Three-way principal balance analysis: algorithm and interpretation

Compositional Data Analysis can be useful for unveiling relative variability patterns among variables describing the parts of a phenomenon. Compositions are often represented as orthonormal balances associated with a sequential binary partition (SBP). Principal bal- ances analysis (PBA) is a tool used to find a meaningful SBP by subsequently maximizing explained variability. The exact estimation of PBA is prohibitive for large datasets; therefore, algorithms providing an acceptable approximation are used instead. For compositional data of third-order, such exploratory search must account for third-mode variability. To this end, this work introduces a three-way adaptation of PBA in which estimation is carried out by Tucker3. A study on the composition of academic recruitment fields by Italian macro-region and gender/role is carried out to illustrate the merits of this procedure.