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On Browne's Solution for Oblique Procrustes Rotation

Published online by Cambridge University Press:  01 January 2025

Elliot M. Cramer
Elliot M. Cramer*
Affiliation:
University of North Carolina
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Abstract

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of Φ — FT with Diag (T'T)= I. Browne's solution involves the eigenvectors and values of F'F and leads to an iterative solution.

This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.

Keywords

Public PolicyFactor StructureEigenvalue ProblemStatistical TheoryComputational Procedure
Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 2 , June 1974 , pp. 159 - 163
Copyright
Copyright © 1974 The Psychometric Society

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Footnotes

1

Part of this work was presented at the April 1969 meetings of the Psychometric Society. The anthor is indebted to Dr. Ledyard Tucker for some helpful discussions. This work was supported in part by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health, and Grant No. GM-12868 from the Institute of General Medical Sciences, Public Health Service.

References

Browne, M. W. On oblique procrustes rotation. Psychometrika, 1967, 32, 125–132.CrossRefGoogle ScholarPubMed
Mosier, C. I. Determining a simple structure when loadings for certain tests are known. Psychometrika, 1939, 4, 149–162.CrossRefGoogle Scholar