24
F. Maggino et al.
different values in the elementary indicators, or similar situations between them. 4
This can lead to misleading conclusions. One possible solution to the weakness of
the aggregative-compensative methods can be to synthesize not necessarily through
aggregation. This need has led the research to focus on developing alternative
methods, the non-aggregative approaches. They respect the ordinal nature of the
data and the process and trends of phenomena (not always linear but more frequently
monotonic) and avoid any aggregation among indicators. One of the most useful
references in this perspective is the Partial Order Theory (see for instance Davey
and Priestley 1990). Non-aggregative approaches are focused not on dimensions
but on profiles, which are combinations of ordinal scores, describing the «status»
of an individual. Profiles can be mathematically described and analyzed through
tools referring to that theory, in particular Partially Ordered Set (POSET). Through
these tools, information can be extracted directly from the relational structure of the
data, obtaining robust results, not based on binding hypotheses. This approach gives
an effective representation of data and their structure. 5 The application of POSET
methodologies lead to conclusions much more meaningful, robust and consistent
than those based upon traditional statistical tools. Moreover, focusing on the profiles
allows having a “synthesis” always representative of the effective combinations
among basic indicators. This avoids flattening the differences between different
combinations in a single numerical value and misinterpreting the results.
2 Overview About Concepts in Partial Order Theory
2.1 Two Basic Approaches
Partial order theory is a mathematical discipline which combines elements of Graph
theory and Combinatorics. In the broadest sense is Graph theory that branch of
Discrete Mathematics which studies relations. Within the context of the analysis of
indicators the relations are defined on the basis of profiles. Depending on the type
of data of the indicators two main lines of analysis approaches can be defined:
1. Ordinal data: The corresponding indicators may have finite and discrete values
of different degrees. Let Q(k) the set of values of the kth indicator, then
Q(k),
k = 1, . . . ,m, m the number of indicators, is the set of all possible value’s
combinations, i.e. of all possible profiles. The analysis of this set as described
4 For instance, in two papers on sustainable development and regional differences in Italy, Alaimo
and Maggino show how similar values in composite indicators assumed by different regions
can represent similar or even completely different combinations in basic indicators (Alaimo and
Maggino 2018, 2020).
5 In particular, the computations performed to assign numerical scores to the statistical units involve
only the ordinal features of data, avoiding any scaling procedure or any other transformation of the
kind.
F. Maggino et al.
different values in the elementary indicators, or similar situations between them. 4
This can lead to misleading conclusions. One possible solution to the weakness of
the aggregative-compensative methods can be to synthesize not necessarily through
aggregation. This need has led the research to focus on developing alternative
methods, the non-aggregative approaches. They respect the ordinal nature of the
data and the process and trends of phenomena (not always linear but more frequently
monotonic) and avoid any aggregation among indicators. One of the most useful
references in this perspective is the Partial Order Theory (see for instance Davey
and Priestley 1990). Non-aggregative approaches are focused not on dimensions
but on profiles, which are combinations of ordinal scores, describing the «status»
of an individual. Profiles can be mathematically described and analyzed through
tools referring to that theory, in particular Partially Ordered Set (POSET). Through
these tools, information can be extracted directly from the relational structure of the
data, obtaining robust results, not based on binding hypotheses. This approach gives
an effective representation of data and their structure. 5 The application of POSET
methodologies lead to conclusions much more meaningful, robust and consistent
than those based upon traditional statistical tools. Moreover, focusing on the profiles
allows having a “synthesis” always representative of the effective combinations
among basic indicators. This avoids flattening the differences between different
combinations in a single numerical value and misinterpreting the results.
2 Overview About Concepts in Partial Order Theory
2.1 Two Basic Approaches
Partial order theory is a mathematical discipline which combines elements of Graph
theory and Combinatorics. In the broadest sense is Graph theory that branch of
Discrete Mathematics which studies relations. Within the context of the analysis of
indicators the relations are defined on the basis of profiles. Depending on the type
of data of the indicators two main lines of analysis approaches can be defined:
1. Ordinal data: The corresponding indicators may have finite and discrete values
of different degrees. Let Q(k) the set of values of the kth indicator, then
Q(k),
k = 1, . . . ,m, m the number of indicators, is the set of all possible value’s
combinations, i.e. of all possible profiles. The analysis of this set as described
4 For instance, in two papers on sustainable development and regional differences in Italy, Alaimo
and Maggino show how similar values in composite indicators assumed by different regions
can represent similar or even completely different combinations in basic indicators (Alaimo and
Maggino 2018, 2020).
5 In particular, the computations performed to assign numerical scores to the statistical units involve
only the ordinal features of data, avoiding any scaling procedure or any other transformation of the
kind.