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Ratio maps and correspondence analysis

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Abstract

We compare two methods for visualising contingency tables and develop a method called the ratio map which combines the good properties of both. The first is a biplot based on the logratio approach to compositional data analysis. This approach is founded on the principle of subcompositional coherence, which assures that results are invariant to considering subsets of the composition. The second approach, correspondence analysis, is based on the chi-square approach to contingency table analysis. A cornerstone of correspondence analysis is the principle of distributional equivalence, which assures invariance in the results when rows or columns with identical conditional proportions are merged. Both methods may be described as singular value decompositions of appropriately transformed matrices. Correspondence analysis includes a weighting of the rows and columns proportional to the margins of the table. If this idea of row and column weights is introduced into the logratio biplot, we obtain a method which obeys both principles of subcompositional coherence and distributional equivalence.

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Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 598.

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Date of creation: Jan 2002
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Handle: RePEc:upf:upfgen:598

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

Related research

Keywords: Biplot; compositional data; contingency tables; distributional equivalence; logratio transformation; singular value decomposition; subcompositional coherence;

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  1. 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.
  2. Michael Greenacre, 2008. "Correspondence analysis of raw data," Economics Working Papers 1112, Department of Economics and Business, Universitat Pompeu Fabra, revised Jul 2009.
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