Abstract
The weighting of subindicators is widely debated in the composite indicator literature. However, these weighting schemes’ effects on the composite indicator’s spatial dependence property are still little known. This research reveals a direct relationship between the weighting scheme of subindicators and the spatial autocorrelation of the composite indicator. The Global Moran's Index (I) of composite indicators built using Datadriven (Moran’s I = 0.636) and Hybrid (Moran’s I = 0.597) weighting schemes is, on average, eleven percent higher than in the Equalweights (Moran's I = 0.549) and Expert opinion (Moran's I = 0.560) weighting schemes. The average score of the composite indicator is higher when they are built by weighting schemes that better describe the spatial dependence. The spatial dependence of subindicators and composite indicators are not related. All fifteen subindicators show lower spatial autocorrelation than the composite indicators built by Expert opinion, Hybrid, and Datadriven weighting schemes. The spatial weighting matrix influences the spatial autocorrelation but does not change the robustness and quality parameters of the composite indicator. The research develops a Datadriven weighting scheme that allows individually or simultaneously considering the opinion of experts and parameters of quality and robustness of the composite indicator. It also offers the means to reduce judgment errors and evaluation biases in Expert opinion subindicator weighting schemes.
This is a preview of subscription content, access via your institution.
Data availability
Martinuci, Oseias da Silva; Machado, Alexei Manso Correa; Libório, Matheus Pereira (2021), “Data for: Timeinspace analysis of multidimensional phenomena”, Mendeley Data, V4, https://doi.org/10.17632/m3y4jncvch.4.
Notes
Population Estimates carried out by the Brazilian Institute of Geography and Statistics in July 2021. Accessible at https://www.ibge.gov.br/estatisticas/sociais/populacao/9103estimativasdepopulacao.
The nature of a subindicator is objectivequantitative when the variable is directly measurable, objectivequalitative when the variables are objectively verifiable by the presence or absence of something even if not directly measurable, and qualitativesubjective when the variables are obtained from Experts opinion (Libório et al. 2022a).
References
Abadie J (1969) Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints. Optimization, 37–47
Adeleke R, Alabede O (2021) Understanding the patterns and correlates of financial inclusion in Nigeria. GeoJournal 87:1–18
Aljoufie M, Tiwari A (2020) Exploring housing and transportation affordability in Jeddah. Housing Policy Debate, 1–27
Anselin L (1996) The Moran scatterplot as an ESDA tool to assess local instability in spatial association. In: Fischer M, Scholten HJ, Unwin D (eds) Spatial analytical perspectives on GIS in environmental and socioeconomic sciences. Taylor and Francis, London, pp 111–125
Badea AC, Tarantola S, Bolado R (2011) Composite indicators for security of energy supply using ordered weighted averaging. Reliab Eng Syst Saf 96(6):651–662
Becker W, Paruolo P, Saisana M, Saltelli A (2017) Weights and importance in composite indicators: mind the gap. Handbook of uncertainty quantification, pp 1187–1216
Bernardes P, Ekel PI, Rezende SF, Pereira Júnior JG, dos Santos AC, da Costa MA, Carvalhais RL, Libório MP (2021) Cost of doing business index in Latin America. Qual Quant 56:1–20
Carley S, Evans TP, Graff M, Konisky DM (2018) A framework for evaluating geographic disparities in energy transition vulnerability. Nat Energy 3(8):621–627
Cartone A, Postiglione P (2021) Principal component analysis for geographical data: the role of spatial effects in the definition of composite indicators. Spat Econ Anal 16(2):126–147
Chauhan N, Shukla R, Joshi PK (2020) Assessing impact of varied social and ecological conditions on inherent vulnerability of Himalayan agriculture communities. Hum Ecol Risk Assess Int J 26(10):2628–2645
Cinelli M, Spada M, Kim W, Zhang Y, Burgherr P (2021) MCDA Index Tool: An Interactive Software To Develop Indices And Rankings. Environ Syst Decis 41(1):82–109
Cutter SL, Finch C (2008) Temporal and spatial changes in social vulnerability to natural hazards. Proc Natl Acad Sci 105(7):2301–2306
