Methods for Constructing Non-Compensatory Composite Indices: A Comparative Study

Abstract

Composite indices are being more and more used to measure multidimensional phenomena in social sciences. Considerable attention has been devoted in recent years to the methodological issues associated with index construction, such as non-compensability and comparability of the data over time. The aim of this paper is to compare two non-additive approaches: the Mazziotta–Pareto Index (MPI) and the Weighted Product (WP) method. The MPI is a nonlinear composite index that rewards the units with ‘balanced’ values of the individual indicators. The WP method implicitly penalizes the ‘unbalance’ and allows building, for each unit, two closely interrelated composite indices: a ‘static’ index for space comparisons and a ‘dynamic’ index for time comparisons. The MPI entails an equal weighting of the indicators, and only relative time comparisons are allowed. The indices based on the WP method give more weight to low values, and allow for both absolute and relative time comparisons. An application to indicators of well-being in the Italian regions in 2006 and 2011 is presented.

Keywords::

• composite indices
• normalization
• aggregation
• ranking

JEL classifications::

• C43
• I31

Notes

¹ Techniques of aggregating indicators that give explicit weights to each indicator and sum the product of each indicator and its weight (the sum of all the weights must be 1). Additive methods are advantageous because of their methodological transparency.

² Note that the theoretical part (definition of the phenomenon and selection of the indicators) is not separate from the statistical/methodological part (normalization and aggregation). In particular, the definition of the aggregation method is not independent of the choice of the individual indicators.

³ A weighting system can be implicitly or explicitly defined. For example, in the case of a simple arithmetic mean, the choice of equal weights is made implicitly by dividing by the total number of individual indicators. However, weighting of indicators also depends on the method of normalization.

⁴ The PCA is a multivariate statistical method that, starting from a large number of individual indicators, allows us to identify a small number of composite indices (factors or components) that explain most of the variance observed. The composite indices so obtained are linear combination of the individual indicators with weights that maximize the variation in the aggregated index values, over all possible choices of weights.

⁵ The geometric mean is an average that summarizes sets of positive values, which are interpreted according to their product and not their sum. It is defined as the nth root of the product of n values. The geometric mean is always less than or equal to the arithmetic mean. It is equal to the arithmetic mean when all the values are equal and it is less when the values are unbalanced. The more unbalanced the values are, the more ‘biased’, i.e. lower, the geometric mean is. Thus, it may be useful for measuring phenomena like development (for example, see the HDI), but not poverty.

⁶ The MCA is a set of tools for comparing and ranking different units (e.g. countries) on the basis of a set of criteria (e.g. indicators). A large number of MCA methods exist depending on the decision rule used (compensatory, partially compensatory or non-compensatory) and the type of data available (quantitative, qualitative or mixed). However, the MCA provides results in terms of ranks, and not of an index, so the researcher can only follow the unit rankings though time.

⁷ The Jevons index is a composite index used in economics to calculate elementary price indices. It is defined as the unweighted geometric mean of the price ratios, which is identical to the ratio of the unweighted geometric mean prices.

⁸ The Canadian Index of well-being is a composite index based on eight domains, each of which includes eight individual indicators. The domains are as follows: ‘Living Standards’, ‘Healthy Populations’, ‘Community Vitality’, ‘Democratic Engagement’, ‘Leisure and Culture’, ‘Time Use’, ‘Education’ and ‘Environment’. Indicators are normalized by indicization, i.e., a transformation in index numbers or percentage changes, from a base year. The arithmetic mean is calculated within each domain, and thereafter the arithmetic mean between the domains. So, the final index is calculated as the arithmetic mean of the 64 normalized indicators.