Geographies of Uneven Development

Author(s) 1

Geographies of Uneven Development Working Paper

NO. 5 12/2014

INEQUALITY WITH ORDINAL DATA CROSS-DISCIPLINARY REVIEW OF

METHODOLOGIES AND APPLICATION TO

LIFE SATISFACTION IN EUROPE JÖRG SCHODER

Geographies of Uneven Development

Author(s) 2

Social and Economic Geography Department of Geography and Geology University of Salzburg

Hellbrunnerstrasse 34 5020 Salzburg Austria - Europe

Inequality with Ordinal Data

Cross-Disciplinary Review of Methodologies and Application to Life

Satisfaction in Europe

Jörg Schoder

Abstract

Ordinal data are ubiquitously available and frequently employed in empirical research. A review of methodological approaches from various scientific fields shows that in different fields of the social sciences rather distinct methods for measuring inequality are applied without appreciating works in other disciplines. Synthesizing these works leads to the proposal of a research design, which combines dominance criteria and inequality measures from different families. This design is applied to the most recent data on life satisfaction in Europe, a field of research continuing to gain political importance but typically building on questionable methodologies. Hence, empirical results presented are valuable in several respects. One important finding is that specific measures are not consistent as re-rankings frequently occur. It seems this is not primarily due to the underlying data being ordinal, but more because dominance can hardly be found. Indeed, the underlying 0-10 Likert-scale appears to cause a dilemma as the Allison-Foster principle of dominance turns out to be restrictive when applied to relatively dense scales. Keywords: Inequality with ordinal data, dominance, problems of cardinalization, cohesion,

life satisfaction, bootstrapped confidence intervals

www.uni-salzburg.at/geo/uneven_development

1 Introduction

Most of the research on inequality has a strong focus on income, not only in economics.Obviously, there is a strong interdependence between income inequality and inequalitiesin other dimensions that are deemed to be relevant in a society (e.g. in health, socialor spatial aspects, to name only a few). But this correlation should not be taken asan argument to limit the scope of studies to income. The critique of GDP as a welfareindicator can easily be translated to the individual sphere. The increasing interest insubjective well-being witnessed in recent years reflects this critique, at least partly.Works in the field of subjective well-being (henceforth life satisfaction, well-being and

happiness are used synonymous) are primarily devoted to the study of levels and thusaverages. One of the most famous results in this strand of the literature is the so-called Easterlin-Paradox, which claims that increasing average income does not raiseaverage well-being.1 Though this is clearly a relevant perspective with important policyimplications (Frey, 2012), this article has a different focus. It is dedicated shiftingthe focus of analysis from levels/averages to inequalities/variations or, more generally,distributional aspects in well-being. Due to the current situation in Europe, and thenewly available data with a specific interest in the effects of times of crises.These inequalities in subjective well-being have only recently been addressed. Among

the first studies is Veenhoven (2005), who questions the primacy of income as anindicator of social aspects and convincingly motivates why inequality in life satisfactionshould also be considered to assess social inequalities. His theoretical arguments aresupported by his empirical findings of a decline in happiness inequality for a numberof European countries in the period 1973-2001. Thus, increases in income inequality asprominently diagnosed by Piketty (2014) do not necessarily carry over to other socialspheres, which indeed casts doubt on the primacy of income as social indicator in thepolitical debate. Similarly, Stevenson and Wolfers (2008) find decreasing happinessinequality in the United States for the period 1972-2006. They claim that, together withthe Easterlin-Paradox, non-pecuniary factors may have gained more importance in thisperiod. Other international studies, such as Clark et al. (2014), Dutta and Foster(2013) confirm the basic finding of a decline in happiness inequality. Madden (2011)comes up with similar results for Ireland, claiming that economic booms are accompaniedby decreasing inequality in happiness.Recalling that Article 174 of the Treaty on European Union and the Treaty on the

Functioning of the European Union declares the promotion of economic, social and terri-torial cohesion as a priority in European Policies,2 the empirical findings on inequality inlife satisfaction are of obvious political relevance. A diverging assessment of individualswith respect to the perception of the quality of their lives has potential impact on socialand also territorial cohesion. Obviously, to serve as an indicator for policy, happinessinequality needs to be measured in a robust way. For the majority of studies mentioned1 There are different versions of the Easterlin-Paradox, as Stevenson and Wolfers (2013) point out.2 The competences in the area of economic, social and territorial cohesion are shared between the

Union and the Member States according to Article 4 of the Treaty on European Union and the Treatyon the Functioning of the European Union.

