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Relationships Among Several Methods of Linearly Constrained Correspondence Analysis

Published online by Cambridge University Press:  01 January 2025

Yoshio Takane ,
Haruo Yanai  and
Shinichi Mayekawa
Yoshio Takane*
Affiliation:
McGill University
Haruo Yanai
Affiliation:
The National Center for University Entrance Examination
Shinichi Mayekawa
Affiliation:
The National Center for University Entrance Examination
*
Requests for reprints should be sent to Yoshio Takane, Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, Montreal, Quebec H3A IB1 CANADA.
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Abstract

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.

Keywords

canonical correlation analysisgeneralized singular value decomposition (GSVD)the method of additive scoringthe second type of quantification method (Q2)canonical correspondence analysis (CCA)ANOVA of categorical datacanonical analysis with linear constraints (CALC)zero average restrictionsKhatri's lemma
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 4 , December 1991 , pp. 667 - 684
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The work reported in this paper has been supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. We wish to thank Carolyn Anderson, Ulf Böckenholt, Henk Kiers, Shizuhiko Nishisato, Jim Ramsay, Tadashi Shibayama, Cajo ter Braak, and Peter van der Heijden for their helpful comments on earlier drafts of this paper.

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