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Social Indicators Research

, Volume 136, Issue 3, pp 1053–1070 | Cite as

A Reduced Posetic Approach to the Measurement of Multidimensional Ordinal Deprivation

  • Marco FattoreEmail author
  • Alberto Arcagni
Article

Abstract

In this paper, we discuss the existence of particular systems of generators for posets associated to multidimensional systems of ordinal indicators and derive a reduced posetic procedure for the measurement of multidimensional ordinal deprivation. The proposal is motivated by the need to lessen the computational complexity of the original posetic procedure described in Fattore (Soc Indic Res 128(2):835–858, 2015), so as to make it applicable to larger multi-indicator systems, particularly to those comprising many variables scored on “short” scales, as typical in deprivation studies. The reduced procedure computes identification and severity functions based only on so-called lexicographic linear extensions. These are a particular generating system for the basic achievement poset, naturally associated to rankings of deprivation attributes. After motivating this choice, both from an interpretative and a computational point of view, the paper provides some simulated examples, comparing the reduced and the non-reduced procedures.

Keywords

Multidimensional deprivation Fuzzy poverty Partial order theory Hasse diagram 

1 Introduction

In some recent papers (Fattore et al. 2011, 2012; Fattore and Maggino 2014, 2015; Fattore 2016), the use of Partial Order Theory in the measurement of multidimensional deprivation has been proposed, as an alternative to more classical composite indicators methodologies or counting approaches (Alkire and Foster 2011a, b; Cerioli and Zani 1990). In principle, the use of poset theory solves most of the conceptual problems that other procedures have, when treating ordinal data. Poset theory is in fact designed to deal with order relations and provides the right tools for computing over multidimensional systems of ordinal attributes. The resulting measurement procedure is completely consistent with the nature of the data, still fitting the logic thread of classical poverty assessment processes. No scaling of ordinal attributes into numerical scores and no introduction of numerical weights to account for attribute importance are necessary, in order to get synthetic, yet non-composite, deprivation indicators. Nevertheless, the evaluation procedure has its main drawback in computational complexity. The computation of the evaluation functions draws upon the extraction of linear extensions from the achievement poset (see next section for a formal definition). In general, the number of these extensions grows extremely fast with the cardinality of the poset, so that in real multi-indicator systems exact computations are usually impossible. The problem is partly circumvented by using sampling procedures and computing evaluation functions on a random subset of linear extensions. This extends the range of applicability of the posetic procedure, but not enough to fit the needs of social scientists completely. In particular, a more usable posetic procedure is needed to deal with multi-indicator systems comprising many attributes, scored on scales with few degrees (e.g. two or four). This is, in fact, typical of modern deprivation or well-being studies, which are based on data extracted from international surveys, like EU-Silc. Moving from these considerations, in the following, we show that the basic achievement poset can be reconstructed by particular families of linear extensions, from which different evaluation functions can be built. These functions are of interest, both from a computational and an interpretative point of view. In fact, they require less computations to be worked out (with respect to the evaluation functions originally proposed in Fattore (2016)) and capture different aspects of multidimensional deprivation. From these evaluation functions, we then derive a reduced evaluation procedure, useful to extend the posetic approach to larger systems of ordinal attributes.

The paper is organized as follows. Section 2 provides an outline of the posetic approach to the evaluation of multidimensional ordinal deprivation. Section 3 discusses the computational issues to face, when applying the procedure to real posets. Section 4 introduces the alternative generating systems for the basic achievement poset and the related evaluation functions. Section 5 presents the reduced evaluation procedure. Section 6 compares it to the non-reduced procedure, on some simple posets. Section 7 concludes. A final appendix reports the proof of a technical result, needed in the text. The paper is rather technical, but it is to be considered as a development and a completion of Fattore (2016), where the foundations of the posetic methodology have been illustrated. Together, the papers provide a valuable and effective setting for the study of multidimensional deprivation.

2 The Partial Order Approach to Deprivation Measurement

In this section, we give a sketch of the posetic approach proposed in (Fattore 2016), where all the details are worked out, and highlight some of its criticalities, so as to motivate the reduced procedure. We limit ourselves to the essentials; elements of poset theory and other technical details can be found in the original paper and in Neggers and Kim (1998), Schröder. (2002).

2.1 Building the Basic Achievement Poset

Let \(v_1^{},\ldots ,v_k^{}\) be k ordinal attributes, possibly scored on scales with different number of degrees \(m_1^{},\ldots ,m_k^{}\). The set P of all the achievement profiles, i.e. of k-dimensional sequences of scores on \(v_1^{},\ldots ,v_k^{}\), partially ordered according to the product order \(\trianglelefteq\) (Neggers and Kim 1998; Schröder. 2002) of the linear orders corresponding to the attributes, is called the basic achievement poset (denoted by \(\varPi =(P, \trianglelefteq )\)). In practice, two profiles \(\varvec{p}=(p_{1}^{},\ldots ,p_{k}^{})\), \(\varvec{q}=(q_{1}^{},\ldots ,q_{k}^{})\) are ordered in \(\varPi\) (i.e. \(\varvec{p}\,\trianglelefteq\, \varvec{q}\)) if and only if \(p_{i}^{}\le q_{i}^{}\) (\(i=1,\ldots ,k\)). Notice that the achievement poset may comprise profiles not realized in the data. In fact, it represents the “evaluation space” and not the space of observed profiles.

