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On the Measurement of Happiness and of its Inequality

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Abstract

Given that happiness or satisfaction with life is generally measured via an ordinal variable, the question arises as to how one should measure average happiness or satisfaction with life in a country, given that there is no arithmetic mean when dealing with ordinal variables. The same issue exists as far as deriving measures of inequality in happiness or in satisfaction with life is concerned, since traditional inequality indices cannot really be used with ordinal variables. The objective of this paper is to adopt recent suggestions made in the literature concerning the distribution of self-assessed health, an ordinal variable, and to propose new measures of the inequality in happiness or satisfaction with life and of the overall achievement in happiness or satisfaction with life. We apply the indices introduced in this literature and compare them with more traditional measures. Our empirical illustration is based on the World Values Surveys for the years 1995–98 (wave 3) and 2010–14 (wave 6), uses the data on satisfaction in life and covers 31 countries.

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Notes

  1. Among the papers that included measures of inequality in happiness without emphasizing too much methodological issues, one may mention those of Ott (2005), Ovaska and Takashima (2010), Van Praag (2011), Balestra and Ruiz (2015), Niimi (2018), Jordá et al. (2019), Zaborskis et al. (2019), Kollamparambil (2020a) (2020b).

  2. A Likert (1932) scale is a rating system aiming at measuring people’s attitudes, opinions, or perceptions. Subjects have a choice between different possible answers to a question or a statement. These possible answers could, for example be: “strongly agree”, “agree”, “am neutral”, “disagree” and “strongly disagree.” In some cases the categories of possible answers are coded numerically.

  3. A Visual Analogue Scale (VAS) is a measurement instrument that attempts to measure a characteristic or attitude that is difficult to measure directly and is assumed to range across a continuum of values. It has often been used in clinical research to measure, for example, the intensity of pain that a patient feels.

  4. Responders are asked to answer to the following question: “Imagine a ladder/mountain with steps numbered from zero at the bottom to ten at the top. Suppose we say that the top of the ladder/mountain represents the best possible life for you and the bottom of the ladder/mountain represents the worst possible life for you. If the top step is 10 and the bottom step is 0, on which step of the ladder/mountain do you feel you personally stand at the present time?”.

  5. A few numerical illustrations of the problems that occur when using cardinal indices when only ordinal variables are available are given in “Appendix 2”.

  6. See, (Leik, 1966; Berry and Mielke, 1992; Blair and Lacy, 2000; Van der Cees, 2001; Van der Eijk, 2001; Van Doorslaer and Jones, 2003; Allison and Foster, 2004; Apouey, 2007; Tastle and Wierman, 2007; Abul Naga and Yalcin, 2008; Zheng, 2008; Abul Naga and Yalcin, 2010; Madden, 2010; Kalmijn and Arends, 2010; Giudici and Raffinetti, 2011; Kobus and Miłos, 2012; Costa Font and Cowell, 2013; Apouey and Silber, 2013; Lazar and Silber, 2013; Gravel et al., 2014; Schoder, 2014; Abul Naga and Stapenhurst, 2015; Lv et al., 2015; Kobus, 2015; Peñaloza, 2016; Yalonetzky, 2016; Cowell and Flachaire, 2017; Allanson, 2017; Schroeder and Yitzhaki, 2017; Cowell et al., 2017; Gravel et al., 2019; Kobus et al., 2019).

  7. This measure was also derived by Apouey (2007).

  8. Lv et al. (2015) stressed the fact (page 469, lines 17–19) that the index \(I_{{LWX1}}\) can be considered as the absolute Gini index of the ranks of the different categories (“happiness” categories in our case). Note that the absolute Gini index is equal to half the mean difference, a well-known measure of absolute dispersion.

    As far as the index \(I_{{LWX2}}\) is concerned, note first that the expression (K−1) appears in the definition of both \(I_{{LWX1}}\) and \(I_{{LWX1}}\). But (K−1) is in fact the maximum value of \(\left| {h - k} \right|\). In the definition of \(I_{{LWX1}}\), we observe that \(\left| {h - k} \right|\) is divided by its maximum value while in the definition of \(I_{{LWX2}}\) it appears that \(\left| {h - k} \right|\) is deducted from its maximum value. It is also easy to check that when the parameter α → 1, \(I_{{LWX2}}\) becomes very close to \(I_{{LWX1}}\). The parameter α allows one therefore to vary the weight we want to give to the gap (K−1) − \(\left| {h - k} \right|\).

