TCA and TLRA: A comparison on contingency tables and compositional data

Abstract: There are two popular general approaches for the analysis and visualization of a contingency table and a compositional data set: Correspondence analysis (CA) and log ratio analysis (LRA). LRA includes two independently well developed methods: association models and compositional data analysis. The application of either CA or LRA to a contingency table or to compositional data set includes a preprocessing centering step. In CA the centering step is multiplicative, while in LRA it is log bi-additive. A preprocessed matrix is double-centered, so it is a residuel matrix; which implies that it affects the final results of the analysis. This paper introduces a novel index named the intrinsic measure of the quality of the signs of the residuals (QSR) for the choice of the preprocessing, and consequently of the method. The criterion is based on taxicab singular value decomposition (TSVD) on which the package TaxicabCA in R is developed. We present a minimal R script that can be executed to obtain the numerical results and the maps in this paper. Three relatively small sized data sets available freely on the web are used as examples.