2.2 Introducing Attribute Relevance

The basic achievement poset is just one possible way to partially order achievement profiles. In a sense, it is the weakest way to order them, respecting the natural concept that if a statistical unit improves one of its achievement without worsening any of the others, his/her achievement profile strictly improves. Attributes are treated on the same ground and no one of them is considered as more relevant than the others (from the point of view of its impact on the deprivation status of a statistical unit). Assuming some attributes as more important than others imposes a different partial order on the profile set. The resulting achievement poset, denoted by \(\varPi ^{ext}_{}\), is necessarily an extension of \(\varPi\), i.e. it is obtained from \(\varPi\) in an “ordering preserving way”, turning some incomparabilities into comparabilities (since, also in \(\varPi ^{ext}\), the natural ordering between profiles cannot be violated). Different ways to turn the exogenous information on the relevance of attributes into an extension of \(\varPi\) may be considered. For our purposes, the fundamental one comes from linking univocally any partial order on the set of attributes to an extension of the basic achievement poset, as described in (Fattore 2016). The extended achievement poset may then be considered as the basic one “made consistent” to the set of socio-economic values expressed assigning a relevance pattern (partial order) to the set of attributes.

2.3 Setting the Threshold

The achievement poset (basic or extended) conveys no information on deprivation, unless a deprivation threshold \(\tau\) is set in it. The threshold must be set by experts or through other exogenous processes, but from a mathematical point of view it must be chosen as an antichain (i.e. a subset of mutually incomparable achievement profiles), whose elements represent profiles “on the edge” of deprivation. Once the achievement poset is endowed with a deprivation threshold, it gets a socio-economic meaning and actually becomes the evaluation space of deprivation.

2.4 Computing the Evaluation Functions

The heart of the evaluation procedure is the computation of the identification function, assigning to each profile in the achievement poset its degree of deprivation. The identification score of a profile \(\varvec{p}\) is to be interpreted, in a fuzzy spirit, as the membership degree of \(\varvec{p}\) to deprivation, according to the threshold selected. It assumes values in [0, 1], where 0 correspondes to “no deprivation at all” and 1 to “complete deprivation”. The identification score of completely or partially deprived profiles can be complemented with a severity score, measuring the “depth” of deprivation. Severity can be measured both in absolute and relative terms, i.e. comparing the absolute scores to that of the most deprived profile, namely the bottom of the achievement poset (details on the formulas and the interpretation of the evaluation functions are given in Fattore (2016)). The identification and severity scores of a profile are finally assigned to each statistical unit sharing that profile.

2.5 Computing Overall Deprivation Indicators

Once each statistical unit has been assessed in terms of identification and severity, score distributions and overall indicators can be computed, in the spirit of the Head Count Ratio or Poverty Gap, at population or subpopulation level.

The posetic procedure schematically sketched above provides a “completely ordinal”way to the evaluation of ordinal deprivation (or well-being, or quality-of-life or the like). It represents a step forward over composite indicators and counting approaches; in fact, it extracts information in an effective way directly from the partial order structure of the multi-indicator system, neither turning ordinal scores into numbers, nor reducing ordinal attributes to binary variables, as in the Alkire–Foster methodology (Alkire and Foster 2011a, b).

3 Computational Issues

To get an idea of the computations involved in the posetic evaluation procedure, let us consider the deprivation identification function (similar considerations hold for the severity functions). The value of the identification function on a profile \(\varvec{p}\) is computed counting the fraction of linear extensions of the achievement poset, classifying \(\varvec{p}\) as deprived (Fattore (2016)), formally:
$$\begin{aligned} idn_{\varPi }^{}(\varvec{p}|\tau )=\frac{1}{|\varOmega (\varPi )|}\sum _{\ell \in \varOmega (\varPi )} idn_{\ell }(\varvec{p}|\tau ) \end{aligned}$$
(1)
where \(\varOmega (\varPi )\) is the set of linear extensions of the achievement poset and \(idn_{\ell }(\varvec{p}|\tau )\) is the deprivation classifier associated to linear extension \(\ell\); it assumes value 1 if \(\varvec{p}\) is on or below the maximum element (in \(\ell\)) of the threshold \(\tau\) and 0 otherwise (for future convenience, here we explicit the role of the threshold \(\tau\) - which may comprise several profiles - as a parameter of the identification function). To give an idea of the computational burden involved by the above formula, consider the class \(\{\varvec{2}^k\}\) (\(k=2,3,4\ldots\)) of product posets generated by \(2,3,4\ldots\) two-elements chains. This kind of achievement posets comes out, for example, when considering material deprivation/non-deprivation in terms of ownership of goods or of service availability, or when ordinal attributes are binarized, as in the Counting Approach of Alkire and Foster. Figure 1 depicts the Hasse diagrams of posets \(\varvec{2}^{2}_{}\), \(\varvec{2}^{3}_{}\) and \(\varvec{2}^{4}_{}\), which comprise 4, 8, and 16 elements respectively. A part from trivial cases, there is no easy way to compute the number of linear extensions of posets (Brightwell and Winkler 1991; Loof et al. 2006). For our purposes, however, it is enough to consider lower bounds to these numbers, so as to get an even rough idea of the computational complexity to manage. Poset \(\varvec{2}^2_{}\) has just 2 linear extensions. Poset \(\varvec{2}^3_{}\) has 48 linear extensions, as can be checked by constructing them directly. In general, if \(|\varOmega (\varvec{2}^{k}_{})|\) denotes the number of linear extensions of \(\varvec{2}^k_{}\), a lower bound for the number of linear extensions of \(\varvec{2}^{k+1}_{}\) is given by (see footnote 4, later in the paper):
$$\begin{aligned} |\varOmega (\varvec{2}^{k+1}_{})|>(k+1)\cdot |\varOmega (\varvec{2}^{k}_{})|^2+\prod _{s=1}^{k+1}\left[ \left( {\begin{array}{c}k+2\\ s\end{array}}\right) !\right] . \end{aligned}$$
(2)
Substituting \(|\varOmega (\varvec{2}^{k}_{})|\) with its lower bound, for \(k=4,5,\ldots\), we finally get the results reported in Table 1, which gives a picture of the computational issue (for a deeper discussion on counting linear extensions of posets, see Brightwell and Winkler (1991)).
Table 1

