Weights in Multidimensional Indices of Wellbeing: An Overview

Abstract

Multidimensional indices are becoming increasingly important instruments to assess the wellbeing of societies. They move beyond the focus on a single indicator and yet they are easy to present and communicate. A crucial step in the construction of a multidimensional index of wellbeing is the selection of the relative weights for the different dimensions. The aim of this article is to study the role of these weights and to critically survey eight different approaches to set them. We categorize the approaches in three classes: data-driven, normative, and hybrid weighting, and compare their respective advantages and drawbacks.

Keywords:

• Composite indicator
• Multidimensional wellbeing index
• Weights

JEL Classification:

• I31
• C43
• O1

ACKNOWLEDGMENTS

We thank Sabina Alkire, Tony Atkinson, Laurens Cherchye, Lucio Esposito, Francisco Ferreira, Alessio Fusco, Marc Fleurbaey, James E. Foster, Ori Heffetz, Nicky Rogge, Luc Van Ootegem, Elsy Verhofstadt as well as participants at the OPHI workshop on Weighting Dimensions (2008). We are especially grateful to Sabina Alkire for encouraging us to write this article. We also received valuable comments and remarks on this or a previous version of the article at the AEA (2009) conference in San Francisco and the HDR 2010 consultation in Rabat. We are very grateful for the helpful comments we received from two anonymous referees and the guest editor of this issue. Finally, María Ana Lugo thanks the UK Economic and Social Research Council for financial support. The usual disclaimer applies.

Notes

In a recent report of the Commission on the Measurement of Economic Performance and Social Progress, chaired by Nobel prize laureates Stiglitz et al. (2009, p. 14), the authors write “To define what wellbeing means, a multidimensional definition has to be used. Based on academic research and a number of concrete initiatives developed around the world, the Commission has identified the following key dimensions that should be taken into account. At least in principle, these dimensions should be considered simultaneously: (i) Material living standards (income, consumption and wealth); (ii) Health; (iii) Education; (iv) Personal activities including work; (v) Political voice and governance; (vi) Social connections and relationships; (vii) Environment (present and future conditions); (viii) Insecurity, of an economic as well as a physical nature. All these dimensions shape people's wellbeing, and yet many of them are missed by conventional income measures.”

The European Commission gives an overview of wellbeing indices used by national and international institutions on http://composite-indicators.jrc.ec.europa.eu/.

We refer the reader to Micklewright (2001) for a detailed treatment of advantages and disadvantages of using a multidimensional wellbeing index.

Note that can also denote shortfall in achievement to a predefined threshold, and the resulting index represents the extent of multidimensional poverty. Indeed, in reviewing the literature on how to set weights (section four), we also include examples where these approaches have been used to construct multidimensional poverty indices.

Blackorby and Donaldson (1982) provide an axiomatic characterization of the weighted mean of order β. In the literature of multidimensional inequality, Maasoumi provides an information-theoretic justification of this class of wellbeing indices for multiple dimensions (Maasoumi, 1986) and for incomes in multiple times (Maasoumi and Jeong, 1985). Further, it belongs to the wider class of wellbeing indices proposed by Bourguignon (1999) while it is similar to the proposal by Foster et al. (2005) for a distribution-sensitive measure of human development. Decancq and Lugo (2012) axiomatically characterize it as part of a multidimensional Gini measure and Decancq et al. (2009) have used it to analyze the trend in multidimensional global inequality.

For a more extensive survey of the various transformation procedures and their properties we refer the reader to Jacobs et al. (2004) and Nardo et al. (2005).

Other commonly used values of β are β = 0 and β → − ∞. The first case is favored by Ebert and Welsch (2004) when dimensions are measured in different measurement units. It implies a unit elasticity of substitution between transformed achievements, in other words, a one percent decrease in one of the dimensions can be compensated by a one percent increase in another dimension. The second case, where β → − ∞, assumes that the elasticity of substitution is equal to zero, which means that there is no possible substitution between dimensions. In this case, the wellbeing index becomes the minimum of the transformed achievements across the dimensions.

Legend: 1 = frequency based; 2 = statistical; 3 = most favorable; 4 = equal or arbitrary; 5 = expert; 6 = price based; 7 = self-stated; 8 = hedonic;

A =identity transformation; B =rescaling; C =linear transformation; D =increasing transformation; E =other transformation; S =sensitivity analysis.

