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Relationships among several methods of linearly constrained correspondence analysis

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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.

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Authors and Affiliations

  1. The National Center for University Entrance Examination, Japan

    Haruo Yanai & Shinichi Mayekawa

  2. Department of Psychology, McGill University, 1205 Dr. Penfield Avenue, H3A 1B1, Montreal, Quebec, Canada

    Yoshio Takane

Authors
  1. Yoshio Takane
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  2. Haruo Yanai
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  3. Shinichi Mayekawa
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Additional information

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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Takane, Y., Yanai, H. & Mayekawa, S. Relationships among several methods of linearly constrained correspondence analysis. Psychometrika 56, 667–684 (1991). https://doi.org/10.1007/BF02294498

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