Number of linear extensions for some posets of the class \(\{\varvec{2}^k_{}\}\)

Poset

Elements

Linear ext.

\(\varvec{2}^2_{}\)

4

2

\(\varvec{2}^3_{}\)

8

48

\(\varvec{2}^4_{}\)

16

\(>423,936\)

\(\varvec{2}^5_{}\)

32

\(>1.8\cdot 10^{17}\)

In real applications, attributes often have more than two degrees, achievement posets become larger and more complex and the number of linear extensions “explodes”. Given that exact computations are unfeasible, the alternative is to sample from the set \(\varOmega (\varPi )\) and to compute identification scores on such a random subset. This is the strategy implemented in PARSEC, the R package implementing the evaluation procedure (Arcagni and Fattore 2014) and freely available in the R ecosystem (R Core Team 2012). Linear extensions are sampled following the Bubley–Dyer algorithm (Bubley and Dyer 1999) which is, to our knowledge, the most efficient available in literature. It is designed in such a way that the asymptotic sampling distribution is uniform so that, in practice, after a sufficiently large number of runs, the extraction probability is uniform over \(\varOmega (\varPi )\). In the original paper of Bubley and Dyer, a formula is reproduced to determine the number of runs to perform, in order to achieve uniformity within a given error \(\epsilon\). Table 2 reports this number for different posets and for a distance \(\epsilon\) from uniformity equal to \(10^{-3}\).
Table 2

Number of runs, of the Bubley–Dyer sampling algorithm, needed to achieve uniformity in the sampling distribution, within an error of \(\epsilon =10^{-3}\), for different posets

Poset

Elements

Runs

\(\varvec{2}^4_{}\)

16

3,359,977

\(\varvec{2}^5_{}\)

32

123,534,106

\(\varvec{2}^6_{}\)

64

4,581,459,610

\(\varvec{2}^7_{}\)

128

168,568,776,870

Fig. 1

Hasse diagrams of posets \(\varvec{2}^2_{}\) (a), \(\varvec{2}^3_{}\) (b) and \(\varvec{2}^4_{}\) (c)

Using standard personal computers, the sampling algorithm allows the evaluation procedure to be applied to posets with up to some hundreds profiles. For example, deprivation measures can be computed on posets like \(\varvec{2}^7_{}\), \(\varvec{4}^4_{}\) (comprising 128 and 256 elements, respectively), \(\varvec{3}^5_{}\) (243 elements) or \(\varvec{5}^3_{}\) (125 elements). In many studies, however, much more attributes are considered and achievement posets may comprise thousands of profiles. In these cases, the posetic evaluation procedure suffers of computational problems and cannot be put to work easily.

4 Alternative Generating Systems for the Basic Achievement Poset

In this section, we show that the basic achievement poset can be generated by particular families of linear extensions and that different evaluation functions can be derived from them. Before turning to formal developments, however, we outline the logic thread followed in the next paragraphs, to ease understanding the technicalities. In the original posetic procedure, the evaluation functions are built as an average of simpler functions associated to linear extensions of the achievement poset. The theoretical justification for this construction is that finite posets are univocally identified by their respective set of linear extensions (see Fattore (2016) for a deeper discussion on this point). More precisely, any finite poset \(\varPi\) is the intersection of the set \(\varOmega (\varPi )\) of its linear extensions, i.e. \(\varOmega (\varPi )\) is a generating system for \(\varPi\). It turns out that different generating systems1, other than \(\varOmega (\varPi )\), can be found (see for example Fattore (2016) and, for more general results, Schröder. (2002)) and that each of them can be used, in principle, to build alternative evaluation functions. When \(\varPi\) is the basic achievement poset built as described in Sect. 2, some particularly simple generating systems can be found. As thoroughly discussed below, these systems lead to evaluation functions which are easily interpretable from the point of view of multi-criteria decision making; moreover, their cardinality is smaller than that of the full set of linear extensions, leading to lighter computations. The technical developments of the following paragraphs are basically devoted to partitioning the set of linear extensions of the basic achievement poset in such a way that the alternative generating systems easily emerge. We first show that \(\varOmega (\varPi )\) can be partitioned as the disjoint union of the subsets of conditional, compensative and mixing linear extensions (the terminology will be explained below); the set of conditional linear extensions will be further partitioned into the subsets of lexicographic and switching linear extensions. The set of lexicographic linear extensions (and hence also the set of conditional linear extensions) turns out to be a generating systems for the achievement poset, and to be of special interest in view of deprivation evaluation. The reduced evaluation procedure, in particular, follows from computing the evaluation functions over lexicographic linear extensions. We introduce these results, by first studying poset \(\varvec{2}^3_{}\), depicted in Fig. 1b. This poset is very simple and allows for effective graphical representations, making the formal arguments much easier to grasp and to generalize to more complex partially ordered sets.

