Product Model for Multitrait-Multimethod Matrices. In: Berkane, M. (eds) Latent Variable Modeling and Applications to Causality. Lecture Notes in Statistics, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1842-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1842-5_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-94917-8

  • Online ISBN: 978-1-4612-1842-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation