# Aggregating Composite Indicators through the Geometric Mean: A Penalization Approach

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Compensability and Balance among Indicator

#### 2.2. The Human Development Index

## 3. A New Reading of the Mazziotta Pareto Index

## 4. The Penalized Geometric Mean

**Proposition**

**1.**

- 1.
- $G{M}_{i}^{+}\ge {\mu}_{0,i}\ge G{M}_{i}^{-}$.
- 2.
- $G{M}_{i}^{+}=G{M}_{i}^{-}={\mu}_{0,i},\phantom{\rule{4.pt}{0ex}}if\phantom{\rule{4.pt}{0ex}}and\phantom{\rule{4.pt}{0ex}}only\phantom{\rule{4.pt}{0ex}}if\phantom{\rule{4.pt}{0ex}}{\tilde{S}}_{0,i}=0$.
- 3.
- $G{M}_{i}^{+}=G{M}_{i}^{-}exp\left\{2\phantom{\rule{0.166667em}{0ex}}{\tilde{S}}_{0,i}^{2}\right\}$.
- 4.
- Given two units k and h ($k\ne h$) with ${\mu}_{p,k}={\mu}_{p,h},$ we have:$$\begin{array}{cc}\hfill G{M}_{k}^{-}>G{M}_{h}^{-}& \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}iff\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\tilde{S}}_{0,h}^{2}>{\tilde{S}}_{0,k}^{2},\hfill \\ \hfill G{M}_{k}^{+}>G{M}_{h}^{+}& \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}iff\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\tilde{S}}_{0,k}^{2}>{\tilde{S}}_{0,h}^{2}.\hfill \end{array}$$
- 5.
- Given two units k and h ($k\ne h$) we have:$$\begin{array}{cc}\hfill G{M}_{k}^{-}>G{M}_{h}^{-}& \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}iff\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\mu}_{0,k}>{\mu}_{0,h}exp\left\{{\tilde{S}}_{0,k}^{2}-{\tilde{S}}_{0,h}^{2}\right\},\hfill \\ \hfill G{M}_{k}^{+}>G{M}_{h}^{+}& \phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}iff\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{\mu}_{0,k}>{\mu}_{0,h}exp\left\{{\tilde{S}}_{0,h}^{2}-{\tilde{S}}_{0,k}^{2}\right\}.\hfill \end{array}$$

**Proof.**

**Proposition**

**2.**

- 1.
- $\frac{\partial G{M}_{i}^{-}}{\partial {z}_{ik}}\ge 0$ for ${z}_{ik}\le {z}_{i}^{-}$ and has a local maximum at the point ${z}_{i}^{-},$
- 2.
- $\frac{\partial G{M}_{i}^{+}}{\partial {z}_{ik}}\ge 0$ for ${z}_{ik}\ge {z}_{i}^{+}$ and has a local minimum at the point ${z}_{i}^{+},$where ${z}_{i}^{\pm}={\mu}_{0,i}exp\left\{\mp \frac{m}{2(m-1)}\right\}.$