4.1 Conditional, Compensative and Mixing Linear Extensions

As stated before, poset \(\varvec{2}^3_{}\) is the product order of three two-element chains, i.e. of three binary attributes \(v_1^{},v_2^{}\) and \(v_3^{}\). As it can be directly checked, \(\varvec{2}^3_{}\) can be represented as the intersection of 4 posets, \(Con_1^{}\), \(Con_2^{}\), \(Con_3^{}\) and Com, as depicted in Fig. 2. Posets \(Con_1^{}, Con_2^{}\) and \(Con_3^{}\) are called conditional, since they order profiles “conditional on one attribute”, i.e. fixing the scores of one attribute, and partially ordering profiles according to the scores on the other attributes. Each of the conditional posets has four linear extensions, called conditional, as well. Poset Com is instead obtained from \(\varvec{2}^3\) adding edges to the original Hasse diagram, in such a way that each element on a level2 covers all of the elements on the level below. Profiles belonging to the same level of Com share the same number of 1s and 0s, so that this poset can be interpreted as a partial order where “the more achievements, the better” (assuming “1” stands for the ownership of a specific good, for example). Since profiles with the same number of achievements are “equivalent” in Com (i.e. they share the same comparabilities and incomparabilities and exchanging two of them would not change the partial order relation), this poset is here called compensative. Linear extensions of Com are easily computed, by stacking permutations of profiles belonging to the same level: they are 36, obviously called compensative. These four posets do not have any linear extension in common, so \(CON_1^{}(\varvec{2}^3_{})=\varOmega (Con_1^{})\), \(CON_2^{}(\varvec{2}^3_{})=\varOmega (Con_2^{})\), \(CON_3^{}(\varvec{2}^3_{})=\varOmega (Con_3^{})\) and \(COM(\varvec{2}^3_{})=\varOmega (Com)\) form a partition of \(\varOmega (\varvec{2}^3_{})\):
$$\begin{aligned} \varOmega (\varvec{2}^3_{})=CON_1^{}(\varvec{2}^3_{})\cup CON_2^{}(\varvec{2}^3_{})\cup CON_3^{}(\varvec{2}^3_{})\cup COM(\varvec{2}^3_{}). \end{aligned}$$
(3)
Fig. 2

Decomposition of \(\varvec{2}^3\) as intersection of conditional and compensative extensions

Fig. 3

Linear extensions of poset \(Con_1^{}\), with the two lexicographic elements \(\ell _{123}\) and \(\ell _{132}\) put in evidence

In more general terms, it can be proved (see Appendix) that an achievement poset \(\varPi\) generated by k ordinal attributes can be represented as the intersection of k conditional posets \(Con_1^{}(\varPi ),\ldots ,Con_k^{}(\varPi )\), one compensative poset \(Com(\varPi )\) and a residual set \(Mix(\varPi )\) of linear extensions which order profiles mixing them, so as to get neither conditional, nor compensative linear extensions (for this reason, here we term them mixing; in the case of \(\varvec{2}^3\), the set of mixing linear extensions is empty). Formally:
$$\begin{aligned} \varPi =\bigcap _{j=1}^{k}Con_j^{}(\varPi )\cap Com(\varPi )\cap Mix(\varPi ). \end{aligned}$$
(4)
Since the components of the above representation have no linear extensions in common, the set of linear extensions of \(\varPi\) can be correspondingly partitioned3 as:
$$\begin{aligned} \varOmega (\varPi )=\bigcup _{j=1}^{k}CON_j^{}(\varPi )\cup COM(\varPi )\cup MIX(\varPi ) \end{aligned}$$
(5)
where \(CON_j^{}(\varPi )\) is the set of linear extensions of \(Con_j^{}(\varPi )\) (\(j=1,\ldots ,k\)), \(COM^{}(\varPi )\) is the set of linear extensions of \(Com^{}(\varPi )\) and \(MIX^{}(\varPi )\) corresponds to \(Mix^{}(\varPi )\) (which is in fact composed of mixing linear extensions). Finally, putting
$$\begin{aligned} CON(\varPi )=\bigcup _{j=1}^{k}CON_j^{}(\varPi ) \end{aligned}$$
(6)
we can restate formula (5) in a more compact form4:
$$\begin{aligned} \varOmega (\varPi )=CON(\varPi )\cup COM(\varPi )\cup MIX(\varPi ) \end{aligned}$$
(7)
putting in evidence the set of conditional linear extensions \(CON(\varPi )\).

4.2 Lexicographic and Switching Linear Extensions

Let us consider again poset \(\varvec{2}^3_{}\) and its conditional components. Figure 3 reports the linear extensions of \(Con_1^{}\). Two of them, \(\ell _{123}^{}\) and \(\ell _{132}^{}\), are obtained by listing achievement profiles lexicographically (i.e. in an “alphabetic” fashion), according to \(v_1^{},v_2^{},v_3^{}\) and \(v_1^{},v_3^{},v_2^{}\) respectively. For this reason, they will be referred to as lexicographic linear extensions. Similarly, \(Con_2^{}\) has two lexicographic linear extensions, ordered according to \(v_2^{},v_1^{},v_3^{}\) and \(v_2^{},v_3^{},v_1^{}\); also \(Con_3^{}\) has two, ordered according to \(v_3^{},v_1^{},v_2^{}\) and \(v_3^{},v_2^{},v_1^{}\). Non-lexicographic linear extensions of \(Con_1\) will be instead referred to as switching, since they exchange the order of \(v_2\) and \(v_3\) at different levels of \(v_1\). The same terminology holds for non-lexicographic linear extensions of \(Con_2\) and \(Con_3\). Collecting all of the lexicographic linear extensions of \(\varvec{2}^3_{}\) into the set \(LEX(\varvec{2}^3_{})\) and all the switching linear extensions into the set \(SWI(\varvec{2}^3_{})\), we can finally partition \(\varOmega (\varvec{2}^3_{})\) as follows (recall that this poset has no mixing linear extensions):
$$\begin{aligned} \varOmega (\varvec{2}^3_{})=LEX(\varvec{2}^3_{})\cup SWI(\varvec{2}^3_{})\cup COM(\varvec{2}^3_{}). \end{aligned}$$
(8)
These results can be directly generalized to any basic achievement poset \(\varPi\), generated by k ordinal attributes; in particular, \(\varPi\) has k! lexicographic linear extensions and \(\varOmega (\varPi )\) can be decomposed as the disjoint union of the sets of lexicographic, switching, compensative and mixing linear extensions:
$$\begin{aligned} \varOmega (\varPi )=LEX(\varPi )\cup SWI(\varPi )\cup COM(\varPi )\cup MIX(\varPi ). \end{aligned}$$
(9)

Remark

For sake of clarity, we summarize the nested partitions of \(\varOmega (\varPi )\) illustrated above, in the following scheme (in bold, we highlight the two subsets of interest in the paper).

4.3 Evaluation Functions for the Basic Achievement Poset

In view of the definition of alternative evaluation functions and the development of a reduced evaluation procedure, the main result is that the set \(LEX(\varPi )\) is a generating system for \(\varPi\). This can be directly checked for poset \(\varvec{2}^3_{}\) and is proved for general basic achievement posets in Fattore (2016). As a consequence, also \(CON(\varPi )\) is a generating system for \(\varPi\), since it comprises \(LEX(\varPi )\). In other words, \(LEX(\varPi )\) and \(CON(\varPi )\) univocally identify \(\varPi\) so that evaluation functions can be computed over them, following the same procedure as for the evaluation functions associated to the whole set \(\varOmega (\varPi )\). Explicitly, we can write
$$\begin{aligned} idn_{lex}(\cdot |\cdot )= & {} \frac{1}{|LEX(\varPi )|} \sum _{\ell \in LEX(\varPi )} idn_{\ell }(\cdot |\cdot ) \end{aligned}$$
(11)
$$\begin{aligned} idn_{con}(\cdot |\cdot )= & {} \frac{1}{|CON(\varPi )|} \sum _{\ell \in CON(\varPi )} idn_{\ell }(\cdot |\cdot ) \end{aligned}$$
(12)
where the threshold \(\tau\) has been left unspecified, for sake of generality (analogous expressions can be given for severity functions, restricting the formulas given in Fattore (2016) to the set of lexicographic or conditional linear extensions). The conditional identification function can be decomposed as the average of the identification functions associated to each of the conditional posets, i.e.:
$$\begin{aligned} idn_{con}(\cdot |\cdot )=\sum _{j=1}^k \frac{|CON_j(\varPi )|}{|CON(\varPi )|}\cdot idn_{con_j}(\cdot |\cdot ) \end{aligned}$$
(13)
where
$$\begin{aligned} idn_{con_j}(\cdot |\cdot )=\frac{1}{|CON_j(\varPi )|} \sum _{\ell \in CON_j(\varPi )} idn_{\ell }(\cdot |\cdot )\quad (i=1,\ldots ,k). \end{aligned}$$
(14)
The conditional and lexicographic identification functions have been introduced in a rather formal way, given a suitable partition of the set of linear extensions of the achievement poset. What makes them of interest in view of deprivation evaluation, however, is their interpretation in terms of multi-criteria decision making. We first discuss the interpretation of the conditional identification function \(idn_{con}(\cdot |\cdot )\). Let us write \(v_i \lhd ^* v_j\) if attribute \(v_j\) is considered as more relevant than attribute \(v_i\) and \(v_i || v_j\) if the two attributes share the same relevance (see Fattore (2016) for details on how to formalize attribute relevance in a posetic framework). The identification function associated to the j-th conditional component assigns scores to deprivation profiles consistently with the following partial order on the set of attributes
$$\begin{aligned} v_i \lhd ^* v_j\text { for all } i\ne j; \quad v_i||v_h, \text { for all }i,h\ne j, \end{aligned}$$
(15)
i.e. picking up attribute \(v_j\) as the most relevant and considering all of the other attributes as equally important. As a whole, \(idn_{con}(\cdot |\cdot )\) computes deprivation scores averaging over “valuations” which differ on the attribute they retain as more relevant. A similar approach can be followed to provide an interpretation of the lexicographic identification function. In this case, classifiers associated to single lexicographic linear extensions assign deprivation scores 1 or 0 consistently with a complete order of attributes by relevance, i.e.:
$$\begin{aligned} v_{i_1} \lhd ^* v_{i_2}\lhd ^*\ldots \lhd ^*v_{i_k}. \end{aligned}$$
(16)
For example, with reference to poset \(\varvec{2}^3_{}\) and Fig. 3, if the threshold is set to 011, \(\ell _{123}\) (which is associated to \(v_3\lhd ^*v_2\lhd ^*v_1\)) classifies as non-deprived profiles with \(v_1=1\) and as deprived profiles with \(v_1=0\); if the threshold is set to 001, it classifies as non-deprived profiles with either \(v_1=1\) or \(v_2=1\) and as deprived profiles with both \(v_1=0\) and \(v_2=0\). Identification function \(idn_{lex}(\cdot |\cdot )\) is thus the average over k! classifiers, covering all of the complete orders on k elements. So, from the perspective of multi-criteria decision making, the difference between conditional and lexicographic identification functions lies in the underlying partial orders defined on the set of attributes. Similar considerations could be done for the severity functions.

