The indicator of a happy, long and sustainable life can be presented as three single indicators, see Sects. 2, 3 and 4, and as a composite indicator. Here we explain how we constructed the indicator of a happy, long and sustainable life as a composite indicator, and we give the results we obtained. As Maggino, (2017) states, this process is a combination of objectivity and subjectivity.
Data
Happiness. We used the 2014 data on happiness from the Pew Research Center in which happiness is measured using the national values for answers to the Cantril ladder question asking respondents to value their lives today on a 0 to 10 scale, with the worst possible life as a 0 and the best possible life as a 10. The Pew Research Center provided free frequency distributions to calculate the negative utilitarianist measure of happiness. We then calculated the negative utilitarianist measure of happiness for each country in 2014, see Sect. 2.3 for the formula.
Note: the Cantril ladder has a weakness. It is a measure of contentment, the degree to which an individual perceives his or her aspirations are met, not a measure of happiness as liking the life one leads. Another weakness of the Cantril ladder is that it anchors in possible lives rather than in wanted lives.
Long life. We used the 2014 data on Potential Years of Life Lost from the OECD.
Sustainability. We collected the data of Global Footprint Network, an independent think tank, on biocapacity per country and ecological footprint per country for 2014. We then calculated the sustainability ratio for each country in 2014.
Data were not available for each country. We got the data we needed to calculate the results of the indicator of a happy, long and sustainable life for 15 countries. This lack of data was due to the fact that PYLL was not evaluated for a lot of countries, and to the fact we did not find a lot of free, valid data on happiness frequency distribution, see Table 2 the summary statistics of the three dimensions of the indicator of a happy, long and sustainable life before normalization.
Table 2 Summary statistics of the three dimensions of the indicator before normalization Measurement model, normalization and aggregation
Measurement model
Two kinds of measurement model exist: the reflective model and the formative model. In the reflective model, causality comes from the latent variable to the single indicators, a change in the latent variable may cause variations in the single indicators. In the formative model, it is the opposite, causality comes from the single indicators to the latent variable, and single indicators can also have positive, negative or zero correlations (Mazziotta & Pareto, 2019).
The choice between the two kinds of model depends on the nature of the phenomenon being measured and its definition (Maggino, 2017). The indicator of a happy, long and sustainable life as a composite indicator includes three single indicators, happiness, long life and sustainability, and any change in one or more of these indicators will cause a change in the composite indicator result. Correlations between single indicators can be positive, negative or null. The measurement model of the indicator of a happy, long and sustainable life is a formative model.
Normalization
Normalization aims at unifying different measurement units when data for all variables do not have a common or equivalent measurement. OECD and JRCFootnote 6, (2008) make a list of nine different methods to normalize data. Three methods could have been used to construct our composite indicator: standardization, min–max method and distance to a reference method. We chose to apply the min–max method. Standardization depends on the sample or population. Distance to a point does not take into account the minimal values.
For each single indicator for a given country, the mathematical formula of the min–max method is as follows:
$$\mathop I\nolimits_{ij} = \frac{{\mathop V\nolimits_{ij} - \mathop V\nolimits_{\min } \, }}{{\mathop V\nolimits_{\max } - \mathop V\nolimits_{\min } }}$$
(3)
where \(\mathop V\nolimits_{ij}\) is the collected data for the dimension \(i\), for the country \(j\), \(\mathop V\nolimits_{min}\) the minimum value and \(\mathop V\nolimits_{max}\) the maximum value.
OECD and JRC, (2008) suggest to use minimum and maximum values across all countries in the min–max method, but indicators such as the HDI were constructed using theoretical and subjectively selected values as minimum and maximum values.
