New tools for the construction of ranking and evaluation indicators in multidimensional systems of ordinal variables

The problem of ranking and evaluation in multidimensional ordinal datasets is one of the most important issue in applied statistics, particularly in the socio-economic field. Unfortunately, dealing with ordinal variables raises many conceptual and methodological issues, particularly when consistent and meaningful indicators are to be defined out of qualitative data. Methodological difficulties rise particularly when single ordinal indicators are to be aggregated into a composite indicator, to get unidimensional scores for comparing and ranking statistical units. Basically, it can be asserted that the issue of ranking and evaluation in an ordinal setting is still an open problem, since the statistical methodologies, applied in the common practice or proposed at theoretical level, are unsatisfactory in many respects. Motivated by these issues and by the relevance of the topic, in this paper, we introduce new tools for addressing the construction of indicators for ranking and evaluation purposes in an ordinal context, with the aim to overcome the main problems of the classical composite indicator approach. The proposed methodology draws upon Partial Order SET (POSET) theory, a branch of discrete mathematics providing concepts and tools, fitting very naturally the needs of ordinal data analysis. POSET theory provides useful technical tools for addressing the evaluation problem. But more important than this, it helps reformulating the ranking and evaluation problem in such a way that satisfactory solution to the issues outlined above can indeed be worked out, fully respecting the qualitative features of data.