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Goodness of fit of biplots and correspondence analysis

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  • K. Ruben Gabriel

Abstract

The present paper examines proportional goodness of fit to variables recorded on individuals, the variances and covariances of the variables, and the form and distances between individuals. No single plot displays all three optimally in the sense of least squares. However, even aspects which are non-optimally fitted by biplots and Benzecri plots often closely preserve the optimal fit. This is shown by means of a preservation-of-fit function which depends on the type of display and on the ratio of the second to the first singular value of the data matrix. This function is never below 0·5, so at least half the fit is always preserved, and it is close to 1 unless the ratio of the singular values is small. That explains the frequently observed similarity of the various biplots and the Benzecri plot and the fact that they usually lead to the same conclusions. It follows that in many applications it is reasonable to use either the symmetric biplot or the Benzecri plot or a compromise maximin preservation plot, and that the difference between these three is usually unimportant. Copyright Biometrika Trust 2002, Oxford University Press.

Suggested Citation

  • K. Ruben Gabriel, 2002. "Goodness of fit of biplots and correspondence analysis," Biometrika, Biometrika Trust, vol. 89(2), pages 423-436, June.
    • Handle: RePEc:oup:biomet:v:89:y:2002:i:2:p:423-436
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    Cited by:

    1. Rozkrut Dominik, 2014. "Measuring Eco-Innovation: Towards Better Policies to Support Green Growth," Folia Oeconomica Stetinensia, Sciendo, vol. 14(1), pages 1-12, June.
    2. Jan Graffelman, 2005. "Enriched biplots for canonical correlation analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(2), pages 173-188.
    3. Michael Greenacre & Paul Lewi, 2005. "Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data," Economics Working Papers 908, Department of Economics and Business, Universitat Pompeu Fabra, revised Aug 2007.
    4. Michael Greenacre & Paul Lewi, 2009. "Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data, Contingency Tables and Ratio-Scale Measurements," Journal of Classification, Springer;The Classification Society, vol. 26(1), pages 29-54, April.
    5. Ulrich Kohler & Magdalena Luniak, 2005. "Data inspection using biplots," Stata Journal, StataCorp LP, vol. 5(2), pages 208-233, June.
    6. Michael Greenacre, 2006. "Tying up the loose ends in simple correspondence analysis," Economics Working Papers 940, Department of Economics and Business, Universitat Pompeu Fabra.

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