We chose to use theoretical and subjectively selected values for three reasons. One, the theoretical and subjectively selected value are not dependent on a sample or a population. As a consequence, the results of an indicator for which we use these kinds of value are not dependent on a sample or a population. Two, using minimal and maximal values across all countries in the min–max method may give a false image of the reality. For example, we saw that a country, Mexico, had a score of 6.2 on 0 to 10 scale and, after a normalization for which we used the minimal and maximal values across countries, a score of 10 on a 0 to 10 scale. The reason for this discrepancy is that Mexico is the country with the highest degree of happiness in our sample when happiness is measured using our negative utilitarianist formula. Three, what matters is the distance between a country and the optimal and the worst conditions, not the distance to the other countries.
We chose the following theoretical and subjectively selected values as minimal and maximal values, see Table 3.
Table 3 The minimal and maximal theoretical and subjectively selected values of the sub-indicators the indicator of a happy, long and sustainable life PYLL is not calculated for a lot of countries. In our sample the country with the worst situation regarding PYLL is South Africa with 17,962 potential years of life lost. It is likely that some other countries have a much higher number of potential years of life lost. According to the World Bank, life expectancy in South Africa was approximately 61 years in 2014, while it was 50.6 in the Central Africa Republic. We can then hypothesize that the number of potential years of life lost is much higher in the Central Africa Republic. We chose 50,000 as the maximal value for PYLL, however, this choice will have to be reassessed, once PYLL has been calculated for most of the countries in the world.
If the sustainability ratio is strictly greater than 1.1, then we replace the result of the ratio by 1.1. See Sect. 4.3 for the reason.
Aggregation
All weighting methods are not compatible with all aggregation methods (OECD and the JRC, 2008). As we chose to use equal weighting, linear aggregation, multi-criteria aggregation and geometric aggregation, could be applied (OECD and the JRC, 2008).
The linear aggregation method is based on an assumption of perfect substitutability between the different components of a composite indicator. When the dimensions of an indicator are complementary, requiring an equal effort, the method resulting from the multi criteria analysis is more appropriate (Munda 2005; OECD and JRC 2008; Dialga and Le 2017; Dialga, 2018). Geometric aggregation allows for some flexibility of compensation between index components at given thresholds. This method of aggregation is increasingly used in the construction of indicators, e.g. the HDI from 2010 (Herrero, Martínez and Villar, 2010) and the Sustainability Index of Mining Countries (Dialga, 2018).
In addition, the three components of the indicator of a happy, long and sustainable life interact with each other. Geometric aggregation is precisely the method that allows us to take interaction effects into account. Moreover, it is reasonable to assume that the three variables selected make a decreasing marginal contribution to the indicator we construct. In other words, small values of these variables make a greater contribution to the composite scores of the countries, but as these variables take on larger values, they contribute less and less to the overall score. This is something valuable in a negative utilitarianist approach.
The mathematical formula of geometric aggregation is as follows:
$$\forall \mathop I\nolimits_{i} \succ 0,$$
$$\mathop {CI}\nolimits_{j} = \prod\limits_{i = 1}^{n} {\mathop {\left( {\mathop I\nolimits_{i} } \right)}\nolimits^{{\mathop w\nolimits_{i} }} }$$
(4)
where \(\mathop {CI}\nolimits_{j}\) denotes the composite score of the indicator for the country \(j\), \(\mathop I\nolimits_{i}\) is the normalized sub-indicator, and \(\mathop w\nolimits_{i}\) is the weight associated with \(\mathop I\nolimits_{i}\). As all dimensions measured by the sub-indicators \(\mathop I\nolimits_{i}\) are assumed to have the same importance, \(\mathop I\nolimits_{i}\) are equally weighted; thus,
$$\mathop w\nolimits_{i} = \frac{1}{n}$$
(5)
with \(n\) the number of sub-indicators.
