Posetic Tools in the Social Sciences: A Tutorial Exposition
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2.5 Mutual Ranking Probabilities
If x i π x j in poset π , then it is also x i λ x j in each linear extension λ of π ;
on the contrary, if x i || π x j , i.e. if the two elements are incomparable in π , then in
some linear extension λ we have x i λ x j and in some other linear extension ρ we
have x j ρ x i . The fraction p ij of linear extensions λ ∈ where x i λ x j is
called the mutual ranking probability (MRP) of x j over x i (De Loof 2009; De Loof
et al. 2006, 2008); informally, the MRP expresses the “degree of dominance” of x j
over x i . MRPs are usually arranged into the n × n mutual ranking probability matrix
M, whose entry ij is p ij . As it will be shown in subsequent paragraphs, M plays a
key role in practical applications.
2.6 Software Resources
Currently, there are two main software resources to perform statistical analysis
on partially ordered data. The PyHasse suite (Koppatz and Bruggemann 2017),
developed in Python, is available at https://pyhasse.org/. It provides a huge number
of modules and procedures for various statistical analyses. A system of functions for
poset manipulation and socio-economic analysis on partially ordered data in the R
environment is provided in the package parsec (Arcagni 2017). Some other useful
routines, mainly to compute mutual ranking probability matrices, are also available
in the package netrankr (Schoch 2017).
3 Posetic Tools in Socio-economics
In this section, we present some of the main posetic tools, for the analysis of ordinal
MISes and partially ordered data in socio-economics. We organize the exposition
around some reference topics, which are of main interest for social scientists. For
each tool, we give the main formulas, summarize its properties and provide a short
example of its use on real data.
3.1 Scoring and Ranking Partially Ordered Data
Perhaps the main problem in the study of multi-dimensional MISes and partially
ordered data is that of ranking statistical units, based on their score profiles or
on their “degree of dominance” with respect to other units. Usually, this requires
computing a non-negative score function s(·) (i.e. a function assigning to each
profile a non-negative real number) and then ranking units based on it. In order to
223
2.5 Mutual Ranking Probabilities
If x i π x j in poset π , then it is also x i λ x j in each linear extension λ of π ;
on the contrary, if x i || π x j , i.e. if the two elements are incomparable in π , then in
some linear extension λ we have x i λ x j and in some other linear extension ρ we
have x j ρ x i . The fraction p ij of linear extensions λ ∈ where x i λ x j is
called the mutual ranking probability (MRP) of x j over x i (De Loof 2009; De Loof
et al. 2006, 2008); informally, the MRP expresses the “degree of dominance” of x j
over x i . MRPs are usually arranged into the n × n mutual ranking probability matrix
M, whose entry ij is p ij . As it will be shown in subsequent paragraphs, M plays a
key role in practical applications.
2.6 Software Resources
Currently, there are two main software resources to perform statistical analysis
on partially ordered data. The PyHasse suite (Koppatz and Bruggemann 2017),
developed in Python, is available at https://pyhasse.org/. It provides a huge number
of modules and procedures for various statistical analyses. A system of functions for
poset manipulation and socio-economic analysis on partially ordered data in the R
environment is provided in the package parsec (Arcagni 2017). Some other useful
routines, mainly to compute mutual ranking probability matrices, are also available
in the package netrankr (Schoch 2017).
3 Posetic Tools in Socio-economics
In this section, we present some of the main posetic tools, for the analysis of ordinal
MISes and partially ordered data in socio-economics. We organize the exposition
around some reference topics, which are of main interest for social scientists. For
each tool, we give the main formulas, summarize its properties and provide a short
example of its use on real data.
3.1 Scoring and Ranking Partially Ordered Data
Perhaps the main problem in the study of multi-dimensional MISes and partially
ordered data is that of ranking statistical units, based on their score profiles or
on their “degree of dominance” with respect to other units. Usually, this requires
computing a non-negative score function s(·) (i.e. a function assigning to each
profile a non-negative real number) and then ranking units based on it. In order to