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

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.

Appendices

Appendix 1

In this Appendix, we provide the proof of the following proposition, which plays a fundamental role in the development of F-FOD (see Sect. 2):

Proposition

Let$\pi$beafiniteposet; then$F\le _{FOD}G$on$\pi$ifandonly if, foreachlinearextension$\ell$of$\pi$, $F\le _{FOD}G$on$\ell$.

Proof

(Necessity) Let $F\le _{FOD}G$ in $\pi$ and let $\ell$ be a linear extension of $\pi$. Consider an element $\varvec{q}$ and its down-set $\delta _q$ in $\ell$; $\delta _q$ is the union of two subsets, namely the subset A of elements lower than or equal to $\varvec{q}$ in $\pi$ (i.e. the down-set $\downarrow \!\varvec{q}$ of $\varvec{q}$ in $\pi$) and the subset B of elements of $\delta _q$ which are incomparable with $\varvec{q}$ in $\pi$ (which is possibly empty). As a consequence, $\delta _q$, as a set, is the same as the down-set C of $\pi$ generated by $B\cup \{\varvec{q}\}$ (i.e. the set of elements $\varvec{p}$ of $\pi$ such that $\varvec{p}\le \varvec{y}$, for at least one $\varvec{y}\in B\cup \{\varvec{q}\}$). Since $F\le _{FOD}G$ in $\pi$, it holds:

$$\begin{aligned} \sum _{\varvec{p}\in \delta _q}G(\varvec{p})=\sum _{\varvec{p}\in C} G(\varvec{p})\le \sum _{\varvec{p}\in C} F(\varvec{p})=\sum _{\varvec{p}\in \delta _q}F(\varvec{p}). \end{aligned}$$

(6)

Since $\varvec{q}$ is arbitrary, and $\ell$ is a linear order, we get $F\le _{FOD}G$ in $\ell$. $\square$

(Sufficiency) Let $F\le _{FOD}G$ for each linear extension $\ell$ of $\pi$; let $\delta$ be a down-set of $\pi$ and let n be the cardinality of $\delta$. There exists at least one linear extension $\ell$ such that the elements of $\delta$ are the first (from below) n elements $\varvec{q}_1,\ldots \varvec{q}_n$ of $\ell$. Since $F\le _{FOD}G$ in $\ell$, it holds

$$\begin{aligned} \sum _{\varvec{p}\in \delta }G(\varvec{p})=\sum _{i=1}^n G(\varvec{q}_i)\le \sum _{i=1}^n F(\varvec{q}_i)=\sum _{\varvec{p}\in \delta }F(\varvec{p}). \end{aligned}$$

(7)

Since this holds for any down-set $\delta$, it is proved that $F\le _{FOD} G$ in $\pi$.

Appendix 2

In this appendix, we report the R code used to elaborate the example pertaining to child well-being in the Democratic Republic of Congo (Sect. 4). Readers may easily replicate the computations and use the following scripts for further applications.

• 1. Definition of the set of binary profiles.

  

• 2. Data reported in Table 9 are put into the data.frameCONGO (notice that the profiles in CONGO must be listed in the same order as in prf).

• 3. Application of the F-FOD procedure.

  

• 4. Function FFOD returns several results (e.g. the matrix $\Delta$ of dominance degrees and the cover matrices of the $\alpha$-cut posets) and feeds the function rank_stability, which computes in each $\alpha$-cut poset the average rank of each profile and the rank intervals with the chosen least coverage probability.

  

• 5. The graphs shown in Fig. 9 are finally produced by:

  

Running the above computations took 5.46 secs on a pc equipped with Intel(R) Core(TM) i7-3632QM CPU @ 2.20GHz, 8GB ram, Windows 10, R version 3.4.3 and R Studio 1.1.382.