Marco Fattore

From Composite Indicators to Partial Orders: Evaluating Socio-Economic Phenomena Through Ordinal Data

In this paper we present a new methodology for the statistical evaluation of ordinal socio-economic phenomena, with the aim of overcoming the issues of the classical aggregative approach based on composite indicators. The proposed methodology employs a benchmark approach to evaluation and relies on partially ordered set (poset) theory, a branch of discrete mathematics providing tools for dealing with multidimensional systems of ordinal data. Using poset theory and the related Hasse diagram technique, evaluation scores can be computed without performing any variable aggregation into composite indicators. This way, ordinal scores need not be turned into numerical values, as often done in evaluation studies, inconsistently with the real nature of the phenomena at hand. We also face the problem of “weighting” evaluation dimensions, to account for their different relevance, and show how this can be handled in pure ordinal terms. A specific focus is devoted to the binary variable case, where the methodology can be specialized in a very effective way. Although the paper is mainly methodological, all of the basic concepts are illustrated through real examples pertaining to material deprivation.

Using Poset Theory to Compare Fuzzy Multidimensional Material Deprivation Across Regions

In this paper, a new approach to the fuzzy analysis of multidimensional material deprivation data is provided, based on partial order theory. The main feature of the methodology is that the information needed for the deprivation assessment is extracted directly from the relational structure of the dataset, avoiding any kind of scaling and aggregation procedure, so as to respect the ordinal nature of the data. An example based on real data is worked out, pertaining to material deprivation in Italy for the year 2004.

On the relationships among latent variables and residuals in PLS path modeling: The formative-reflective scheme

A new approach for the estimation and the validation of a structural equation model with a formative-reflective scheme is presented. The basis of the paper is a proposal for overcoming a potential deficiency of PLS path modeling. In the PLS approach the reflective scheme assumed for the endogenous latent variables (LVs) is inverted; moreover, the model errors are not explicitly taken into account for the estimation of the endogenous LVs. The proposed approach utilizes all the relevant information in the formative manifest variables (MVs) providing solutions which respect the causal structure of the model. The estimation procedure is based on the optimization of the redundancy criterion. The new approach, entitled redundancy analysis approach to path modeling (RA-PM) is compared with both traditional PLS Path Modeling and LISREL methodology, on the basis of real and simulated data.

PARSEC: An R Package for Poset-Based Evaluation of Multidimensional Poverty

The paper introduces PARSEC, a new software package implementing basic partial order tools for multidimensional poverty evaluation with ordinal variables. The package has been developed in the R environment and is freely available from the authors. Its main goal is to provide socio-economic scholars with an integrated set of elementary functions for multidimensional poverty evaluation, based on ordinal information. The package is organized in four main parts. The first two comprise functions for data management and basic partial order analysis; the third and the fourth are devoted to evaluation and implement both the poset-based approach and a more classical counting procedure. The paper briefly sketches the two evaluation methodologies, illustrates the structure and the main functionalities of PARSEC, and provides some examples of its use.

Partial Orders in Socio-economics: A Practical Challenge for Poset Theorists or a Cultural Challenge for Social Scientists?

In this “position paper” we discuss the potential role of partial order theory in socio-economic statistics and social indicators construction. We maintain that the use of concepts and tools from poset theory is needed and urgent to improve currently adopted methodologies, which often prove ineffective for exploiting ordinal data. We also point out that the difficulties in spreading partial order tools are cultural in nature, and that some open-mindedness is needed among social scientists. We address these issues introducing some examples of open questions in socio-economic data analysis: (i) the problem of multidimensional poverty evaluation, (ii) the problem of assessing inequality and societal polarization, and (iii) the problem of clustering in multidimensional ordinal datasets.

Measuring Structural Dissimilarity Between Finite Partial Orders

In this paper, we address the problem of measuring structural dissimilarity between two partial orders with n elements. We propose a structural dissimilarity measure, based on the distance between isomorphism classes of partial orders, and propose an interpretation in terms of graph theory. We give examples of structural dissimilarity computations, using a simulated annealing algorithm for numerical optimization.

A least squares approach to latent variables extraction in formative–reflective models

In this paper, we propose a new least-squares based procedure to extract exogenous and endogenous latent variables in formative-reflective structural equation models. The procedure is a valuable alternative to PLS-PM and Lisrel; it is fully consistent with the causal structure of formative-reflective schemes and extracts both the structural parameters and the factor scores, without identification or indeterminacy problems. The algorithm can be applied to virtually any kind of formative-reflective scheme, with unidimensional and even multidimensional formative blocks. To show the effectiveness of the proposal, some simulated examples are discussed. A real data application, pertaining to customer equity management, is also provided, comparing the outputs of our approach with those of PLS-PM, which may produce inconsistent results when applied to formative-reflective schemes.

