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Published in: Social Indicators Research 1/2019

11-12-2018

F-FOD: Fuzzy First Order Dominance Analysis and Populations Ranking Over Ordinal Multi-Indicator Systems

Authors: Marco Fattore, Alberto Arcagni

Published in: Social Indicators Research | Issue 1/2019

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Abstract

In this paper, we develop a new statistical procedure for the comparison of frequency distributions on systems of ordinal indicators, based on a multidimensional fuzzy extension of the first order dominance (FOD) criterion. The procedure, named fuzzy-first order dominance (F-FOD), employs concepts and tools from partially ordered set theory and from fuzzy relational calculus and is designed to overcome the main limitations of previously developed algorithms for FOD analysis. In particular, F-FOD produces full pairwise comparison matrices, allows for partial orderings and rankings of the statistical units to be derived from the input data, is computationally sufficiently light to be applied in most cases of practical interest and is freely available in the R package PARSEC. To illustrate its effectiveness, we also show F-FOD in action on two real datasets concerning health in Denmark and child well-being in the Democratic Republic of Congo.

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Appendix
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Footnotes
1
We stress, however, that the problem of information loss and dimensionality reduction on partially ordered domains is not confined to the study of ordinal MISes; in fact, the same issue occurs whenever evaluation and ranking involve multidimensional profiles built on systems of weakly interrelated variables, even of a cardinal type.
 
2
A partially ordered set is a set endowed with a reflexive, anti-symmetric and transitive binary relation Davey and Priestley (2002).
 
3
A down-set\(\delta\) of a poset \(\pi\) is a subset of \(\pi\) such that if \(\varvec{p}\in \delta\) and \(\varvec{q}\le \varvec{p}\), then \(\varvec{q}\in \delta\). Dually, an up-setu of a poset \(\pi\) is a subset of \(\pi\) such that if \(\varvec{p}\in u\) and \(\varvec{p}\le \varvec{q}\), then \(\varvec{q}\in u\). The down-set\(\downarrow \!\!\varvec{p}\) of an element \(\varvec{p}\) of \(\pi\) is the same as the set of elements equal to or smaller than \(\varvec{p}\): \(\downarrow \!\!\varvec{p}=\{x\in \pi : x\le \varvec{p}\}\); similarly, the up-set\(\uparrow \!\!\varvec{p}\) of \(\varvec{p}\) is the same as the subset of elements equal to or higher than \(\varvec{p}\): \(\uparrow \!\!\varvec{p}=\{x\in \pi : x\ge \varvec{p}\}\). In Arndt et al. (2012), down-sets are called comprehensive sets, but this is not standard posetic terminology.
 
4
By “ equivalent position” of two poset elements, we mean that the poset is invariant upon exchanging their labels.
 
5
The length of a chain is defined as the number of edges in it Davey and Priestley (2002).
 
6
Dealing with discrete variables, it is in general not possible to build rank intervals covering exactly the desired probability level.
 
7
The Bubley-Dyer algorithm is, to our knowledge, the most efficient algorithm for quasi-uniform sampling from the set of linear extensions of a poset. Based on it, an algorithm for quasi-uniform sampling of LLEs could be easily derived.
 
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Metadata
Title
F-FOD: Fuzzy First Order Dominance Analysis and Populations Ranking Over Ordinal Multi-Indicator Systems
Authors
Marco Fattore
Alberto Arcagni
Publication date
11-12-2018
Publisher
Springer Netherlands
Published in
Social Indicators Research / Issue 1/2019
Print ISSN: 0303-8300
Electronic ISSN: 1573-0921
DOI
https://doi.org/10.1007/s11205-018-2049-2

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