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Use and Misuse of PCA for Measuring Well-Being

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Abstract

The measurement of well-being of people is very difficult because it is characterized by a multiplicity of aspects or dimensions. Principal Components Analysis (PCA) is probably the most popular multivariate statistical technique for reducing data with many dimensions and, often, well-being indicators are reduced to a single index of well-being by using PCA. However, PCA is implicitly based on a reflective measurement model that is not suitable for all types of indicators. In this paper, we discuss the use and misuse of PCA for measuring well-being, and we show some applications to real data.

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Notes

  1. For the sake of simplicity, only linear models will be considered.

  2. Some authors exclude the error term so that Eq. (2) reduces to a weighted linear combination of the Xi (Diamantopoulos 2006).

  3. Experts suggest that weights could be determined a priori, according to the theoretical contribution of the indicators to the concept (Howell et al. 2007). For Cadogan and Lee (2013), if there is no theory suggesting the contrary, individual indicators should have equal weightings.

  4. Because the formative measurement model is based on a multiple regression, the stability of the coefficients λi is affected by the strength of the indicator intercorrelations. Therefore, multicollinearity must be avoided. (Diamantopoulos and Winklhofer 2001).

  5. Individual indicators must have at least an interval level of measurement. For variables measured on nominal or ordinal scale, we recommend the use of Categorical Principal Components Analysis (CATPCA). For a introduction and application of CATPCA, see Linting et al. (2007).

  6. Often, the use of the first principal component as the ‘only’ composite index is a bad practice that reduces the PCA potentials.

  7. Normalization aims to make individual indicators comparable, as they often have different measurement units and/or different polarities. Normalized indicators are calculated by transforming individual indicators into pure, dimensionless, numbers, with positive polarity (Mazziotta and Pareto 2017).

  8. Principal components can be real features of the data, or more or less convenient fictions and summaries. That they are real is a hypothesis for which PCA can provide only a very weak evidence (Shalizi 2009).

  9. STATIS is an exploratory technique of multivariate data analysis for handling three-way matrices, where the same units have measures on a set of indicators under a number of conditions (Lavit et al. 1994).

  10. Several robust PCA methods have been introduced in the literature (Filzmoser 1999; Hubert et al. 2005), but they make the analysis resistant to outlying observations.

  11. There are two types of FA: exploratory and confirmatory. In this paper, we consider exploratory factor analysis (Fabrigar and Wegener 2011).

  12. Individual indicators were normalized as z-scores. The signs were reversed if the polarity is negative.

  13. The first factor of PCA accounts for 72.4% of the variance in the data.

  14. Note that, for constructing a composite index, all the normalized indicators must have positive polarity, so that an increase in each of them corresponds to an increase in the composite index (Mazziotta and Pareto 2013).

  15. Influence Analysis is a particular case of Uncertainty Analysis that aims to empirically quantify the ‘weight’ of each individual indicator in the calculation of the composite index (Mazziotta and Pareto 2017).

  16. Note that only individual indicators are released by Istat at the provincial level.

  17. A pillar describes a particular aspect—not directly observable—of the latent phenomenon by a set of individual indicators which are assumed to be related to it.

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Acknowledgements

The paper is the result of the common work of the authors: in particular M. Mazziotta has written Sects. 2.1, 3.2, 4 and A. Pareto has written Sects. 1, 2.2, 2.3, 2.4, 3.1.

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Authors and Affiliations

  1. Italian National Institute of Statistics, Viale Oceano Pacifico 171, 00144, Rome, Italy

    Matteo Mazziotta

  2. Italian National Institute of Statistics, Via A. Depretis 77, 00184, Rome, Italy

    Adriano Pareto

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  1. Matteo Mazziotta
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  2. Adriano Pareto
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Correspondence to Adriano Pareto.

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Mazziotta, M., Pareto, A. Use and Misuse of PCA for Measuring Well-Being. Soc Indic Res 142, 451–476 (2019). https://doi.org/10.1007/s11205-018-1933-0

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