Program VII International Workshop on Proximity Data

Chair: Laura Vicente-González

Title: "New perspectives in analysing income inequality: discrepancies and similarities among European countries"

Abstract:
Income inequality remains a central topic in economic and social research, as disparities in the distribution of resources can have far-reaching implications for well-being, social cohesion and economic development. Traditional approaches often rely on synthetic measures, such as the Gini inequality index or Theil indices, to summarize income concentration in a population. While these indicators are undeniably useful and widely adopted, they provide only a partial picture.
A systematic framework is proposed for analysing income inequality by exploiting the rich information content of share densities, functions directly linked to the Lorenz curves (Lorenz, 1905) and capable of describing multiple aspects of how income is distributed. When comparing income concentration across groups or geographic areas, synthetic indicators such as the Gini or Theil indices provide useful initial insights, but often fail to reveal differences in the underlying structure of inequality. Indeed, populations with similar Gini values may exhibit very different distributive patterns. Share densities, defined as the derivative of the Lorenz curve (Farris, 2010), quantify how income shares are allocated across percentiles of the population, offering a probabilistic and fine-grained interpretation of income concentration. Despite their descriptive potential, they are rarely employed explicitly in empirical analyses.
This study adopts a parametric approach, modelling directly the probability that a randomly selected unit of income belongs to a specific percentile range. From this perspective, a weighted maximum likelihood function is derived to estimate share densities and the corresponding Lorenz curves. Particular attention is devoted to parametric families widely used for modelling income distributions (Kleiber and Kotz, 2003), with a focus on the Dagum model (Dagum, 1977), whose associated share density belongs to the Generalized Beta distribution of the first kind (GB1). The likelihood function is constructed using ordered income data and their associated population shares, allowing for the estimation of parameters that determine inequality measures such as the Gini and Theil indices (Rohde, 2008).
EU-SILC 2021 data for the 27 European countries are considered to explore the potential of the proposed method. The application shows that the GB1 model provides an excellent approximation of the empirical share densities and Lorenz curves. The close correspondence between observed and fitted Gini and Theil values confirms the effectiveness of the parametric approach in capturing the main features of income inequality. To compare the inequality structures of different countries, the Jensen–Shannon (JS) divergence is employed as a measure of dissimilarity between densities.
The JS divergence captures differences across the entire distribution, reflecting the full behaviour of share densities. As a result, any type of deviation, such as increased inequality in the tails or greater concentration around the centre of the income distribution, contributes to the measure. This makes the JS divergence particularly suitable for comparing populations characterised by heterogeneous inequality patterns. On the basis of JS dissimilarities, both hierarchical (Condino, 2023a) and non-hierarchical (Condino, 2023b) clustering algorithms for unconventional data are applied. The hierarchical method starts from the JS dissimilarity matrix among pairs of countries and produces clear groupings of countries with similar inequality structures. The Dynamic Clustering Algorithm (DCA) further refines these results by iteratively minimising within-cluster divergence and representing each cluster through a prototype, defined as the mixture of the share densities of countries in that cluster. Three distinct clusters emerge, corresponding respectively to low, intermediate and high inequality regimes. Moreover, the proposed framework enables the decomposition of total divergence into within-cluster and between-cluster components. This allows the evaluation of the obtained partition quality through the ratio DJS W / DJS T, which quantifies the degree of homogeneity of countries in terms of income inequality within clusters.
References
Condino, F. (2023a). Hierarchical Clustering of Income Data Based on Share Densities. In: Grilli, L., Lupparelli, M., Rampichini, C., Rocco, E., & Vichi, M. (eds), Statistical Models and Methods for Data Science. Springer, Cham.
Condino, F. (2023b). Share density-based clustering of income data. Statistical Analysis and Data Mining, 16(4), 336–347.
Dagum, C. (1977). A new model of personal distribution: specification and estimation. Economie Appliquée, 30, 413–437.
Farris, F.A. (2010). The Gini Index and Measures of Inequality. The American Mathematical Monthly, 117(10), 851-864.
Kleiber, C., Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. John Wiley & Sons.
Lorenz, M.O. (1905). Methods of Measuring the Concentration of Wealth. Publications of the American Statistical Association, 9(70), 209–219.
Rohde, N. (2008). Lorenz Curves and Generalised Entropy Inequality Measures. In: Chotikapanich, D. (ed), Modeling Income Distributions and Lorenz Curve. Springer.

Keywords: income inequality, share density, Lorenz curve, Jensen-Shannon divergence, unconventional data.

Chair: Ana Belén Nieto Librero

Title: "Logistic Biplots for Ordinal Variables based on alternating gradient descent on the cumulative probabilities"

Abstract:
Biplot methodology provides a joint low-dimensional representation of the row and column entities of a data matrix. Classical formulations are based on Principal Component Analysis (PCA) and are therefore restricted to continuous variables under linear model assumptions. Subsequent extensions to binary and nominal data, known as logistic biplots (LBs), replace the linear structure with generalized linear models using logistic link functions. Nevertheless, these approaches do not adequately accommodate ordinal data, which exhibit inherent ordering but violate both metric and nominal assumptions.
In this work, we generalize the biplot framework to ordinal responses through the introduction of the ordinal logistic biplot (OLB). The proposed model assumes a cumulative logit formulation in which row scores act as latent traits generating ordinal responses across multiple dimensions, while column parameters define category thresholds and discrimination vectors. The resulting response functions correspond to ordinal logistic surfaces which, when evaluated over the latent space, induce a linear biplot representation. This construction is embedded within a latent variable model and yields a multidimensional structure formally equivalent to Samejima’s graded response model in Item Response Theory (IRT).
We analyze the geometric structure of the OLB, characterizing the relationship between category boundaries, discrimination parameters, and the induced partitioning of the latent space. For parameter estimation, we develop an iterative optimization scheme based on alternating gradient descent, jointly updating row scores and item parameters under identifiability constraints. Additionally, we derive prediction directions that enable coherent visualization of ordinal response probabilities within the biplot space.
The OLB can thus be interpreted as a multidimensional IRT model augmented with a biplot-based graphical representation, enhancing both interpretability and exploratory capabilities. Its objective is to recover latent structures and variable relationships in ordinal data through a unified geometric and probabilistic framework.
The methodology is illustrated through an application to job satisfaction data from PhD holders in Spain. The estimated latent structure reveals two dominant dimensions, interpretable as intellectual satisfaction and extrinsic job conditions (e.g., salary and benefits). Comparative evaluation against alternative methods demonstrates improved discriminatory capacity of the proposed model with respect to item parameters.