Mathematical constraint \(\forall \mathop I\nolimits_{i} \succ 0\) must be fulfilled because, if one of the values is 0 in a geometric aggregation, then the result of the geometric aggregation is 0. As the indicator falls within a negative utilitarianist approach, it would not be an issue if this theoretical possibility occurred. In practice, none of the three dimensions of the composite indicator can take the value 0 after normalization. No country has each person of its whole population declaring they are at the lowest degree of a scale used to measure happiness. Sadly, no country has a population for which each individual reaches the age of 70. A country is by definition a territory where human beings live and, as a consequence, have an ecological footprint.
Note: despite its success, the aggregative approach, that includes linear and geometric aggregative methods, has been criticized as inappropriate and often inconsistent from both conceptual and methodological perspectives (Freudenberg, 2003; Maggino, 2017). The non-aggregative approach, mainly multi-criteria analysis, is increasingly used by the experts because of assumptions of non-compensability between dimensions. The non-aggregative approach allows some limits of the aggregative one to be overcome, as recently highlighted successively in Alaimo et al, (2020a), Alaimo et al, (2020b) and Alaimo et al, (2020c). Since the indicator of a happy, long and sustainable life is based on a negative utilitarianist approach, the geometric aggregation method is more appropriate because this method values (overestimates) lower values more than higher values. Geometric aggregation respects the conceptual framework of the indicator.
Controlling the relevance of our choices using the Mazziotta and Pareto flow chart
In order to use the Mazziotta and Pareto flow chart (2013), we had to answer 4 questions:
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Are our single indicators substitutable or non-substitutable? The components of a composite indicator are called ‘substitutable’ if a deficit in one component may be compensated by a surplus in another.
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Is our aggregation simple or complex? An aggregation method is called ‘simple’ if its mathematical function is easily understandable.
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Are our comparisons absolute or relative? The comparisons are called ‘absolute’ if the definition of extreme values is independent from the data.
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Are our weights objective or subjective? The weights are called ‘subjective’ if they are set by people rather using a mathematical method.
As our single indicators were non-substitutable, our aggregation simple, our comparisons absolute and our weights subjective, the min–max method was chosen.
The formula of the composite indicator of a happy, long and sustainable life
After normalization, the formula to determine the indicator of a happy, long and sustainable life as a composite indicator is:
Indicator of a happy, long and sustainable life = (Negative utilitarianist measure of happiness) x (1 - Potential Years of Life Lost per 100 000 inhabitants) x (Biocapacity/Ecological footprint)
Some of the brackets are mathematically useless, but they allow us to show the three components of the indicator: a measure of happiness, life duration and sustainability.
See the negative utilitarianist measure of happiness we chose in Sect. 2.3.
Potential Years of Life Lost works in the opposite way to the negative utilitarianist measure of happiness and the sustainability ratio. This is why the formula to take it into account for the PYLL is (1-PYLL).
If the sustainability ratio is strictly greater than 1.1, then we replace the result of the ratio by 1.1. See Sect. 4.3 for the reason.
We finally put a cubic root over the results of the formula to take into account the fact that the formula is the multiplication of three components.
The developed form of the composite indicator is:
$$\sqrt[3]{{\frac{{\frac{{\sum\limits_{{i = 1}}^{{i = 10}} {\mathop {\left( {\frac{{\mathop i\nolimits_{j} }}{{10}}} \right)}\nolimits^{2} } }}{{\sum {\mathop n\nolimits_{{ij}} } }} - \mathop V\nolimits_{{\min k}} }}{{\mathop V\nolimits_{{\max k}} - \mathop V\nolimits_{{\min k}} }}x\left( {1 - \frac{{\mathop {PYLL}\nolimits_{j} - \mathop V\nolimits_{{\min k}} }}{{\mathop V\nolimits_{{\max k}} - \mathop V\nolimits_{{\min k}} }}} \right)x\frac{{\frac{{\mathop {BIO}\nolimits_{j} }}{{\mathop {EF}\nolimits_{j} }} - \mathop V\nolimits_{{\min k}} }}{{\mathop V\nolimits_{{\max k}} - \mathop V\nolimits_{{\min k}} }}{\text{ }}}} $$
(6)
where i the degree of happiness assigned by the respondent, j the country, k the dimension of the indicator, happiness, long life and sustainability, PYLL the value of the Potential Years of Life Lost, BIO the biocapacity of the country j, and EF the ecological footprint of the country j.