  9. The measure they propose is derived axiomatically using five axioms: a continuity axiom, an anonymity axiom, a scale invariance axiom, an independence axiom allowing them to define their index as a function of a sum of functions of status and a monotonicity in distance axiom according to which modifications in status that increase the distance between the status and the reference point are assumed to increase inequality.

  10. A referee drew our attention to some possible advantage of the index proposed by Cowell and Flachaire (2017) when compared to the Abul Naga and Yalcin index. Consider a 3 point scale with proportions as follows: (0.5, 0.0, 0.5). In this case the index \({\mathrm{I}}_{\mathrm{A}\!\mathrm{Y}}\) is equal to 1. Add now extra blank steps at the extremes to have proportions (0.0, 0.5, 0.0, 0.5, 0.0). It then turns out that \({\mathrm{I}}_{\mathrm{A}\mathrm{Y}}=0.5\) while the index \({I}_{\alpha}\left(s,e\right)\) introduced by Cowell and Flachaire (2017) would keep the same value.

  11. The surveys we use included also data on self-assessment of happiness but for this question the respondents were asked to use a four-point scale. We therefore thought that it would be better to focus on the answers given to questions on satisfaction in life as in this case respondents had a choice between ten possible answers.

  12. Reardon (2009) did not derive his index axiomatically but Lv et al. (2015) showed that the index of Reardon that we use is actually equivalent to the index they derived axiomatically and which is shown in Eq (8).

  13. We thank an anonymous referee for suggesting to add to the Gini, Atkinson and Theil indices the standard deviation and the coefficient of variation.

  14. We computed bootstrap 5%-95% confidence intervals.

  15. The Pearson correlation between the rankings of countries is identical to the Spearman rank correlation.

  16. Most of the results presented in Atkinson (1970) appear in fact in Kolm (1969), but Atkinson was not aware of this.

  17. The cumulative relative frequency \(F_{k} \left( s \right)\) is defined as \(F_{k} \left( s \right) = \mathop \sum \nolimits_{{h = 1}}^{k} p_{h} \left( s \right)\)

  18. Remember that equations (11) to (15) ignore the axioms of equity and proportion equality.

  19. We will suppose henceforth that whenever \( \alpha \to 1, \) α will be assumed to be equal to 0.999.

  20. It does not matter whether they label the ordinal categories 1,2,3,4,5 or 1,3,5,7,9 or 1,6,11,16,21, etc…

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Authors and Affiliations

  1. University of Toulon, LEAD, Toulon, France

    Valérie Bérenger

  2. Université Côte d’Azur, CNRS, GREDEG, Sophia Antipolis, France

    Valérie Bérenger

  3. Department of Economics, Bar-Ilan University, Ramat Gan, Israel

    Jacques Silber

  4. LISER (CEPS/INSTEAD), Esch-sur-Alzette, Luxembourg

    Jacques Silber

  5. Centro Camilo Dagum, Advanced Statistics for Equitable and Sustainable Development, Tuscan Interuniversity Centre, Florence, Italy

    Jacques Silber

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  1. Valérie Bérenger
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Correspondence to Jacques Silber.

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Appendices

Appendix 1

See Table 8.

Table 8 List of countries
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Appendix 1.1 A Simple Illustration on the Limits of Cardinal Indices when Working with Ordinal Data

Assume now that we use the Gini index when analyzing ordinal data. Let us suppose first that individuals who are asked to give a self-assessment of their satisfaction with life have a choice between four possible answers, category 1 referring to the lowest and category 4 to the highest level of satisfaction with life. The numbers associated with these four categories are hence 1, 2, 3 and 4. Assume that there are 9 individuals who choose category 1, that no individual chooses category 2 or 3 and that one individual chooses category 4. It is then easy to show that the traditional Gini index of the distribution of levels of satisfaction with life will be equal to 0.208.

Assume now that in another survey covering this time 100 individuals, rather than 10, 99 individuals choose level 1, no one chooses levels 2 or 3 and one individual chooses level 4. It is then easy to check that the Gini index will now be equal to 0.0288.

