Non-compensatory Composite Indices for Measuring Changes over Time: A Comparative Study

Transcription

1 Non-compensatory Composite Indices for Measuring Changes over Time: A Comparative Study Matteo Mazziotta and Adriano Pareto Italian National Institute of Statistics

2 Introduction In the recent years a large number of composite indices of socioeconomic development and well-being have been proposed in literature. Important issues concerning the composite inde construction are: non-compensability among the components of the inde; comparability of the data over time. Compensability = a deficit in one dimension can be compensated by a surplus in another (e.g., arithmetic mean) Comparability = a comparison over time may be absolute or relative. Comparability firstly depends on the normalization rule In this work, we compare two unbalance adustment methods: Adusted Mazziotta-Pareto Inde (AMPI); Mean-Min Function (MMF). 1

3 Composite inde construction Main steps Selecting a group of elementary indicators Normalizing elementary indicators to make them comparable Aggregating the normalized indicators by composite indices Problems Finding data Losing information Researcher arbitrariness Advantages Unidimensional measurement of the phenomenon Immediate availability Simplification of the data analysis 2

4 Adusted MPI (AMPI) Basic idea The MPI is a non-compensatory composite inde based on a normalization of the data, at the reference time, that converts the indicators to a common scale with mean=100 and standard deviation=10. Therefore, only relative comparisons over time are allowed. In order to perform absolute comparisons over time, we propose a re-scaling of the data in the range (70; 130) according to two goalposts, i.e., a minimum and a maimum value which represent the possible range of each indicator for all time periods and for all units. Requirements Space-time comparisons Simplicity of computation Easy interpretation 3

5 Adusted MPI (AMPI) Formula 1) Normalization Given the matri X={ i } with n rows (units) and m columns (indicators), we calculate the normalized matri R={r i } as follow: r i ( i (Ma Min Min ) ) (1) 4 where i is the value of the indicator for the unit i; Min and Ma are the goalposts for the indicator. If the indicator has negative polarity, then the complement of (1) with respect to 200 is computed.

6 Adusted MPI (AMPI) 2) Aggregation M ri S ri Denoting with and, respectively, the mean and the standard deviation of the normalized values for the unit i, the generalized form of the Adusted MPI is given by: AMPI /- i M r i S r i cv i where cvi Sr i Mr i is the coefficient of variation for the unit i and the sign ± depends on the kind of phenomenon to be measured. For a positive composite inde (e.g., development), the AMPI - is used; for a negative composite inde (e.g., poverty), the AMPI + is used. 5

7 Adusted MPI (AMPI) 6 How to set the goalposts To facilitate the interpretation of results, we suggest to choose the goalposts so that 100 represents a reference value (e.g., the average in a given year). A simple procedure for setting the goalposts is the following. Let Inf and Sup be the overall minimum and maimum of the indicator across all units and all years. Denoting with Ref the reference value for the indicator, the goalposts are defined as: where Δ (Sup Inf ) / 2. Min Ma Ref Δ Δ The R values will fall approimately in the range (70; 130). Ref

8 Mean-Min Function (MMF) MMF formula Given a normalized matri Z={z i }, the MMF is defined as: MMF i M zi (M zi min{ z i 2 }) 2 where the parameters 0 1 and 0 are respectively related to the intensity of penalization of unbalance and intensity of complementarity between indicators. The function reduces to the arithmetic mean for 0 (no unbalance is penalized) and to the minimum function for 1 and 0 (unbalance has maimum penalization). 7

9 An application to real data AMPI vs. MMF In order to compare AMPI and MMF, we consider a set of 5 indicators of development in Italy, at the regional level, in 2004 and Since we are measuring the development, the AMPI - is used A MMF with proportional compensability ( 0 0 (MMF1) 0.5 (MMF2) 1 (MMF3) ) is considered, for: The normalization procedure for calculating the MMF is given by: z i ( i (Ma Min Min ) where the goalposts were set so that: 100 = Italy s value in ) 8

10 An application to real data Indicators of development I1) Life epectancy at birth - years (positive polarity); I2) Income distribution inequality - Gini coefficient (negative polarity); I3) Employment rate for people aged % (positive polarity); I4) People aged with low education level - % (negative polarity); I5) Greenhouse gas emissions - CO 2 equivalent tons per capita (negative polarity). 9

11 An application to real data Individual indicators of development in the Italian regions - Years 2004, 2011 Region I1 I2 I3 I4 I5 I1 I2 I3 I4 I5 Piemonte Valle d'aosta Liguria Lombardia Bolzano/Bozen Trento Veneto Friuli-V.G Emilia-Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna Italia

12 An application to real data Composite Indices of development in the Italian regions - Year 2004 Regions AMPI MMF1 MMF2 MMF3 Value Rank Value Rank Value Rank Value Rank Difference of rank MMF1 MMF2 MMF3 11 Piemonte Valle d'aosta Liguria Lombardia Bolzano/Bozen Trento Veneto Friuli-V.G Emilia-Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna Italia Mean absolute difference Spearman's rank correlation

13 An application to real data Composite Indices of development in the Italian regions - Year 2011 Regions AMPI MMF1 MMF2 MMF3 Value Rank Value Rank Value Rank Value Rank Difference of rank MMF1 MMF2 MMF3 12 Piemonte Valle d'aosta Liguria Lombardia Bolzano/Bozen Trento Veneto Friuli-V.G Emilia-Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna Italia Mean absolute difference Spearman's rank correlation

14 An application to real data Composite Indices of development in the Italian regions - Years (Variation) Regions AMPI MMF1 MMF2 MMF3 Value Rank Value Rank Value Rank Value Rank Difference of rank MMF1 MMF2 MMF3 13 Piemonte Valle d'aosta Liguria Lombardia Bolzano/Bozen Trento Veneto Friuli-V.G Emilia-Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Sardegna Italia Mean abs. difference Spearman's correlation

15 Conclusions The construction of a composite inde requests the respect of specific constraints that are derived from the obectives of the study. The need to make comparisons over time obliges the researcher to use only a few normalization methods of the original indicators: in this contet, the Min-Ma function satisfies this need. If you add the additional constraint of non-substitutability of the individual indicators, it is necessary to model the composite inde both in the normalization and aggregation step. The application shows that the AMPI respects both the constraint of comparisons over time and the non-compensability by using an easier and more transparent methodology than the MMF. 14

16 References Casadio Tarabusi E., Guarini G. (2012). An Unbalance Adustment Method for Development Indicators. Soc. Indic. Res., doi: / s De Muro P., Mazziotta M., Pareto A. (2011). Composite Indices of Development and Poverty: An Application to MDGs. Soc. Indic. Res., 104: 1-18 Mazziotta M., Pareto A. (2013). A Non-compensatory Composite Inde for Measuring Well-being over Time. Cogito. Multidisciplinary Research Journal, V, 4: OECD (2008). Handbook on Constructing Composite Indicators. Methodology and user guide. Paris: OECD Publications 15