The above discussion can be summarized as follows. In the posetic approach, the intrinsic nuances and the fuzziness of multidimensional deprivation reflect into the existence of incomparabilities among achievement profiles. These can be seen as due to the existence of different linear extensions, i.e. of different profile rankings, whose intersection determines the basic achievement poset. In turn, some of these different rankings can be seen as due to difference in the partial or complete order of attributes, in terms of importance with respect to the evaluation of deprivation. These “components” of deprivation nuances and fuzziness are linked to conditional components and lexicographic linear extensions and are measured by the respective identification functions.

5 The Reduced Evaluation Procedure

The aims behind the definition of a reduced posetic procedure for deprivation evaluation can be summarized as follows: (i) we want to preserve the posetic nature of the approach (the procedure must be completely ordinal and lead to a fuzzy description of deprivation), (ii) we want to keep the structure of the evaluation process (the procedure must follow the steps 1–5 outlined in Sect. 2) and (iii) we require computations to be lighter (i.e. the procedure must be applicable to larger posets). In particular, we want the posetic approach to be effective when one deals with several attributes scored on a few degrees (as typical in deprivation studies). Actually, the critical point is the third. In principle, there are three ways to modify the evaluation procedure to achieve it: (i) developing more efficient linear extensions sampling algorithms, (ii) finding out mathematical formulas to analytically approximate evaluation functions (Loof et al. 2008) or (iii) restricting the class of linear extensions involved in the computations. We focus on the third alternative, i.e. we restrict the computation of the evaluation functions to a subset of linear extensions.

The basic idea is to choose a suitable generating system for the basic achievement poset and to build the evaluation functions over it. Given the above discussion, we could choose both the conditional and the lexicographic systems. However, the number of conditional linear extensions still grows too fast, as the number of attributes increases, for computations to be performed on many real posets. Hence, we define the reduced evaluation procedure computing the evaluation functions over the lexicographic generators, whose number is k!. In fact, although k! grows fast with k, it stays much smaller than the lower bounds given in Eq. (2) for the cardinality of \(\varOmega (\varPi )\). A comment on this point is in order. The choice of the lexicographic generating system is not to be considered primarily as an attempt to provide approximations to the non-reduced evaluation functions. In the reduced procedure, basic classifiers assign profiles to deprivation or to non-deprivation differently, since each of them reflect a different attribute ranking. It is the existence of such alternative rankings that “generates” deprivation fuzziness. Consequently, the reduced procedure should be employed whenever deprivation measurement is addressed sharing this point of view. Considering the subset of lexicographic linear extensions only, implies disregarding other linear generators, in particular the switching and the mixing ones. The cardinality of these classes of extensions is greater than that of the lexicographic ones (which is “only” k!); but since there is no link between the way they order profiles and attribute rankings, in the reduced procedure switching and mixing generators are not considered and do not enter the evaluation exercise. This does not mean that these linear extensions cannot be of interest in other cases. For example, if one assumes that attribute rankings can differ, conditional on the level of one attribute, then switching linear extensions could come into play. Indeed, other reduced procedures could be developed, associating evaluation functions to other families of generators, according to different “conceptual models” or “representations” of the way deprivation fuzziness originates. In other words, the existence of different poset representations allows for alternative and non-equivalent evaluation functions (if they were equivalent, there would be no interest in the existence of such different representations). Some representations can be considered more useful than others, and the choice of a specific one should be indeed motivated on some grounds (theoretical or practical; for example, a representation can prove useful, since it is effective in capturing deprivation sufficiently well, providing useful hints for policy-making). The point is thus the following: is the reduced procedure proposed in this paper meaningful and useful? Its meaning is clear, in terms of link to attribute rankings. Its usefulness should be tested in applications, but there are evidences, that the reduced procedure is indeed effective (see Sect. 6, for more precise statements in this perspective).

