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Some Results on Proper Eigenvalues and Eigenvectors with Applications to Scaling

Published online by Cambridge University Press:  01 January 2025

Roderick P. McDonald,
Yukihiko Torii and
Shizuhiko Nishisato
Roderick P. McDonald*
Affiliation:
The Ontario Institute for Studies in Education
Yukihiko Torii
Affiliation:
Ontario Ministry of Community and Social Services
Shizuhiko Nishisato
Affiliation:
The Ontario Institute for Studies in Education
*
Requests for offprints should be sent to R. P. McDonald, Department of Measurement, Evaluation and Computer Applications, The Ontario Institute for Studies in Education, 252 Bloor St. W., Toronto, Ontario, M5S 1V6, Canada.
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Abstract

Conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations. It is shown that two classes of optimal weighting problems, based respectively on the grouping of variables and on the grouping of observations, satisfy these conditions. The classical treatment of optimal scaling of forced-choice multicategory data is extended for these cases. It is shown that previously suggested methods based on reparameterization will work only under very special conditions.

Keywords

eigenvalueseigenvectorsoptimal scaling

Information

Type
Original Paper
Information
Psychometrika , Volume 44 , Issue 2 , June 1979, pp. 211 - 227
Copyright
Copyright © 1979 The Psychometric Society

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References

Reference Notes

de Leeuw, J. HOMALS, 1976, New Jersey: Murray Hill.Google Scholar
Nishisato, S.Optimal scaling as applied to different forms of data. Department of Measurement and Evaluation, The Ontario Institute for Studies in Education, 1976.Google Scholar
Nishisato, S.Analysis of categorical data: Dual scaling and its applications, manuscript in preparation, 1978.Google Scholar

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