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above this may, however, be doubted due to highly questionable methodologies. Withthe notable exception of Dutta and Foster (2013) and Madden (2011), data in thesestudies are typically analyzed with standard measures of dispersion (or inequality). Thesemeasures require cardinal data, but subjective well-being – like many other variables insocial surveys – is typically measured on an ordinal scale. Consequently, the findings ofdeclining happiness inequality cited above may in fact be only artifacts. This is not onlyproblematic with respect to the scientific quest for truth, but also with regard to thepossibility of political exploitation.This paper aims to the existing literature in methodological and also empirical terms.

One of the major contributions with respect to the methodological issue is to review,how ordinal data are addressed in various fields of the social sciences: happiness re-search, health economics, sociology, political sciences, mathematics and statistics as wellas information science. In these fields, diverse approaches are applied, ranging fromrather simple cardinalization, via dominance to parametric measures specifically devel-oped for the use with ordinal data. With respect to the latter, a problem, is that thesespecific measures rather coexist without being systematically compared and categorized.It is exactly in this respect, where another methodological contribution of this paper ismade. Measures defined in fields such as sociology (Blair and Lacy, 2000) or politicalscience (Van der Eijk, 2001) are put in relation to works in other fields by evaluat-ing certain properties. These properties are also discussed in detail. One such propertyis the median-preserving spread principle defined in health economics by Allison andFoster (2004). These property-based classification allow to get a better idea of which ofthese (unidimensional)3 measures are rather substitutes or complements for each other.This seems especially useful for the applied researcher. These methodological issues arediscussed in an integrated approach, together with important empirical findings in thefield of happiness research.The methodological insights are applied then applied to recent happiness data for

European countries to empirically contribute to the following questions: Can the declinefound in the studies mentioned above be confirmed, irrespective of the methodologyemployed? Or, in other words, do different approaches come up with the same qualitativeresults? Due to the possibility to use more recent data it is also interesting to see, whetherthe ongoing EU debt crises has maybe led to an increase in inequality. This wouldbe the logic consequence of the above result of booms being accompanied by declininginequality. To this end, bootstrapped confidence intervals were calculated, constitutinganother novelty in happiness studies. Finally, the empirical results will directly feed backinto methodological considerations. It will turn out, that dominance, especially of theAllison-Foster type is restrictive, when ordinal scales are rather dense.The remainder of this paper is organized as follows. Section 2 provides the integrated

discussion of methodology and empirical results – the latter primarily in the realm of3 This perspective is of course narrow, but as will be shown yields important insights. Nevertheless

two-dimensional measures, as recently developed by Zheng (2011) and Apouey and Silber (2013)promise to be interesting for studying inequality in happiness relative to other dimensions (incomesuggests itself). The application of measures for two- or more dimensions will however be left forfuture research.

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happiness research for obvious reasons. The application of diverse inequality statistics onhappiness data in the European Social Survey (henceforth ESS) for the period 2002-2012is then presented in section 3. Among the measures considered are concepts that weresuggested only recently. One of them has not yet been applied to happiness data: the(ranks-based) Gini coefficient developed by Giudici and Raffinetti (2011). Anothermeasure suggested by Cowell and Flachaire (2012) has already been applied, butnot to data covering times of crises. The conclusion (section 4) summarizes the mainfindings and offers some lines of potential future research.