Davino C, Gherghi M, Sorana S, Vistocco D (2021) Measuring social vulnerability in an urban space through multivariate methods and models. Soc Indic Res 157(3):1179–1201
Dialga I, Le Giang TH (2017) Highlighting methodological limitations in the steps of composite indicators construction. Soc Indic Res 131(2):441–465
Ekel P, Pedrycz W, Pereira J Jr (2020) Multicriteria decisionmaking under conditions of uncertainty: a fuzzy set perspective. Wiley, Chinchester
Ekel P, Bernardes P, Vale GMV, Libório MP (2022) South American business environment cost index: reforms for Brazil. Int J Bus Environ 13(2):212–233
El Gibari S, Gómez T, Ruiz F (2019) Building composite indicators using multicriteria methods: a review. J Bus Econ 89(1):1–24
Fusco E, Vidoli F, Sahoo BK (2018) Spatial heterogeneity in composite indicator: a methodological proposal. Omega 77:1–14
Geary RC (1954) The contiguity ratio and statistical mapping. Inc Stat 5(3):115–146
Getis A (1999) Spatial statistics. Geogr Inf Syst 1:239–251
Goodchild MF (1991) Geographic information systems. Prog Hum Geogr 15(2):194–200
Greco S, Ishizaka A, Tasiou M, Torrisi G (2019) On the methodological framework of composite indices: a review of the issues of weighting, aggregation, and robustness. Soc Indic Res 141(1):61–94
Harris P, Clarke A, Juggins S, Brunsdon C, Charlton M (2015) Enhancements to a geographically weighted principal component analysis in the context of an application to an environmental data set. Geogr Anal 47(2):146–172
IBGE (2010) Censo demográfico 2010. https://censo2010.ibge.gov.br/. Accessed 23 Dec 2022
Jha RK, Gundimeda H (2019) An integrated assessment of vulnerability to floods using composite index–a district level analysis for Bihar, India. Int J Disaster Risk Reduct 35:101074
Kallio M, Guillaume JH, Kummu M, Virrantaus K (2018) Spatial variation in seasonal water poverty index for Laos: an application of geographically weighted principal component analysis. Soc Indic Res 140(3):1131–1157
Katumba S, Cheruiyot K, Mushongera D (2019) Spatial change in the concentration of multidimensional poverty in Gauteng, South Africa: evidence from quality of life survey data. Soc Indic Res 145(1):95–115
KucCzarnecka M, Lo Piano S, Saltelli A (2020) Quantitative storytelling in the making of a composite indicator. Soc Indic Res 149(3):775–802
Lasdon LS, Fox RL, Ratner MW (1974) Nonlinear optimization using the generalized reduced gradient method. Revue française d’automatique, informatique, recherche opérationnelle. Recherche Opérationnelle 8(V3):73–103
Libório MP, Martinuci ODS, Ekel PI, Hadad RM, Lyrio RDM, Bernardes P (2021) Measuring inequality through a noncompensatory approach. GeoJournal 87:1–18
Libório MP, Laudares S, Abreu JFD, Ekel PY, Bernardes P (2020a) Property tax: dealing spatially with economic, social, and political challenges. Urbe. Revista Brasileira de Gestão Urbana, p 12
Libório MP, Martinuci ODS, Laudares S, Lyrio RDM, Machado AMC, Bernardes P, Ekel P (2020b) Measuring intraurban inequality with structural equation modeling: a theorygrounded indicator. Sustainability 12(20):8610
Libório M, Abreu JF, Martinuci ODS, Ekel PI, Lyrio RDM, Camacho VAL, Melazzo ES (2022a) Uncertainty analysis applied to the representation of multidimensional social phenomena. Papers in Applied Geography, 1–24
Libório MP, Ekel PY, Martinuci ODS, Figueiredo LR, Hadad RM, Lyrio RDM, Bernardes P (2022b) Fuzzy set based intraurban inequality indicator. Qual Quant 56(2):667–687
Libório MP, Martinuci ODS, Machado AMC, Ekel PI, Abreu JFD, Laudares S (2022c) Representing multidimensional phenomena of geographic interest: benefit of the doubt or principal component analysis?. The Professional Geographer, pp 1–14
Lindén D, Cinelli M, Spada M, Becker W, Gasser P, Burgherr P (2021) A framework based on statistical analysis and stakeholders’ preferences to inform weighting in composite indicators. Environ Model Softw 145:105208
Maricic M, Egea JA, Jeremic V (2019) A hybrid enhanced Scatter Search—Composite IDistance Indicator (eSSCIDI) optimization approach for determining weights within composite indicators. Soc Indic Res 144(2):497–537
Marzi S, Mysiak J, Essenfelder AH, Amadio M, Giove S, Fekete A (2019) Constructing a comprehensive disaster resilience index: the case of Italy. PLoS ONE 14(9):e0221585
