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Section 6 specializes the methodology to the fundamental case of binary variables. Section 7 concludes. The aim of this chapter is primarily methodological, leaving to future works the systematic application of the evaluation procedure to real data. Nevertheless, for sake of clarity, all the basic concepts and the key ideas behind the methodology are introduced by examples, all of which pertain to material deprivation and multidimensional poverty. 2 The Composite Indicator Approach and Its Critical Issues Addressing the complexity of socio-economic phenomena for evaluation aims is a complex task, often requiring the definition of large systems of indicators.
Consider two variables v1, v2 each recorded on a three-grade scale: 0, 1 and 2. The corresponding profile poset is depicted in Fig. 7a. Suppose that, based on some exogenous considerations, profile 12 is regarded as more deprived than profile 21. e. 21 ᭠ 12). Unfortunately, P È (21, 12) is not a poset, since it does not satisfy the transitivity axiom. In fact, since 20 ᭠ 21 in P, from 21 ᭠ 12 it follows that also 20 ᭠ 12 must be added to P, so as to restore transitivity and to get a partial order P which is an extension of P.
We address the comparison of profiles to benchmarks in a multidimensional setting by using tools and results from partially ordered set theory (poset theory, for short). Indeed, through poset tools, sequences of scores can be assessed without involving any aggregation of the underlying variables since the evaluation is performed by exploiting the relational structure of the data, which involves solely the partial ordering of the profiles. The remainder of this chapter is organized as follows. Section 2 gives a brief account of the composite indicator approach, highlighting its main criticalities, particularly in the ordinal case.