**Proof.**

**Proposition**

**3.**

**Proof.**

## 5. Empirical Findings

#### 5.1. The HDI: A Brief Introduction and Its Computation

#### 5.2. The Computation of the pHDI

#### 5.3. A Comparison between the Two Approaches

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof of Proposition**

**2.**

**Proof of Proposition**

**3.**

## References

- OECD/European Union/EC-JRC. Handbook on Constructing Composite Indicators: Methodology and User Guide; OECD Publishing: Paris, France, 2008; Available online: https://www.oecd-ilibrary.org/economics/handbook-on-constructing-composite-indicators-methodology-and-user-guide_9789264043466-en (accessed on 28 February 2022).
- Berger, R.L.; Casella, G. Deriving Generalized Means as Least Squares and Maximum Likelihood Estimates. Am. Stat.
**1992**, 46, 279–282. [Google Scholar] - Mazziotta, M.; Pareto, A. A non-compensatory approach for the measurement of the quality of life. In Quality of Life in Italy; Springer: Dordrecht, The Netherlands, 2012; pp. 27–40. [Google Scholar]
- Ijiri, Y. Fundamental Queries in Aggregation Theory. J. Am. Stat. Assoc.
**1971**, 66, 766–782. [Google Scholar] [CrossRef] - UNDP. Human Development Report 2010: The Real Wealth of Nations—Pathways to Human Development; UNDP: New York, NY, USA, 2010; Available online: http://hdr.undp.org/en/content/human-development-report-2010 (accessed on 28 February 2022).
- UNDP. Human Development Report 2007/2008-Fighting Climate Change: Human Solidarity in a Divided World; UNDP: New York, NY, USA, 2007; Available online: https://hdr.undp.org/sites/default/files/reports/268/hdr_20072008_en_complete.pdf (accessed on 28 February 2022).
- Box, G.E.P.; Cox, D.R. An analysis of transformations. J. R. Stat. Soc. Ser. B
**1964**, 26, 211–252. [Google Scholar] [CrossRef] - Casadio Tarabusi, E.; Palazzi, P. An index for sustainable development. BNL Q. Rev.
**2004**, 229, 185–206. [Google Scholar] - Casadio Tarabusi, E.; Guarini, G. An unbalance adjustment method for development. Soc. Indic. Res.
**2013**, 112, 19–45. [Google Scholar] [CrossRef] - El Gibari, S.; Gomez, T.; Ruiz, F. Building composite indicators using multicriteria methods: A review. J. Bus. Econ.
**2019**, 89, 1–24. [Google Scholar] [CrossRef] - Lai, E.; Lundie, S.; Ashbolt, N.J. Review of multi-criteria decision aid for integrated sustainability assessment of urban water systems. Urban Water J.
**2008**, 5, 315–327. [Google Scholar] [CrossRef] - Azapagic, A.; Perdan, S. An integrated sustainability decision-support framework Part II: Problem analysis. Int. J. Sustain. Dev. World Ecol.
**2005**, 12, 112–131. [Google Scholar] [CrossRef] - Charnes, A.; Cooper, W.W.; Rhodes, E.L. Measuring the efficiency of decision making units. Eur. J. Oper. Res.
**1978**, 2, 429–444. [Google Scholar] [CrossRef] - Diaz-Balteiro, L.; Gonzalez-Pachon, J.; Romero, C. Measuring systems sustainability with multicriteria methods: A critical review. Eur. J. Oper. Res.
**2017**, 258, 607–616. [Google Scholar] [CrossRef] - Roy, B. The outranking approach and the foundations of ELECTRE methods. Theory Decis.
**1991**, 31, 49–73. [Google Scholar] [CrossRef] - Brans, J.P.; Vincke, P.; Mareschal, B. How to select and how to rank projects: The PROMETHEE methods. Eur. J. Oper. Res.
**1986**, 24, 228–238. [Google Scholar] [CrossRef] - Greco, S.; Ishizaka, A.; Tasiou, M.; Torrisi, G. On the methodological framework of composite indices: A review of the issues of weighting, aggregation, and robustness. Soc. Indic. Res.
**2019**, 141, 61–94. [Google Scholar] [CrossRef] [Green Version] - Mazziotta, M.; Pareto, A. Un indicatore sintetico di dotazione infrastrutturale: Il metodo delle penalità per coefficiente di variazione. In Proceedings of the Lo Sviluppo Regionale nell’Unione Europea-Obiettivi, Strategie, Politiche, Atti della XXVIII Conferenza Italiana di Scienze Regionali, Bolzano, Italy, 28–28 September 2007; Available online: https://aisre.it/images/old_papers/Mazziotta-Pareto.pdf (accessed on 28 February 2022).
- Mazziotta, M.; Pareto, A. On a generalized non-compensatory composite index for measuring socioeconomic phenomena. Soc. Indic. Res.
**2016**, 127, 983–1003. [Google Scholar] [CrossRef] - Noorbakhsh, F.A. modified human development index. World Dev.
**1998**, 26, 517–528. [Google Scholar] [CrossRef] - Paul, S. A modified human development index and international comparison. Appl. Econ. Lett.
**1996**, 3, 677–682. [Google Scholar] [CrossRef] - Jha, R.P.; Bhattacharyya, K.; Mishra, D.; Pedgaonkar, S.P. Health Adjusted Human Development Index: A Modified Measure of Human Development. Int. J. Health Sci. Res.
**2017**, 7, 207–220. [Google Scholar] - Prados de la Escosura, L. Improving human development: A long-run view. J. Econ. Surv.
**2010**, 24, 841–894. [Google Scholar] [CrossRef] [Green Version] - Chakravarty, S.R. A generalized human development index. Rev. Dev. Econ.
**2003**, 7, 99–114. [Google Scholar] [CrossRef] - Alkire, S.; Foster, J.E. Designing the Inequality-Adjusted Human Development Index; OPHI Working Paper Series; 2010; Volume WP37, Available online: https://www.ophi.org.uk/wp-content/uploads/ophi-wp37.pdf (accessed on 28 February 2022).
- Mazziotta, C.; Mazziotta, M.; Pareto, A.; Vidoli, F. La sintesi di indicatori territoriali di dotazione infrastrutturale: Metodi di costruzione e procedure di ponderazione a confronto. Riv. Econ. Stat. Territ.
**2010**, 1, 7–33. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Distribution of the Health indicator ($LEI$). (

