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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
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.

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