Mavhura E, Manyangadze T, Aryal KR (2021) A composite inherent resilience index for Zimbabwe: an adaptation of the disaster resilience of place model. Int J Disaster Risk Reduct 57:102152
Mazziotta M, Pareto A (2017) Synthesis of indicators: the composite indicators approach. In: Complexity in society: from indicators construction to their synthesis. Springer, Cham, pp 159–191
Mazziotta M, Pareto A (2018) Measuring wellbeing over time: the adjusted MazziottaPareto index versus other noncompensatory indices. Soc Indic Res 136(3):967–976
Moran PA (1950) Notes on continuous stochastic phenomena. Biometrika 37(1/2):17–23
Musa HD, Yacob MR, Abdullah AM (2019) Delphi exploration of subjective wellbeing indicators for strategic urban planning towards sustainable development in Malaysia. J Urban Manag 8(1):28–41
Nardo M, Saisana M, Saltelli A, Tarantola S (2005) Tools for composite indicators building. Eur Com Ispra 15(1):19–20
Otoiu A, Pareto A, Grimaccia E, Mazziotta M, Terzi S (2021) Open issues in composite indicators. A starting point and a reference on some stateoftheart issues, vol 3. Roma TrEPress, Rome
Pedrycz W, Ekel P, Parreiras R (2011) Fuzzy multicriteria decisionmaking: models, methods and applications. Wiley, Chichester
Powell SG, Batt RJ (2008) Modeling for insight. Wiley, Hoboken
Rufat S, Tate E, Emrich CT, Antolini F (2019) How valid are social vulnerability models? Ann Am Assoc Geogr 109(4):1131–1153
Saaty TL (1988) What is the analytic hierarchy process?. In: Mathematical models for decision support. Springer, Berlin, Heidelberg, pp 109–121
Saisana M, Tarantola S (2002) Stateoftheart report on current methodologies and practices for composite indicator development, vol 214. Ispra: European Commission, Joint Research Centre, Institute for the Protection and the Security of the Citizen, Technological and Economic Risk Management Unit
Saisana M, Saltelli A, Tarantola S (2005) Uncertainty and sensitivity analysis techniques as tools for the quality assessment of composite indicators. J R Stat Soc A Stat Soc 168(2):307–323
Saltelli A (2007) Composite indicators between analysis and advocacy. Soc Indic Res 81(1):65–77
Tobler WR (1970) A computer movie simulating urban growth in the Detroit region. Econ Geogr 46(sup1):234–240
Van Laarhoven PJ, Pedrycz W (1983) A fuzzy extension of Saaty’s priority theory. Fuzzy Sets Syst 11(1–3):229–241
Funding
This work was carried out with the support of the Coordination for the Improvement of Higher Education Personnel—Brazil (CAPES)—Financing Code 0001 and the National Council for Scientific and Technological Development of Brazil (CNPq)—Productivity Grant, Grant 311922/20210 and 151518/20220.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Human and/or animal rights
No human participants and/or animals are involved in this research.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: weighting of subindicators by expert opinion
Appendix: weighting of subindicators by expert opinion
1.1 Selection of experts
Three criteria were used to select the experts. First, have a doctorate in human and social sciences, such as sociology and geography. Second, have a publication in scientific journals on social phenomena such as inequality, social exclusion, poverty, and social vulnerability. Third, know the social aspects of the study area.
The selection of experts from this set of criteria has advantages and disadvantages. On the one hand, this set of criteria limits the number of experts qualified to carry out the assessments. On the other hand, this set of criteria favors the quality and homogeneity of the evaluations.
Four experts were selected based on the virtual curriculum system created and maintained by the Brazilian National Council for Scientific and Technological Development. Different evaluation formats and the degree of consensus were adopted to avoid these experts' judgment errors and evaluation biases.
1.2 Assessment of alternatives
The assignment of weights by Expert opinion is commonly associated with the problem of judgment errors (Greco et al. 2019). One way to reduce judgment errors is to offer experts the possibility to choose the format for assessing alternatives (subindicators) that best suits them. Alternative assessment formats are not without limitations, but the flexibility of choosing the assessment format gives the expert psychological comfort that reduces judgment errors (Ekel et al. 2020).