2 Measuring Inequality with Ordinal Data - RelatedLiterature and Methodological Considerations

One of the first concepts undergraduate students are being taught in statistics courses, isthat of scale. In applied works, it turns out to be nontrivial problem already, to simplydetermine “whether a particular scale is ordinal, less than ordinal, or more than ordi-nal.”4 Beyond teaching the pure existence of different scales, it is typically emphasizedthat the type of scale determines which transformations of the data are appropriate,and accordingly, which interpretations/measures are meaningful.5 Standard measures ofinequality like the Variance, the Gini Coefficient, Measures of the Atkinson Family orGeneral Entropy Measures, are relying on cardinal data. This is due to the fact thatthese measures are mean-based , i.e. they conceptualize inequality as deviation from the(arithmetic) mean or use the mean for normalization purposes (Allison and Foster,2004). However, with ordinal data, all order-preserving transformations are allowed, sothat the mean becomes a meaningless concept. Consequently, standard (mean-based)measures are not generally applicable in the case of ordinal data. Blair and Lacy(2000) demonstrate this using an illustrative example of a frequency distribution of avariable with five categories. The (variance-based) inequality ordering for two groups(male vs. female) can be reversed, when the variance is calculated under the assumptionof a log-scale instead of a linear scale.This stylized fact of reversed inequality rankings shows that results may not only dif-

fer quantitatively, but qualitatively and underlines that the issue should not only botherpedants. However, this consequence is not generally reflected in the growing empiricalliterature employing subjective data – typically measured on an ordinal scale.6 In fact,in the different fields of the social sciences with contributions to this body of literaturethe ordinal nature of the data has been addressed rather heterogeneously in terms ofmethodology. In this context, it seems interesting from a philosophy of science perspec-tive, that there are fields of research (or to employ Thomas Kuhns’ terminology: scientificcommunities), which internally evolve rather homogeneously with respect to the methods

4 Knapp (1990), p. 121.5 See Knapp (1990) for a brief discussion of the ordinal-interval-controversy and references on the

history of conflicting views about scales and statistics.6 For Blair and Lacy (2000) the increased availability of survey data contributed to ordinal data

being almost ubiquitous these days.

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employed, but by and large do not account for methodological insights and findings inother fields (or scientific communities). Examples in support of this supposition will begiven in sections 2.2.1 and 2.2.2. The heterogeneity of approaches is obviously not a prob-lem per se – quite the contrary. However, it will be shown that most of these studies donot check for the robustness of their results. Robustness is not an issue specific to ordinaldata, but has been extensively investigated in the well-developed literature on cardinalinequality measures (with income being the most prominent underlying variable).

2.1 Comparing Distributions based on Dominance Criteria

A powerful concept for assessing robustness is stochastic dominance.7 However, the well-developed framework for the case of cardinal data can not be directly transferred to theordinal case. In fact, the only concept that more or less directly carries over to theordinal case, is first order dominance. In their seminal paper, Allison and Foster(2004) reinterpret some standard results of stochastic dominance analysis.8

In the ordinal case, first order (stochastic) dominance can be defined in close analogyto the cardinal case by comparing cumulative population shares in the different cat-egories of an ordinal variable. For the lower k = 1, . . . , n categories of a distributionx = (x1, x2, x3, . . . , xn), these cumulative population shares are defined as Xk =

Fx(k)X ,

where Fx(k) = ∑ki=1 xi is the cumulative distribution function (henceforth CDF) and

X = ∑ni=1 xi represents total population. Following Allison and Foster (2004) dis-

tribution x first order dominates distribution y = (y1, y2, y3, . . . , yn), if the populationshare in the lower k categories Xk ≤ Yk for all k = 1, . . . , n. Hence, first order dominancerequires that the population share in the (merged) lower k categories of the dominatingdistribution (here: x) is no greater than in the dominated distribution (here: y) for everyk. This implies that the mean of y is no greater than the mean of x for every scale c.Therefore distribution orderings based on the mean are independent of the specific scale,when first order dominance applies. This property is useful, as it allows to assess whetherthe overall level of an ordinal variable has increased or decreased. However, first orderdominance defined in this way does not relief doubts about the adequacy of standardmean-based inequality measures. The reason why such caution is advisable is that are-scaling may alter the relative sizes of the means, which in turn affects (mean-based)inequality measures (Allison and Foster, 2004).In the cardinal case, second order dominance is determined by comparing the areas

under the cumulative distribution functions (henceforth CDFs). But in the ordinal case,the scale is not well-defined, so that a comparison of areas is arbitrary, because re-scaling is associated with a change of the width of steps/intervals.9 Hence, second (andhigher) order dominance is not directly applicable with ordinal data. However, whatcan be exploited to ensure scale independence is the fact, that the height of cumulative

