  Collins(2002, 2011) raised a number of issues with regards to correspondence
analysis (CA), such as: qualitative information in a CA map versus quantitative
information in the relevant contingency table; the interpretation of a CA map
is difficult and its relation with the \% of inertia (variance) explained. We
tackle these issues by considering CA and taxicab CA (TCA) as a stepwise
Hotelling/Tucker decomposition of the cross-covariance matrix of the row and
column categories into four quadrants. The contents of this essay are: First,
we review the notion of quality/quantity in multidimensional data analysis as
discussed by Benz\'{e}cri, who based his reflections on Aristotle. Second, we
show the importance of unravelling the interrelated concepts of
dependence/heterogeneity structure in a contingency table; and to picture them
two maps are needed. Third, we distinguish between intrinsic and extrinsic
quality of a principal dimension; the intrinsic quality is based on the signs
of the residuals in the four quadrants, hence to the interpretability.
Furthermore, we provide quantifications of the intrinsic quality and use them
to uncover structure in particular in sparse contingency tables. Finally, we
emphasize the importance of looking at the residual cross-covariance values at
each iteration.
