Mixing compositions and other scales
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Theory of compositional data analysis is often focused on the composition only. However in practical applications we often treat a composition together with covariables
with some other scale. This contribution systematically gathers and develop statistical tools for this situation. For instance, for the graphical display of the dependence
of a composition with a categorical variable, a colored set of ternary diagrams might
be a good idea for a first look at the data, but it will fast hide important aspects if
the composition has many parts, or it takes extreme values. On the other hand colored scatterplots of ilr components could not be very instructive for the analyst, if the
conventional, black-box ilr is used.
Thinking on terms of the Euclidean structure of the simplex, we suggest to set up
appropriate projections, which on one side show the compositional geometry and on the
other side are still comprehensible by a non-expert analyst, readable for all locations and
scales of the data. This is e.g. done by defining special balance displays with carefully-
selected axes. Following this idea, we need to systematically ask how to display, explore,
describe, and test the relation to complementary or explanatory data of categorical, real,
ratio or again compositional scales.
This contribution shows that it is sufficient to use some basic concepts and very few
advanced tools from multivariate statistics (principal covariances, multivariate linear
models, trellis or parallel plots, etc.) to build appropriate procedures for all these combinations of scales. This has some fundamental implications in their software implementation, and how might they be taught to analysts not already experts in multivariate
analysis
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