If instead of 99 individuals choosing category 1, we had now 999 individuals choosing this category, in a survey covering 1000 individuals, the observations in the other 3 categories being as in the two previous examples, we would find that the Gini index is equal to 0.00299.

We therefore observe that when one individual is in the highest category and all the other individuals in the lowest category, the Gini index will be lower, the higher the number of individuals who are in the lowest category. Such a conclusion is in complete contradiction with what we know about the Gini index as will now be shown. Assume 10 individuals, 9 of them have a zero income and one has an income of 10. The Gini index is then equal to 0.9. If we now suppose that 99 individuals have a zero income and one an income of 10, the Gini index will be equal to 0.99. And if there are 999 individuals with a zero income and one with an income of 10, the Gini index will be equal to 0.999. The same results would evidently be obtained if the only individual with a positive income had an income of 100, 1000 or whatever positive number one selects. In short when all the individuals but one are in the worst state (with a zero income) and one individual only has a positive income, the Gini index is higher, the higher the number of individuals in the worst income state.

Assume that there are four happiness categories and that the cardinal numbers assigned to these categories are 0, 1 2 and 3, rather than 1, 2, 3 and 4. We would then discover that if 9 individuals are in the lowest category (given a value of 0) and 1 individual is in the highest category (no one is in the other categories), the Gini index will be equal to 0.9. If 99 individuals are in the lowest category and one in the highest (and still no one in the other categories) the Gini index will be equal to 0.99. Finally if 999 individuals are in the lowest category and one in the highest (and still no one in the other categories) the Gini index will be equal to 0.999. In other words when all the individuals but one are in the lowest category and one in the highest, the Gini index is now higher, and not lower, the higher the number of individuals in the lowest category. Which value to assign to each category is hence a crucial issue and, depending on the values selected, one can reach completely opposite conclusions.

We can also look at another critical illustration, one where half the individuals are in the lowest category and half in the highest. Following Abul Naga and Yalcin (2008), all the indices of inequality introduced in the literature for the case of ordinal variables (see, for example, Reardon, 2009, Lazar and Silber, 2013, Lv et al., 2015) have assumed that inequality will be maximal when half the individuals are in the lowest category and half in the highest. Let us now see what happens when we use the Gini index and assume, as before, equidistance between the different categories. Assuming that half of the individuals are in the lowest category and half in the highest and that the four categories receive respectively the values 1, 2, 3 and 4 (or 10, 20, 30 40; or 100. 200. 300, 400, etc…). We will then find that the Gini index is equal to 0.3. If we assign to the four categories the values 0, 1 2 and 3 (or, for example, 0, 100, 200 and 300) and still assume that half of the individuals are in the lowest and half in the highest category, the Gini index will be equal to 0.5. If we assign to the four categories the values 2, 3, 4, and 5 the Gini index will be equal to 0.214 and if, for example, the four categories receive respectively the values 97, 98, 99 and 100, the Gini index will be equal to 0.00761. In other words, when half of the population is in the lowest category and half in the highest, assuming equidistance between the different categories, the Gini index will be lower, the more equal in relative terms the values assigned to the different categories.

To summarize what we have stressed, we can first say that the implicit assumptions lying behind the use of the Gini index when dealing with ordinal variables are quite different from the assumptions made by the inequality measures that have been proposed to deal with cardinal variables. Second the Gini index (or any other cardinal measure of inequality) will be very sensitive to the values assigned to the different categories whereas the inequality indices introduced in the literature when working with ordinal variables do not depend at all on the values assigned to the different categories.

Appendix 2

See Tables 9, 10.

Table 9 Inequality in satisfaction with life: wave 3 (1995-1998)
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Table 10 Inequality in satisfaction with life wave 3 (1995–1998)
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Appendix 2.1

See Tables 11, 12, 13 and 14.

Table 11 Inequality in satisfaction with life: wave 6 (2010–2014)
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Table 12 Inequality in satisfaction with life (cardinal indices): wave 6 (2010–2014)
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Table 13 Country ranking comparisons for satisfaction with life (wave 6)
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Table 14 Correlation between the various indices (wave 6)
Full size table

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Bérenger, V., Silber, J. On the Measurement of Happiness and of its Inequality. J Happiness Stud 23, 861–902 (2022). https://doi.org/10.1007/s10902-021-00429-7

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