Socio-economic Statistics for a Complex World: Perspectives and Challenges in the Big Data Era

This chapter addresses a topic which is gaining increasing interest in socio-economic statistics and that will play a central role in what in the future could possibly be called “information-based policy-making”. The topic is that of big data and data science and of their potential effects on next future socio-economic statistics (Landefeld 2014). Although no relevant applications have been produced yet at “official level”, the use of big data and the applications of data science methodologies are in fact opening new avenues to the way socio-economic statisticians may extract information from different data sources and provide it to decision-makers. It is surely not easy to write a chapter on this theme. The topic, “big data and data science”, is in fact a broad concept and cannot be considered as a scientific discipline yet, though it is attracting research efforts from many different sectors and many people are contributing to its development. It can be addressed from many points of view and different aspects (technological, methodological, epistemological…) could be underlined, giving different alternative pictures of the argument. Given the aim of the book, here we simply outline some basic concepts pertaining to big data, to clarify why socio-economic statisticians should be interested in this area, to help them realize its potentialities and criticalities and to stress the conceptual differences with respect to “traditional” statistical analysis. The chapter is somehow different from others in this book. It is non-technical and is based on reflections and experiences of the Author, who has been involved in didactical activities and in real projects pertaining to big data analysis and data science. As a consequence, the text may seem more “subjective” than other contributions in the volume. This is true and partly unavoidable: the attempt is to collect and share what I could learn on the topic in the last years, motivating why I think big data can open new horizons to applied statistics, in the socio-economic field.

Synthesis of Indicators: The Non-aggregative Approach

The need for new tools in synthetic indicators construction in the social sciences is deeply related to the problem of describing and understanding increasingly complex societal facts. On the one hand, official surveys, administrative data, web data, open data, to say a few, are now easily available to social scientists in the form of wide and complex multidimensional indicator systems. On the other hand, as data complexity grows, the need to get effective synthetic views, capable to enhance decision-making, increases as well. New procedures for data treatment are necessary, to overcome the limitations of older approaches that are designed for simpler data systems, are based on the “synthesis-as-aggregation” paradigm and employ composite indicators as their main statistical tool. To make a concrete example, consider the “beyond GDP” perspective to well-being and to societal evaluation. Going “beyond GDP” invariably requires dealing with multidimensional systems of ordinal data (e.g. pertaining to ownership of goods, access to services, self-perception of health and economic status…), ruling out the possibility to directly apply the composite indicator approach to measurement (Fattore 2015). In this and similar contexts, two main issues arise. 1. Ordinal attributes cannot be aggregated through linear combinations, averages or other functionals, designed for numerical variables. In fact, ordinal scores cannot be summed, multiplied by scalars or composed in other ways. For this reason, they are often transformed into numerical scores, through more or less sophisticated scaling tools, before aggregation. Unfortunately, there are evidences that such procedures may lead to controversial results (Madden 2010). Moreover, one could legitimately ask why concepts naturally conceived in ordinal terms should be forced into numerical settings. Is the idea of ordinal scores as rough manifestations of underlying continuous traits always well founded? Or is it actually motivated by the lack of consistent and effective procedures for the treatment of ordinal data? Such problems go beyond the setting of well-being measurement and arise in many other fields as well. For example, in marketing and customer segmentation, in ecological and environmental studies, in risk management and, more generally, in ordinal multi-criteria decision-making (Bruggemann et al. 1999; Annoni and Bruggemann 2009; Bruggemann and Patil 2010, 2011; Bruggemann and Voigt 2012; Bruggemann and Carlsen 2014; Carlsen and Bruggemann 2014). It is in fact a feature of modern information society that most of data we deal with are of a discrete and qualitative kind. The absence of statistical tools and procedures to manage such data types consistently may well turn into severe limitations in our capability to exploit the great amount of information they convey. 2. Independently of their nature, it is a matter of fact that many data systems available to social scientists often comprise weakly interdependent attributes. The absence of strong interconnections in a multi-indicator system prevents from achieving effective dimension reductions through aggregation procedures. Consequently, and independently of the models or of the procedures they are computed from (e.g. latent variables or structural equation models, PLS path modeling or other formative aggregation tools), composite indicators are inherently inappropriate in these situations, being aggregative and compensative. This leads to a fundamental question: is attribute aggregation the only road to synthesis? The answer to this question motivates most of the present chapter.

Functionals and Synthetic Indicators Over Finite Posets

In this paper, we propose an axiomatic theory of real functionals on frequency distributions over finite posets. The theory links the properties of the functionals to the classical theory of quasi-arithmetic means. In particular, it is shown that, given the frequency distribution, the values assumed by any “well-behaved” functional on a poset π can be expressed as a quasi-arithmetic mean of the values assumed over the linear extensions of π. This result plays a central role in view of the construction of synthetic indicators for multidimensional system of ordinal indicators, as shown through an example pertaining to multidimensional bi-polarization.