, Volume 48, Issue 3, pp 315–342 | Cite as

A second generation nonlinear factor analysis

  • Jamshid Etezadi-Amoli
  • Roderick P. McDonald


Nonlinear common factor models with polynomial regression functions, including interaction terms, are fitted by simultaneously estimating the factor loadings and common factor scores, using maximum-likelihood-ratio and ordinary-least-squares methods. A Monte Carlo study gives support to a conjecture about the form of the distribution of the likelihood-ratio criterion.

Key words

nonlinear factor analysis polynomial regressions maximum-likelihood-ratio estimators dimensions of aphasia 


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Reference notes

  1. McDonald, R. P. PROTEAN—a comprehensive CD3200/3600 program for non-linear factor analysis. Research Memorandum RM-67-26, 1967, Educational Testing Service, Princeton.Google Scholar
  2. Etezadi-Amoli, J. Nonlinear factor analysis using spline functions. Unpublished M.A. thesis, University of Toronto, 1978.Google Scholar


  1. Anderson, T. W. & Rubin, H. Statistical inference in factor analysis.Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1956,5, 111–150.Google Scholar
  2. Buse, A. & Lim, L., Cubic spline as a special case of restricted least squares.Journal of the American Statistical Association, 1977,72, 64–68.Google Scholar
  3. Carroll, J. D. Polynomial factor analysis. Proceedings of the 77th Annual Convention of the American Psychological Association, 1969,4, 103–104.Google Scholar
  4. Fletcher, R. & Reeves, C. M. Function minimization by conjugate gradients.Computer Journal, 1964,7, 149–154.Google Scholar
  5. Gnanadesikan, R. & Wilk, M. B. Data analytic methods in multivariate statistical analysis. In P. R. Krishnaiah (Ed.)Multivariate Analysis: II. New York, Academic Press, 1969.Google Scholar
  6. Guttman, L. The determinacy of factor score matrices with implications for five other basic problems of common factor theory.British Journal of Statistical Psychology, 1955,8, 65–82.Google Scholar
  7. Jöreskog, K. G. & Sörbom, D. EFAP-II,Exploratory Factor Analysis Program. Uppsala, Sweden: Department of Statistics, 1978.Google Scholar
  8. Kestelman, H. The fundamental equation of factor analysis.British Journal of Psychology, Statistical Section, 1952,5, 1–6.Google Scholar
  9. Kruskal, J. B. & Shepard, R. N. A nonmetric variety of linear factor analysis.Psychometrika, 1974,39, 123–157.Google Scholar
  10. Lawley, D. N. Further investigations in factor estimation.Proceedings of the Royal Society, Edinburgh, 1942,62, 176–185.Google Scholar
  11. McDonald, R. P. A general approach to nonlinear factor analysis.Psychometrika, 1962,27, 397–415.Google Scholar
  12. McDonald, R. P. Difficulty factors and nonlinear factor analysis.British Journal of Mathematical and Statistical Psychology, 1965,18, 11–23.Google Scholar
  13. McDonald, R. P. Nonlinear factor analysis.Psychometric Monograph No. 15, 1967a.Google Scholar
  14. McDonald, R. P. Numerical methods for polynomial models in nonlinear factor analysis.Psychometrika, 1967b,32, 77–112.Google Scholar
  15. McDonald, R. P. Factor interaction in nonlinear factor analysis.British Journal of Mathematical and Statistical Psychology, 1967c,20, 205–215.Google Scholar
  16. McDonald, R. P. The simultaneous estimation of factor loadings and scores.British Journal of Mathematical and Statistical Psychology, 1979,32, 212–228.Google Scholar
  17. McDonald, R. P. Exploratory and confirmatory nonlinear factor analysis. In H. Wainer & S. Messick (Eds.) Festschrift for F. M. Lord, Hillside, N.J.: Erlbaum,in press.Google Scholar
  18. McDonald, R. P. & Burr, E. J. A comparison of four methods of constructing factor scores.Psychometrika, 1967,32, 381–401.Google Scholar
  19. McDonald, R. P. & Swaminathan, H. A simple matrix calculus with applications to multivariate analysis.General Systems. 1973,18, 37–54.Google Scholar
  20. Neyman, J. & Scott, Elizabeth L. Consistent estimates based on partially consistent observations.Econometrika, 1948,16, 1–32.Google Scholar
  21. Powell, M. J. D. Restarting procedure for the conjugate gradient method.Mathematical Programming, 1975,12, 241–254.Google Scholar
  22. Schuell, H., Jenkins, J. J. & Carroll, J. B. A factor analysis of the Minnesota Test for differential diagnosis of aphasia.Journal of Speech and Hearing Research, 1962,5, 349–369.Google Scholar

Copyright information

© The Psychometric Society 1983

Authors and Affiliations

  • Jamshid Etezadi-Amoli
    • 1
  • Roderick P. McDonald
    • 2
  1. 1.The Ontario Institute for Studies in EducationUSA
  2. 2.School of EducationMacquarie UniversityNorth RydeAustralia

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