Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

Michael Greenacre and Paul Lewi

Economics Working Papers from Department of Economics and Business, Universitat Pompeu Fabra

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.

Keywords: Association models; biplot; compositional data; contingency tables; correspondence analysis; distributional equivalence; log-ration transformation; ratio-scale data; singular value decomposition (search for similar items in EconPapers)
JEL-codes: C19 C88 (search for similar items in EconPapers)
Date: 2005-12, Revised 2007-08
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