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Collins(2002, 2011) raised a number of issues with regards to correspondenceanalysis (CA), such as: qualitative information in a CA map versus quantitativeinformation in the relevant contingency table; the interpretation of a CA mapis difficult and its relation with the \% of inertia (variance) explained. Wetackle these issues by considering CA and taxicab CA (TCA) as a stepwiseHotelling/Tucker decomposition of the cross-covariance matrix of the row andcolumn categories into four quadrants. The contents of this essay are: First,we review the notion of quality/quantity in multidimensional data analysis asdiscussed by Benz\'{e}cri, who based his reflections on Aristotle. Second, weshow the importance of unravelling the interrelated concepts ofdependence/heterogeneity structure in a contingency table; and to picture themtwo maps are needed. Third, we distinguish between intrinsic and extrinsicquality of a principal dimension; the intrinsic quality is based on the signsof the residuals in the four quadrants, hence to the interpretability.Furthermore, we provide quantifications of the intrinsic quality and use themto uncover structure in particular in sparse contingency tables. Finally, weemphasize the importance of looking at the residual cross-covariance values ateach iteration.

Quantification of intrinsic quality of a principal dimension in correspondence analysis and taxicab correspondence analysis