Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-07T15:11:06.440Z Has data issue: false hasContentIssue false
  • English
  • Français

A Second Generation Nonlinear Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Jamshid Etezadi-Amoli  and
Roderick P. McDonald
Jamshid Etezadi-Amoli
Affiliation:
The Ontario Institute for Studies in Education
Roderick P. McDonald*
Affiliation:
Macquarie University
*
Requests for reprints should be addressed to Roderick P. McDonald, School of Education, Macquarie University, North Ryde, N.S.W. 2113, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

nonlinear factor analysispolynomial regressionsmaximum-likelihood-ratio estimatorsdimensions of aphasia
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 3 , September 1983 , pp. 315 - 342
Copyright
Copyright © 1983 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.)

Footnotes

The research reported in this paper was partly supported by Natural Sciences and Engineering Research Grant No. A6346.

References

Reference Notes

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 Unpublished M.A. thesis, University of Toronto, 1978.Google Scholar

References

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