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Factor Analysis for Clustered Observations

Published online by Cambridge University Press:  01 January 2025

N. T. Longford  and
B. O. Muthén
N. T. Longford*
Affiliation:
Educational Testing Service
B. O. Muthén
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to N. T. Longford, 21-T, Educational Testing Service, Rosedale Rd., Princeton, NJ 08541.
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Abstract

Classical factor analysis assumes a random sample of vectors of observations. For clustered vectors of observations, such as data for students from colleges, or individuals within households, it may be necessary to consider different within-group and between-group factor structures. Such a two-level model for factor analysis is defined, and formulas for a scoring algorithm for estimation with this model are derived. A simple noniterative method based on a decomposition of the total sums of squares and crossproducts is discussed. This method provides a suitable starting solution for the iterative algorithm, but it is also a very good approximation to the maximum likelihood solution. Extensions for higher levels of nesting are indicated. With judicious application of quasi-Newton methods, the amount of computation involved in the scoring algorithm is moderate even for complex problems; in particular, no inversion of matrices with large dimensions is involved. The methods are illustrated on two examples.

Keywords

factor analysisFisher scoring algorithmmaximum likelihoodmultilevel analysis

Information

Type
Original Paper
Information
Psychometrika , Volume 57 , Issue 4 , December 1992 , pp. 581 - 597
Copyright
Copyright © 1992 The Psychometric Society

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Footnotes

Suggestions and corrections of three anonymous referees and of an Associate Editor are acknowledged. Discussions with Bob Jennrich on computational aspects were very helpful. Most of research leading to this paper was carried out while the first author was a visiting associate professor at the University of California, Los Angeles.

References

Aitkin, M., Longford, N. T. (1986). Statistical modeling issues in school effectiveness studies. Journal of the Royal Statistical Society, Series A, 149, 419–461.CrossRefGoogle Scholar
Cronbach, L. J. (1976). Research on classrooms and schools: Formulation of questions, design, and analysis. Unpublished manuscript, Stanford University, Stanford Evaluation Consortium, School of Education.Google Scholar
Crosswhite, F. J., Dossey, J. A., Swafford, J. O., McKnight, C. C., Cooney, T. J. (1985). Second International Mathematics Study: Summary report for the United States, Champaign, IL: Stipes.Google Scholar
de Leeuw, J., Kreft, I. (1986). Random coefficient models for multilevel analysis. Journal of Educational Statistics, 11, 57–85.CrossRefGoogle Scholar
Goldstein, H. (1986). Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika, 73, 43–56.CrossRefGoogle Scholar
Goldstein, H. (1987). Multilevel models in educational and social research, London, U.K.: C. Griffin & Co..Google Scholar
Goldstein, H., McDonald, R. P. (1988). A general model for the analysis of multilevel data. Psychometrika, 53, 455–467.CrossRefGoogle Scholar
Härnquist, K. (1978). Primary mental abilities of collective and individual levels. Journal of Educational Psychology, 70, 706–716.CrossRefGoogle Scholar
Harville, D. A. (1974). Bayesian inference for variance components using only error contrasts. Biometrika, 61, 383–385.CrossRefGoogle Scholar
Jamshidian, M., Jennrich, R. I. (1988). Conjugate gradient methods in confirmatory factor analysis (UCLA Statistics Series 8), Los Angeles: UCLA.Google Scholar
Jennrich, R. I., Schluchter, M. D. (1986). Unbalanced repeated-measures models with structured covariance matrices. Biometrics, 42, 805–820.CrossRefGoogle ScholarPubMed
Jöreskog, K. G. (1977). Factor analysis by least-squares and maximum-likelihood methods. In Enstein, K., Ralston, A., Wilf, H. S. (Eds.), Statistical methods for digital computers (pp. 125–153). New York: John Wiley & Sons.Google Scholar
Jöreskog, K. G., Sörbom, D. (1979). Advances in factor analysis and structural equation models, Cambridge, MA: Abt Books.Google Scholar
Lawley, D. N., Maxwell, A. E. (1971). Factor analysis as a statistical method 2nd ed.,, London: Butterworth & Co..Google Scholar
Lee, S. Y. (1990). Multilevel analysis of structural equation models. Biometrika, 772, 763–772.CrossRefGoogle Scholar
Lee, S. Y., Jennrich, R. I. (1979). A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika, 43, 99–113.CrossRefGoogle Scholar
Longford, N. T. (1987). A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects. Biometrika, 74, 817–827.CrossRefGoogle Scholar
Longford, N. T. (1988). VARCL. Software for variance component analysis of data with hierarchically nested random effects (maximum likelihood), Princeton, NJ: Educational Testing Service.Google Scholar
Longford, N. T. (1990). Multivariate variance component analysis: An application in test development. Journal of Educational Statistics, 15, 91–112.CrossRefGoogle Scholar
Luenberger, D. G. (1984). Linear and nonlinear programming 2nd ed.,, Reading, MA: Addison-Wesley.Google Scholar
Mason, W. M., Wong, G. Y., Entwisle, B. (1984). Contextual analysis through the multilevel linear model. In Leinhardt, S. (Eds.), Sociological methodology (1983–84) (pp. 72–103). San Francisco: Jossey Bass.Google Scholar
McDonald, R. P., Goldstein, H. (1989). Balanced versus unbalanced designs for linear structural relations in two-level data. British Journal Mathematical and Statistical Psychology, 42, 215–232.CrossRefGoogle Scholar
Muthén, B. O. (1989). Latent variable modeling in heterogeneous populations. Psychometrika, 54, 557–585.CrossRefGoogle Scholar
Muthén, B. O. (1991). Multilevel factor analysis of class and student achievement components. Journal of Educational Measurement, 28, 338–354.CrossRefGoogle Scholar
Muthén, B. O. (in press). Mean and covariance structure analysis of hierarchical data. Journal of Educational Statistics.Google Scholar
Muthén, B. O., Satorra, A. (1989). Multilevel aspects of varying parameters in structural models. In Bock, R. D. (Eds.), Multilevel analysis of educational data (pp. 87–99). New York: Academic Press.Google Scholar
Patterson, H. D., Thompson, R. (1971). Recovery of interblock information when block sizes are unequal. Biometrika, 58, 545–554.CrossRefGoogle Scholar
Raudenbush, S. W., Bryk, A. S. (1985). A hierarchical model for studying school effects. Sociology of Education, 59, 1–17.CrossRefGoogle Scholar