Results and robustness
Why we prefer to use alphabetical order of countries over ranking
When a characteristic is measured by country, countries are often ranked from the most successful country to the least. We prefer to use alphabetical order to present the indicator of a happy, long and sustainable life. This is a philosophical choice, and we can give at least five reasons to support this choice. One, we think that living a happy, long and sustainable life should be for a country, and humanity as a whole, the main goals of public policy, however, some people and countries could disagree, and they would be right to do so because they could have a different answer to the question of what is really important for ourself. Two, even if countries have the same goals, it is possible to disagree on how hard and high to strive to reach each goal. Three, even if countries have the same goals and want to reach the same level for each goal, they may not have the same initial conditions and the same means. It is biased to compare a result without taking into account initial conditions and the means. Four, when it comes to happiness, it is a better strategy to focus on our own happiness rather than to make social comparison to be happier. Five, research on between-countries variations bring useful information and is always welcome, however, when ranking goes out the academic world, the understanding of rankings tend to become an underlying competition, and we want to avoid such a competition.
Results by country
The data we used allow us to show the cross-sectional results for 15 countries, see Table 4. We did not have enough data to show how the indicator of a happy, long and sustainable life evolves for a specific country, although we consider that this is the best way to present data.
Table 4 The data on the components of the indicator of a happy, long and sustainable life in 15 countries in 2014 and the results for the indicator A country where the inhabitants are very happy and live long may have bad result because its ecological footprint is much higher than its biocapacity. This is the case of Israel. People living in Israel are among the happiest in a negative utilitarianist approach among 15 countries, and have one of the lowest Potential Years of Life Lost per 100,000 inhabitants among the same 15 countries, but its sustainability ratio is the worst one and very close to zero.
Robustness
Dialga and Le, (2017) use, at different steps, the most used statistical methods to construct a composite indicator, compare the results they obtained, and demonstrate that these different statistical methods lead to different results. In their illustrative case, scores and ranking results fluctuate, although their results are robust.
We compared the results we obtained using the min–max method with the results of the distance to reference method. The results of the two methods were exactly the same. We explain this by the choice of our minimal values: all are 0. We did not compare with the standardization because this method is based on the features of the samples, not the choice of theoretical and subjective minimal and maximal values. We compared the results we obtained using geometric aggregation with the results of the linear aggregation, see Table 5. The correlation between results obtained using geometric aggregation and those obtained using linear aggregation is 0.93. Results using linear aggregation were always higher, particularly for the countries with the worst results, which gives a less bad perception of the countries with the worst results. Geometric aggregation is a better tool in a negative utilitarianist perspective, because it does not mitigate the worst situations.
Table 5 Comparison of two methods of aggregation, geometric and linear aggregations Results within country and results by individual
We presented the results for the indicator of a happy, long and sustainable life at the country level, however, it is possible to evaluate the indicator of a happy, long and sustainable life at a within-country level and at the individual level. Happiness is measured at the individual level and it is always possible to calculate a negative utilitarianist measure of happiness for any level we would like to do so, including within-country levels. Potential Years of Life Lost is usually evaluated at the country level. The OECD also provides data per gender. It should be possible to provide PYLL at a within-country level and, for an individual, by taking into account his or her characteristics such as gender, education and work. Biocapacity and ecological footprint can be calculated at a within-country level. For the individual level, it may be possible to evaluate a proxy of the ecological footprint using income. The higher an individual income is, the higher the ecological footprint is likely to be. Spending would probably be a better proxy than income, but it is easier to determine income than spending. The biocapacity of an individual is the biocapacity of his or her country divided by the number of inhabitants of the country.