Power Transformations in Correspondence Analysis

Abstract

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power the original data and perform a correspondence analysis - this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to the contingency ratios, that is the values in the table relative to expected values based on the marginals - this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

Keywords: Box-Cox transformation, chi-square distance, contingency ratio, correspondence analysis, log-ratio analysis, power transformation, ratio data, singular value decomposition, spectral map

JEL Classification: C19, C88