Compositionally relevant post hoc tests of compositional linear models

The geochemistry of surficial Earth materials, e.g. (weathered) rocks, soils, plants, peatlands, waterbodies or snow, is influenced by many processes and environmental parameters. Often, however, it is unknown which of the observed environmental variables, like soil pH, precipitation, slope, underlying bedrock type, to name but a few, have a significant influence on the geochemical pattern of the respective material. For this purpose, one can test hypotheses about the influence of variables on the (ilr or alr transformed) composition, e.g. by means of linear regression. These tests can either be rejected or accepted. If the result of the statistical analysis shows that the variable(s) tested could have an influence on the overall composition of the surficial material, the question immediately arises for geochemists as to which of the elements or element groups change in relation to this variable. In the case of non-compositional and univariate data, there are the classical post-hoc tests, e.g. tests according to Scheffé or Bonferroni, that allow to check which of certain interpretable subhypothesis are responsible for significant result. However, as far as the authors are aware, there are no such tests developed for multivariate situations and geochemically or compositionally relevant sub-hypotheses, such as those that consider sub-compositions or balances of specific elements.
The contribution provides a systematic compilation of compositionally and geochemically potentially relevant subhypotheses and corresponding post-hoc tests. A particular challenge is that the number of compositional hypotheses is much larger than the number of subhypotheses (all subcompositions and balances) than in a contrast’s situation (all pairs). A Bonferroni approach is therefore impractical. We thus extend Scheffé’s principle of post-hoc testing to the situation where any number of additional subhypotheses can be tested simultaneously without the need for additional p-value corrections. Geochemists in this system of post-hoc tests can then either test geochemically interpretable relevant sub-hypotheses or test the model for the least explanatory sub-hypotheses and interpret these results based on a hierarchy of implied hypotheses.
The application of the method will be demonstrated on a snow data set, which was collected for exploration purposes on a Co-Au deposit.