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Alternative Measures of Fit for the Schönemann-Carroll Matrix Fitting Algorithm

Published online by Cambridge University Press:  01 January 2025

James C. Lingoes  and
Peter H. Schönemann
James C. Lingoes
Affiliation:
The University of Michigan
Peter H. Schönemann
Affiliation:
Purdue University
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Abstract

In connection with a least-squares solution for fitting one matrix, A, to another, B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure, e, and a refined similarity measure, es, which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix, B, e.g., e(A, kB) ≠ e(A, B) for k ≠ 1. Therefore, both measures are useless for answering questions of the type: “Does A fit B better than A fits C?”. In this note two new measures of fit are suggested which do not depend upon the norms of A and B, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.

Type
Original Paper
Information
Psychometrika , Volume 39 , Issue 4 , December 1974 , pp. 423 - 427
Copyright
Copyright © 1974 The Psychometric Society

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Footnotes

*

This research in nonmetric techniques is supported in part by a grant from the National Science Foundation (GS-2850) to the University of Michigan.

References

Lingoes, J. C. The Guttman-Lingoes Nonmetric Program Series, 1973, Ann Arbor, Michigan: Mathesis Press.Google Scholar
Schönemann, P. H. and Carroll, R. M. Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika, 1970, 35, 245–255.CrossRefGoogle Scholar