7 Davidson (2006) provides a concise introduction to stochastic dominance.8 For Madden (2014, p.206) their work is a “breakthrough in analyzing inequality with such [ordinal]

data[.]”9 Put differently, a comparison of integrals requires that the arguments of the CDFs can be summed

in a meaningful way (Madden, 2014).

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distribution is scale invariant (Abul Naga and Yalcin, 2008). This insight forms thebasis of the main contribution in Allison and Foster (2004) and is easily seen inthe graphical representation of the so-called S-curve (figure 2 in Allison and Foster(2004)). The S-curves flips “the portion of the CDFs that lie below the median [..] acrossthe median [..]” ((Allison and Foster, 2004, p.512)) and thus illustrates the ideaof a so-called “median-preserving spread” constituting a partial ordering. Given twodistributions x and y possessing the same median category m, then distribution x is saidto have greater spread compared to y if Xk ≥ Yk for all k <m and Yk ≥Xk for all k ≥m.This can actually be reformulated in terms of first order stochastic dominance – x firstorder dominates y below the median and y first order dominates x above the median.Thus according to the Allison-Foster partial ordering, the more unequal distribution isthe one exhibiting more probability mass in the tails “unambiguously further away fromthe median” (Allison and Foster, 2004, p.512).Obviously, the idea of a median-preserving spread is in direct analogy to the mean-

preserving spread (also: Pigou-Dalton transfer)10 used for assessing Lorenz dominance.The reason for suggesting the median as a point of reference for a partial ordering inthe ordinal case is obviously the arbitrariness of the mean. Recalling that second orderstochastic dominance is equivalent to generalized Lorenz dominance citepShor:1983,11

the key contribution of Allison and Foster (2004) can be seen in the development ofa second order dominance concept for the ordinal case.12 If this partial ordering applies,inequality rankings are independent of scale, thus respecting ordinality and yielding un-ambiguous inequality orderings. This insight forms the basis for a variety of parametricmeasures that have since been developed (see 2.2.2). Also, it allows to assess how robustfindings of the existing studies discussed below are.One study in which dominance criteria have been applied in the realm of well-being is

Madden (2011). He uses first order dominance and the Allison-Foster ordering to gaininsights on the effects of economic booms on various domains of life satisfaction (work,finance, leisure, housing and health). To this end he compares pairs of distributions forthe years 1994-2001. Dominance concepts are partial orderings, which naturally impliesthat not all possible pairs of distributions can be ranked. Thus it is not surprising thatnot all pairs of annual distributions can be ranked on these criteria. Also, first orderdominance expectedly occurs less frequently than the Allison-Foster ordering in the datafor Ireland. However, this does not generally hold, as the data on the finance domainshows. Here, first order dominance is found in 23 out of 28 pairs of distributions. Thus,

10 The Pigou-Dalton transfer is an important concept in research on income inequality. A Pigou-Daltontransfer is defined as a (lump-sum) transfer of resources (typically income) from a rich person to apoor person, that leaves the ordering (before and after transfer) and the total income unchangedwhile reducing the distance between the rich and the poor person. A sensible inequality measureshould thus indicate lower inequality for the distribution that results after a Pigou-Dalton transfer.

11 See Davidson (2006) for a brief and intuitive explanation of the Shorrocks-result.12 Since Allison and Foster (2004) progress in this research area has been made e.g. by Yalonetzky

(2013). He derives conditions for stochastic dominance with ordinal data not only for the univariatebut also for the multivariate case. For the bivariate case (income and health) Zheng (2011) developsan inequality measure (health and income) also drawing on dominance criteria.

