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Contribution biplots

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Abstract

In order to interpret the biplot it is necessary to know which points – usually variables – are the ones that are important contributors to the solution, and this information is available separately as part of the biplot’s numerical results. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic directly into the graphical display, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses such as correspondence analysis, principal component analysis, log-ratio analysis and the graphical results of a discriminant analysis/MANOVA, in fact to any method based on the singular-value decomposition. In the contribution biplot one set of points, usually the rows of the data matrix, optimally represent the spatial positions of the cases or sample units, according to some distance measure that usually incorporates some form of standardization unless all data are comparable in scale. The other set of points, usually the columns, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that usually only one common scale for row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, this version of the biplot also solves the problem in correspondence analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution.

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Bibliographic Info

Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 1162.

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Date of creation: Jul 2009
Date of revision: Jan 2011
Handle: RePEc:upf:upfgen:1162

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Web page: http://www.econ.upf.edu/

Related research

Keywords: biplot; contributions; correspondence analysis; discriminant analysis; log-ratio analysis; MANOVA; principal component analysis; scaling; singular value decomposition; weighting.;

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  1. Michael Greenacre, 2008. "Correspondence analysis of raw data," Economics Working Papers 1112, Department of Economics and Business, Universitat Pompeu Fabra, revised Jul 2009.
  2. John Aitchison & Michael Greenacre, 2002. "Biplots of compositional data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 375-392.
  3. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer, vol. 1(3), pages 211-218, September.
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Cited by:
  1. Michael Greenacre, 2011. "The contributions of rare objects in correspondence analysis," Economics Working Papers 1278, Department of Economics and Business, Universitat Pompeu Fabra.
  2. Michael Greenacre, 2012. "Fuzzy Coding in Constrained Ordinations," Working Papers 640, Barcelona Graduate School of Economics.

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