Remark

The reduced procedure is completely consistent with the way attribute relevance is introduced into the posetic computations, when partial orders on the set of attributes are defined (see Fattore (2016) for details). This means that it can be applied not only to the basic achievement poset, but to any achievement poset obtained filtering some lexicographic linear extensions out, from the original lexicographic generating system. In this case, the lexicographic evaluation functions will be computed on the remaining lexicographic linear extensions.

6 Comparison Between Reduced and Non-Reduced Evaluation Procedures

In this section, we present some simulation studies to compare lexicographic and full evaluation functions on some posets. The shapes of these functions depend in a non-trivial way upon the partial order relation defining the achievement poset and upon the choice of the threshold. For this reason, here we just provide some first investigations on the lexicographic functions, in order to get some insights and to drive future deeper and more systematic analyses. For each poset, we compute both the identification and the relative severity5 functions for the reduced and the non-reduced procedures, comparing their values and the corresponding computation times (Table 3).

Remark

In the following pictures, profiles are listed on the abscissa in increasing order of their scores, computed according to the non-reduced identification, or severity, function.

Before illustrating the simulations, we give a few general comments:
  1. 1.

    Completely deprived profiles and completely non-deprived profiles are preserved by \(idn_{lex}^{}(\cdot |\cdot )\), i.e. \(idn_{lex}^{}(\cdot |\cdot )\) equals 1 or 0 if and only if \(idn_{\varPi }(\cdot |\cdot )\) does. In fact, to get scores 0 or 1, profiles must be classified as non-deprived or deprived (respectively) by each linear extension, and thus also by the lexicographic ones.

     
  2. 2.

    The degree of fuzziness of deprivation depends upon the number of attributes considered (“multidimensionality effect”) and upon the number of degrees of the scales they are scored against (“coarse-graining effect”). The number of lexicographic linear extensions equals k! (where k is the number of attributes), and there are \(k!+1\) different values that the lexicographic identification function can assume (they correspond to the possible fractions of lexicographic linear extensions, classifying a profile as deprived). When the number of attributes is high, the lexicographic identification function may be similar to the full identification function. On the contrary, when the number of attributes is low, the lexicographic procedure cannot account for fuzziness due to the “coarse-graining” effect. As a result, we expect the reduced posetic procedure to be mostly effective when attributes are scored on short scales.

     
  3. 3.

    As it can be checked in the following simulations, even if the difference between the reduced and the full procedures can be relevant sometimes, the overall picture of deprivation need not be so different (e.g. in terms of the existence and identification of individuals with “high” or “low” values of the evaluation functions). So the reduced procedure need not necessarily imply relevant losses of “meaningful” information (e.g. of information with policy-making implications).

     
  4. 4.

    The reduced evaluation procedure still greatly improves over the Alkire–Foster Counting Approach, which provides just a binary (yes/no) assessment of multidimensional deprivation.

     
  5. 5.

    Using the reduced procedure, the computation time decreases dramatically. Clearly, when k grows enough, listing all of the lexicographic linear extensions may still be problematic. However, applying the sampling approach used in the full procedure to the lexicographic generating system, it is possible to apply the reduced procedure to multi-indicator systems with tens of attributes. This is the strategy being implemented in the next release of the R package PARSEC (Arcagni and Fattore 2014).

     
Given the above considerations, the adoption of the reduced procedure appears as highly justifiable, providing social scientists with an effective tool for deprivation analysis. We now turn to the simulations.
Poset \(\varvec{2}^3\). This is the poset depicted in Fig. 1b. As stated, it has 48 linear extensions and six of them are lexicographic. Figure 4 compares the reduced and non-reduced evaluation functions, setting threshold \(\tau\) as (010); as it can be noticed, the lexicographic procedure provides results quite similar to those of the full procedure. Clearly, this happens since attributes have only two scores. In both cases, evaluation functions are computed in less than 1 second.
Fig. 4

Reduced and full evaluation functions for poset \(\varvec{2}^3\)

Posets \(\varvec{2}^2\times \varvec{3}^2\), \(\varvec{3}^3\) and \(\varvec{3}^4\). As in the previous example, these three posets belong to the class of “few-attributes/few-degrees” product posets. Although they are of a small size, full evaluation functions cannot be computed exactly and sampling algorithms have to be adopted. An analytical comparison of the computational burdens of the reduced and full procedures on these posets is reported in Table 3, while Figs. 5, 6 and 7 depict the resulting evaluation functions. As it can be seen, the degree of similarity of the lexicographic evaluation functions to the full ones changes from case to case and, in general, it is better for the severity function than for the identification function. One notice, also, that the lexicographic and full functions order profiles differently (i.e they are not “co-monotone”, so as to say) and that the reduced ones tend to be “more linear” than the full ones.