Ordering of Alternatives (\(OA\)) an array \(O = \left\{ {o\left( {x_{1} } \right),o\left( {x_{2} } \right), \ldots , o\left( {x_{k} } \right), \ldots , o\left( {x_{n} } \right)} \right\}\), where \(o\left( {x_{k} } \right)\) is a permutation function that defines the position of the alternative \(x_{k}\) among the integer values {1,2, …, k,…, n}.

Multiplicative Preference Relations (\(MR\)) reflect the preference intensity ratio between the alternatives \(x_{k}\) and \(x_{l}\), being understood as \(x_{k }\) is \(m\left( {x_{k} ,x_{l} } \right)\) times as good as \(x_{l}\).

Utility Values (\(UV\)) the preferences in \(X\) are given as a set of n \(UV\): \(U = \{ u\left( {x_{1} } \right),u\left( {x_{2} } \right), \ldots ,u\left( {x_{k} } \right), \ldots ,u\left( {x_{n} } \right)\)}, where \(u\left( {x_{k} } \right) \in \left[ {0,1} \right]\) represents the \(UV\) assigned to the alternative \(x_{k}\).

Fuzzy Estimates (\(FE\)) a fuzzy number that can be specified directly or through a linguistic variable in which the elements of \({\text{X}}\) are directly assessed by experts using a set of estimates \(L = \left\{ {l\left( {x_{1} } \right),l\left( {x_{2} } \right), \ldots ,l\left( {x_{k} } \right), \ldots ,l\left( {x_{n} } \right)} \right\}\), where \(l\left( {x_{k} } \right)\) is the \(FE\) associated with the alternative \(x_{k}\).

NonReciprocal Fuzzy Preference Relations (\(FR\)) indicate the degree to which the alternative \(x_{k}\) is, at least, as good as the alternative \(x_{l}\), employing its membership function \(0 \le {\upmu }\left( {x_{k} ,x_{l} } \right) \le 1.\)
The experts assessed the weights of the subindicators through the Utility Values (two experts), Ordering of Alternatives, and Multiplicative Preference Relations evaluation formats. The assessments were standardized in the NonReciprocal Fuzzy Preference Relations format using the preference format transformation functions:
for the transformation \({\text{OA}}\) → \({\text{FR}}\);
for the transformation \({\text{MR}}\) → \({\text{FR}}\);
for the transformation \({\text{UV}}\) → \({\text{FR}}\).
Applying the transformation functions is to obtain the evaluations in a single format. Then, it is possible to calculate the weights of the subindicators according to the group of experts, applying the following expression:
where \(x_{k} \in X,\, 0 \le w_{y} \le 1,\) for \(y = 1,2, \ldots ,v,\) taking into account the condition \(\sum\nolimits_{y = 1}^{v} {w_{y} = 1}\).
Table
10 shows the weights assigned by each expert, the highest and lowest weights per subindicator, and the weights according to the group of experts.
1.3 Individual and group degree of consensus
Another common problem of weighting by Expert opinion is associated with evaluation bias. This problem is especially relevant when the number of experts is small because biased assessments more strongly influence the results (Musa et al. 2019).
The degree of individual consensus measures how much an expert opinion diverges from the collective opinion, signaling possible evaluation biases. The individual degree of consensus is obtained applying the expression:
Evaluation biases can be identified by considering the degree of consensus between the expert's assessments and the expert group's assessment. The acceptance threshold of the degree of consensus is a subjective measure defined by the decisionmaker (Ekel et al. 2020). When the acceptance threshold of the degree of consensus is not reached, the decisionmaker may request the reassessment of the alternatives or disregard the expert's assessments (Pedrycz et al. 2011). After identifying and eliminating evaluation biases, it is possible to calculate the degree of consensus of the group by the following expression:
The research adopts the threshold of 0.70 as the individual and group degree of consensus. Table 11 shows that the four experts reached the adopted degree of consensus threshold.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Libório, M.P., de Abreu, J.F., Ekel, P.I. et al. Effect of subindicator weighting schemes on the spatial dependence of multidimensional phenomena. J Geogr Syst (2022). https://doi.org/10.1007/s1010902200401w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s1010902200401w
Keywords
 Composite indicators
 Spatial dependence
 Subindicators weighting
 Moran's Index
JEL Classification
 C02
 C43
 C44
 D6
 I3