Partially Ordered Sets in Socio-Economic Data Analysis

A partially ordered set (or a poset, for short) is a set endowed with a partial order relation, i.e., with a reflexive, anti-symmetric, and transitive binary relation. As mathematical objects, posets have been intensively studied in the last century, coming to play essential roles in pure mathematics, logic, and theoretical computer science. More recently, they have been increasingly employed in data analysis, multi-criteria decision-making, and social sciences, particularly for building synthetic indicators and extracting rankings from multidimensional systems of ordinal data. Posets naturally represent systems and phenomena where some elements can be compared and ordered, while others cannot be and are then incomparable. This makes them a powerful data structure to describe collections of units assessed against multidimensional variable systems, preserving the nuanced and multi-faceted nature of the underlying domains. Moreover, poset theory collects the proper mathematical tools to treat ordinal data, fully respecting their non-numerical nature, and to extract information out of order relations, providing the proper setting for the statistical analysis of multidimensional ordinal data. Currently, their use is expanding both to solve open methodological issues in ordinal data analysis and to address evaluation problems in socio-economic sciences, from multidimensional poverty, well-being, or quality-of-life assessment to the measurement of financial literacy, from the construction of knowledge spaces in mathematical psychology and education theory to the measurement of multidimensional ordinal inequality/polarization.