Venn Diagrams

Venn diagrams are a diagrammatic representation that shows all possible logical relations between a finite collection of sets. The name of Venn diagrams refers to John Venn who formally introduced the tool (Venn 1880), although the tool pre-dates this and was known—as Venn himself mentions—as Eulerian circles (in fact, although Euler used them, there were uses of similar representations even before Euler).^[82] Venn diagrams consist of a collection of closed figures, such as circles and ellipses, that include, exclude, or intersect one another such that each compartment is associated with a class.^[83]

3.2.1 THE METHODOLOGY AND APPLICATIONS

Applied to the analysis of multidimensional poverty measurement, the interior of each closed figure in a Venn diagram can be used with a set of indicators and associated deprivation cutoffs to represent the number of people who are deprived in a certain dimension.

Table 3.1 Joint distribution of deprivations in two dimensions

Figure 3.1. Venn diagram of joint distribution of deprivations in two dimensions

When there are only two dimensions, a Venn diagram provides a diagrammatic representation of a 2 ? 2 contingency table, introduced in section 2.2.3.

The Venn diagram is particularly useful when two to four dimensions are involved, because the visual representation is easy to interpret. A three-dimension Venn diagram is shown in Figure 3.2. The diagram depicts the frequencies for all the possible combinations of deprivations using the notation n_j∙_{1 j}∙_2j∙₃, such that j_ι = 1 signals deprivation in dimension j_ι and j_ι = 0 signals non-deprivation in dimension j_ι for all ι = 1,2,3. Thus, for example, n₁₁₁ in the intersection of the three circles denotes the number of people who are deprived in all three dimensions, n₀₁₀ denotes the number of people who are deprived in the second dimension only, and so on for other combinations.

Figure 3.2.

In empirical work, the Venn diagram has been used as an exploratory tool to understand the overlapping deprivations in various dimensions and to draw attention to mismatches between them (Ferreira and Lugo 2013). For example, Atkinson et al. (2010) use a three-dimension Venn diagram to depict joint deprivations in income poverty, severe material deprivation, and joblessness. Naga and Bolzani (2006) employ a three-dimension Venn diagram to show how there are disagreements on which households are identified as poor when three different definitions based on income, consumption, and predicted permanent income are used. Venn diagrams have also been selected to capture how different poverty measures or multidimensional targeting instruments agree with each other. For example, Roelen, Gassman, and de Neubourg (2009) created a two-dimension Venn diagram to present the mismatch between the monetary poor and the multidimensionally poor; Alkire and Seth (2013a) used Venn diagrams to portray the mismatches and overlaps between multidimensional poverty targeting instruments; and Decancq, Fleurbaey, and Maniquet (2014) evaluated the degree of overlap between measures of poverty based on expenditures, counting, and preference sensitivity.^[84]

3.2.2 Acriticalevaluation

Venn diagrams are simple and intuitive, yet powerful and information-rich visual graphics. They depict the level of deprivation by dimension (the relative size of the circles) as well as the matches and mismatches across deprivations. By presenting the

Figure 3.3. Venn diagrams of deprivations for four and five dimensions

joint distribution, Venn diagrams provide more information than dashboard measures or composite indices. Additionally, although Venn diagrams do not identify who is poor, they organize the information on the joint distribution in such a way that one could graphically outline an equally weighted identification function of the poor.

3.3