Distributional Equivalence and Subcompositional Coherence in the Analysis of Compositional Data,Contingency Tables and Ratio-Scale Measurements |
| |
Authors: | Michael Greenacre Paul Lewi |
| |
Institution: | (1) Departament d’Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain;(2) Katholieke Universiteit Leuven and Vrije Universiteit Brussel, Brussel, Belgium |
| |
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 analyze the ratios of the data values. A common 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 and can be applied
to a wider class of methods. 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 modeling.
The weighted log-ratio methodology is used here to visualize frequency data in linguistics and chemical compositional data
in archeology.
The first author acknowledges research support from the Fundación BBVA in Madrid as well as partial support by the Spanish
Ministry of Education and Science, grant MEC-SEJ2006-14098. The constructive comments of the referees, who also brought additional
relevant literature to our attention, significantly improved our article. |
| |
Keywords: | Association models Biplot Correspondence analysis Log-ratio analysis Singular value decomposition Spectral mapping |
本文献已被 SpringerLink 等数据库收录! |
|