^#afterwards an increasing transformation of the index is applied;

^†Indicators;

^‡Dimensions.

A striking example can be found in the work by Becker et al. (1987). The authors studied the quality of life in 329 metropolitan areas of the U.S. by ordering them according to standard variables such as quality of climate, health, security, and economical performance. The authors find that, depending on the weighting scheme chosen, there were 134 cities that could be ranked first, and 150 cities that could be rank last. Moreover, there were 59 cities that could be rated either first or last, using the same data, but by selecting alternative weighting schemes.

See also Anand and Sen (1997) for a similar approach.

Graphically, the MRS _{j 1, j 2} reflects the slope of the iso-wellbeing curves in the (x _{j 1} , x _{j 2} ) space.

Note, however, that the rate at which dimensions are traded off, measured in its original units and not transformed, is constant (though not perfect) between the pair health-education, and non-constant for health-income and education-income pairs. Due to the log transformation employed to per capita GDP, the trade-off between, say, per capita GDP and life expectancy also depends on the level of income the country achieves. In particular, the amount of money required to compensate for a year more of life expectancy is increasing in income; for a rich country such as Belgium an extra year of life expectancy is valued at nearly 7,000 US$ (in purchasing power parity terms) which for a relative poor country, such as Cote d'Ivoire this is merely 300 US$. Therefore, contrary to the claim above, the Human Development Index does indeed allow for compensation between dimensions, even when this compensation might vary across levels.

We thank one of the referees for making us aware of this literature.

Deutsch and Silber (2005, p. 150) give the following example: “If owning a refrigerator is much more common than owning a dryer, a greater weight should be given to the former indicator so that if an individual does not own a refrigerator, this rare occurrence will be taken much more into account in computing the overall degree of poverty than if some individual does not own a dryer, a case which is assumed to be more frequent.”

For instance, Srinivasan (1994) reports a correlation coefficient of about 0.8 between the dimensions of the Human Development Index. Whether double counting is really a problem for constructing multidimensional indices of wellbeing or not is a matter open for discussion. One could argue that, indeed, the existence of correlation between the dimensions of wellbeing in a society reflects an important aspect of the society's situation and, as such, it should be included, and not eliminated, from the analysis. The pluralistic egalitarian notion of Walzer (1983), for instance, considers that the correlation between the dimensions is one of the essential characteristics of a society. From that perspective, correcting for correlation between the dimensions might be deemed inappropriate.

In the case that β equals one, this problem reduces to linear programming problem. Cherchye et al. (2008) provide technical details. For β equal to one (the multiplicative case) the same methodology can be applied after a logarithmic transformation of the data (Cherchye et al., 2007a, footnote 11). In a recent article, Zhou et al. (2010) use a multiplicative wellbeing index and compute most favorable weights as well as the least favorable weights and all convex combinations of these exteme cases.

Despotis (2005b) proposes a way of using the individual most favorable weights to construct a common weighting structure. This is done by minimizing the distance between the individual (country)-specific weight and the global weights.

Strictly speaking, equal weighting assigns 1/m weight to all m dimensions included in the wellbeing index and zero weight to all dimensions of wellbeing not included.

One may be concerned that respondents do not take a sufficiently ‘publicly-oriented’ point of view when considering minimal standards of living, but rather that they are influenced by their own possessions. Using a data set for Belgium, Van den Bosch (1998) studies the relation between the individual possessions and perceptions of necessity and finds that respondents can reasonably well distinguish between what they themselves have and what should be considered a necessity for others.

On the other hand, multicollinearity does not bias the results and, in particular, the reliability of the prediction of overall wellbeing for each individual.

For a detailed treatment of sensitivity analysis, we refer the reader to Saltelli et al. (2004) and Saltelli et al. (2008). An early example of sensitivity of results to the choice of weights (and other parameters) can be found in Maasoumi and Nickelsburg (1988) – also in Maasoumi and Jeong (1985) and Maasoumi and Zandvakili (1986) though applied to multiple periods instead of multiple dimensions of wellbeing. More recent examples within the multidimensional wellbeing literature include Justino (2005) and Zhou et al. (2010) who use ranges of weights—see cases marked with ‘S’ in the last column of table. In a few other cases, two or three weighting schemes were computed for robustness—see, for instance, Battiston et al. (forthcoming), Deutsch and Silber (2005), Maasoumi and Lugo (2008), Osberg and Sharpe (2002).