Poset \(\varvec{10}^2\). This poset is of a “few-attributes/many-degrees” kind. It has been reported here, as an example of a case where the lexicographic evaluation functions are necessarily very dissimilar from the full ones. In fact, since there are just two lexicographic linear extensions, the lexicographic identification function can assume just three values, namely 0, 0.5 and 1 (Fig. 8).

Poset \(\varvec{2}^7\). This poset belongs to the class of “many-attributes/few-degrees” partial orders. The computation of the full evaluation functions is very heavy, as can be checked by figures reported in Table 3. The lexicographic identification function is quite dissimilar from the full one, when the latter assumes values below 0.5; for greater values, the full identification function is “more linear” and the lexicographic one is more similar to it (Fig. 9). The shape of the reduced severity function, instead, globally resembles that of the full one.
Table 3

Summary of the computational burden comparison. Time is in seconds

\(\varPi\)

\(\tau\)

Full procedure

Reduced procedure

Runs

Time (s)

\(|LEX(\varPi )|\)

Time (s)

\(\mathbf {3}^3\)

\(\{111\}\)

50,962,721

13.12

6

\(\simeq\) 0

\(\mathbf {2}^2\times \mathbf {3}^2\)

\(\{1010, 0101\}\)

228,284,063

73.65

24

0.02

\(\mathbf {3}^4\)

\(\{1110, 0111, 1101\}\)

15,619,852,978

10416.83

24

0.02

\(\mathbf {10}^2\)

\(\{24, 42, 33\}\)

46,742,477,388

38668.22

2

\(\simeq\) 0

\(\mathbf {2}^7\)

\(\begin{array}{r} \{1010101,0101010,\\ 0001111,1111000\} \end{array}\)

168,568,776,860

183337.88

5040

2.95

Fig. 5

Poset \(\mathbf {2}^2\times \mathbf {3}^2\) with threshold \(\tau =\{1010, 0101\}\)

Fig. 6

Poset \(\mathbf {3}^3\) with threshold \(\tau =\{111\}\)

Fig. 7

Poset \(\mathbf {3}^4\) with threshold \(\tau =\{1110, 0111, 1101\}\)

Fig. 8

Poset \(\mathbf {10}^2\) with threshold \(\tau =\{24, 42, 33\}\)

Fig. 9

Poset \(\mathbf {2}^7\) with threshold \(\tau =\{1010101, 0101010, 0001111, 1111000\}\)

7 Conclusion

In this paper, we have shown that different evaluation functions can be computed, selecting different generating systems for the basic achievement poset, and we have proposed a reduced posetic procedure for the evaluation of multidimensional deprivation. The procedure preserves the power and the logic of the original approach; it lessens the computational burden and can be much more easily applied to highly dimensional multi-indicator systems, in particular when many attributes scored on few degrees are considered. Moreover, it still improves conceptually and practically over the Counting Approach of Alkire and Foster, which provides a simple deprived/non-deprived classification of statistical units. In conjunction with algorithms for sampling linear extensions, the reduced procedure definitely extends the range of applicability of the posetic evaluation procedure. Simulations show that the lexicographic identification function tends to be more “linear” than the non-reduced one, while reduced and non-reduced severity functions tend to be more similar. The reduced procedure, however, is not primarily to be intended as an approximation to the non-reduced one; rather, coherently with the final comment in Sect. 5, it provides exact measures of specific aspects of deprivation fuzziness and severity, namely those directly originating from the lack of a complete ranking, in terms of relevance, of deprivation attributes. In the next future, the reduced procedure is going to be implemented in the R package PARSEC (Arcagni and Fattore (2014)), so providing social scientists with a practical and effective evaluation tool, to deal with multi-indicator systems and posets.

Footnotes

  1. 1.

    Generating systems can have different cardinalities; the smallest cardinality of a generating system of a poset \(\varPi\) is called the dimension of \(\varPi\).

  2. 2.

    In a generic finite poset, the level (sometimes called co-level) of an element can be defined as the length (i.e. the number of edges) of the shortest chain connecting it to a minimal element. See Patil and Taillie (2004) for more details on levels and co-levels.

  3. 3.

    Notice that here we are decomposing sets of linear extensions as union of specific subsets; posets are instead described in terms of intersections of linear extensions or, more generally, of other posets.

  4. 4.

    This explains formula (2), which is a recursive count of the number of linear extensions of the conditional and compensative components of poset \(\varvec{2}^{k+1}\).

  5. 5.
    Here, we consider relative severity as defined in formula (6) of Fattore (2016), i.e.
    $$\begin{aligned} svr^*(\varvec{p})=\frac{1}{|\varOmega (\varPi )|}\sum _{\ell \in \varOmega (\varPi )}\frac{svr_{\ell }(\varvec{p})}{svr_{\ell }(\varvec{p}_{\bot })}, \end{aligned}$$
    where \(svr_\ell (\varvec{p})\) is the severity of profile \(\varvec{p}\) in linear extension \(\ell\) and \(\varvec{p}_{\bot }\) is the minimum element of the achievement poset \(\varPi\); \(svr_\ell (\varvec{p})\) is the distance of \(\varvec{p}\) from the highest element of the threshold, in \(\ell\).

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.University of Milano-BicoccaMilanItaly

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