Distributional Equivalence and Subcompositional Coherence in the Analysis of Contingency Tables, Ratio-Scale Measurements and Compositional Data

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Distributional Equivalence and Subcompositional Coherence in the Analysis of Contingency Tables, Ratio-Scale Measurements and Compositional Data

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Michael Greenacre ()
Paul Lewi

Abstract

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to “spectral mapping”, a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

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Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 908.

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Date of creation: Dec 2005
Date of revision: Aug 2007
Handle: RePEc:upf:upfgen:908

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Find related papers by JEL classification:
C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Other
C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

This paper has been announced in the following NEP Reports:

NEP-ALL-2005-12-20 (All new papers)

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K. Ruben Gabriel, 2002. "Goodness of fit of biplots and correspondence analysis," Biometrika, Oxford University Press for Biometrika Trust, vol. 89(2), pages 423-436, June.
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. [Downloadable!] (restricted)

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