Logratio approach to statistical analysis of 2×2 compositional tables

Abstract

Compositional tables represent a continuous counterpart to well-known contingency tables. Their cells contain quantitatively expressed relative contributions of a whole, carrying exclusively relative information and are popularly represented in proportions or percentages. The resulting factors, corresponding to rows and columns of the table, can be inspected similarly as with contingency tables, e.g. for their mutual independent behaviour. The nature of compositional tables requires a specific geometrical treatment, represented by the Aitchison geometry on the simplex. The properties of the Aitchison geometry allow a decomposition of the original table into its independent and interactive parts. Moreover, the specific case of 2×2 compositional tables allows the construction of easily interpretable orthonormal coordinates (resulting from the isometric logratio transformation) for the original table and its decompositions. Consequently, for a sample of compositional tables both explorative statistical analysis like graphical inspection of the independent and interactive parts or any statistical inference (odds-ratio-like testing of independence) can be performed. Theoretical advancements of the presented approach are demonstrated using two economic applications.

Keywords:

• contingency tables
• compositional data
• Aitchison geometry on the simplex
• isometric logratio transformation
• balances

Acknowledgements

The authors gratefully acknowledge the support of the Operational Program Education for Competitiveness – European Social Fund (project CZ.1.07/2.3.00/20.0170 of the Ministry of Education, Youth and Sports of the Czech Republic).