**b**) Distribution of the Education indicator ($EI$). (

**c**) Distribution of the Economic indicator ($II$).

**Figure 6.**Scatter plot of the harmonic mean version (in green), the arithmetic mean version (in black) of the HDI and the pHDI values (in red) vs. the HDI values.

**Figure 7.**Scatter plot of the ranking obtained with the harmonic mean version of HDI (in green), with the arithmetic mean version of HDI (in black) and with the pHDI (in red) vs. the ranking obtained with HDI ranking.

Variable | Definition | Unit | Range |
---|---|---|---|

LE | Life expectancy at birth | years | 53.3–84.9 |

EYS | Expected years of schooling | years | 5.0–22.0 |

MYS | Mean years of schooling | years | 1.6–14.2 |

GNIpc | GNI per capita (PPP international dollars) | dollars | 754.0–131,032.0 |

Variable | Mean | St. Dev | Median | CV | Skew | Kurtosis |
---|---|---|---|---|---|---|

$LE$ | $72.71$ | $7.39$ | $74.0$ | $10.16\%$ | $-0.55$ | $-0.41$ |

$EYS$ | $13.33$ | $2.94$ | $13.2$ | $22.06\%$ | $-0.11$ | $0.08$ |

$MYS$ | $8.73$ | $3.08$ | $9.0$ | $35.28\%$ | $-0.31$ | $-0.99$ |

$GNIpc$ | $20,219.76$ | $21,229.08$ | $12,707.0$ | $104.99\%$ | $1.76$ | $3.90$ |

Indicator | Mean | St. Dev | Median | Min | Max | Range | Skew | Kurtosis | CV |
---|---|---|---|---|---|---|---|---|---|

Health ($LEI$) | $0.81$ | $0.11$ | $0.83$ | $0.51$ | $1.00$ | $0.49$ | $-0.55$ | $-0.41$ | $13.58\%$ |

Education ($EI$) | $0.66$ | $0.17$ | $0.68$ | $0.25$ | $0.95$ | $0.69$ | $-0.35$ | $-0.77$ | $25.76\%$ |

Economic ($II$) | $0.71$ | $0.17$ | $0.73$ | $0.31$ | $1.00$ | $0.69$ | $-0.24$ | $-0.89$ | $23.94\%$ |

Comp Ind | Mean | St. Dev | Median | Min | Max | Range | Skew | Kurtosis | CV |
---|---|---|---|---|---|---|---|---|---|

HDI | $0.72$ | $0.15$ | $0.74$ | $0.39$ | $0.96$ | $0.56$ | $-0.32$ | $-0.92$ | $20.83\%$ |

pHDI | $0.71$ | $0.16$ | $0.74$ | $0.34$ | $0.96$ | $0.62$ | $-0.37$ | $-0.85$ | $22.54\%$ |

Country | HDI | pHDI | Difference |
---|---|---|---|

Maldives | 95 | 110 | −15 |

Syrian Arab Republic | 151 | 161 | −10 |

Qatar | 45 | 53 | −8 |

Lebanon | 92 | 100 | −8 |

Cuba | 70 | 77 | −7 |

Armenia | 81 | 75 | 6 |

Mongolia | 99 | 93 | 6 |

Lesotho | 165 | 159 | 6 |

Guinea-Bissau | 177 | 170 | 7 |

Nigeria | 161 | 153 | 8 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mariani, F.; Ciommi, M.
Aggregating Composite Indicators through the Geometric Mean: A Penalization Approach. *Computation* **2022**, *10*, 64.
https://doi.org/10.3390/computation10040064

**AMA Style**

Mariani F, Ciommi M.
Aggregating Composite Indicators through the Geometric Mean: A Penalization Approach. *Computation*. 2022; 10(4):64.
https://doi.org/10.3390/computation10040064

**Chicago/Turabian Style**

Mariani, Francesca, and Mariateresa Ciommi.
2022. "Aggregating Composite Indicators through the Geometric Mean: A Penalization Approach" *Computation* 10, no. 4: 64.
https://doi.org/10.3390/computation10040064