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in the years under consideration, it is fair to claim that satisfaction in the domain of lifesatisfaction clearly increased. In contrast, for work and leisure only 2 out of 28 pairs arecharacterized by first order dominance, so that the increase suggested by the mean (usinga linear scale) for basically all domains of life satisfaction may be spurious.13 Applyingthe Allison-Foster ordering to the data allows to rank more, but not all, pairs of distri-butions. Again, the number of pairs that can be ranked vary between the domains underconsideration. In general, the spread-criterion suggests, that inequality had decreasedover the years in the domains work, housing and leisure satisfaction and health. Thesefindings directly raise the question addressed at the end of this paper: Is the oppositealso true, meaning that economic busts are typically accompanied by rising inequality inlife satisfaction?

2.2 Comparing Distributions based on Ordinal Inequality Measures

Irrespective of the nature of the data (cardinal or ordinal), a complete ordering canonly be provided by specific (typically parametric) measures. The obvious drawback ofrankings based on such specific measures is however the lack of generality/robustnesscompared to the dominance approach (Madden, 2014). Nevertheless, it seems fair toclaim that the majority applied work employing subjective ordinal data (be it on life sat-isfaction or self-assessed health etc.) for the assessment of social or economic inequalities,is based on the application of specific measures.With respect to the application of specific measures, two general strategies can be

observed. One strategy is the application of existing or newly developed ordinal measures(2.2.2). The other strategy, which seems to be favored by most happiness researchers,is the cardinalization of the ordinal data to then apply standard cardinal measures ofinequality (2.2.1).Regarding the first strategy, the development of specific ordinal measures has been

spurred by the Allison-Foster ordering, which allows to distinguish between measures ofdispersion and genuine inequality measures. The latter being characterized by the fact,that they decrease, when Pigou-Dalton type transfers occur.14 Nevertheless, the varietyof available indexes for ordinal data is still far behind that in the cardinal case. Conse-quently, the possibilities of performing sensitivity analyses are rather limited (Madden,2014). However, this should not be taken as an argument for following the second strategy(cardinalization), which is accompanied by serious problems to be discussed in section2.2.1.When defining new measures, authors hardly engage in relating their new measure

to those already existing. This shortcoming will, among others, be addressed in thefollowing discussion, by aiming to categorize ordinal inequality measures. One new find-ing of this discussion will be, that the Indexes defined by Blair and Lacy (2000) and

13 The only exception being the General Health Question, for which the mean slightly decreases overthe period 1994-2001.

14 The problem of Pigou-Dalton transfers in the ordinal case will be discussed in more detail in section2.2.2. However, as has been argued in section 2.1, the median-preserving spread is, to date, the mostsensible reinterpretation of the cardinal Pigou-Dalten principle available for ordinal data.

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Van der Eijk (2001) actually satisfy the property of an Allison-Foster ordering. Apartfrom these methodological aspects, the discussion is also meant to provide an overview ofempirical results on inequality in life satisfaction. With respect to the Philosophy of Sci-ence perspective already mentioned, it will also be carved out that variation in empiricalstrategies can be observed between, but seldom within different fields of research.

2.2.1 Cardinalization Approaches

Cardinalization can be done in a rather simple, or in a more sophisticated way. Thediscussion above should however have pointed out, that any cardinalization approach ishighly questionable. Only if the employed scale is “correct”, will inequality assessmentsfollowing these strategies be valid. As long as there is uncertainty with respect to thescale, inequality orderings need to be considered with caution, given the dominanceresults from above. A certain degree of confidence in the findings may be gained bychecking whether the Allison-Foster partial orderings apply. However, in the studiesdiscussed in this section, robustness is hardly assessed – neither by checking the Allison-Foster criterion, nor by using alternative scales to calculate the same indexes.

Simple Cardinal Interpretation The simplest cardinalization strategy is to convert thenumerical identifier of a (ordered) category into a cardinal variable directly and one to one– and then to apply standard measures of central tendency and dispersion to the newlycardinalized data. That this approach predominates in happiness research is impressivelydocumented by the brief summary of previous works on happiness dispersion in Kalmijnand Veenhoven (2005, p. 359): “All these analyses measured inequality of happinessin nations using the standard deviation as the descriptive statistic. This statistic is alsoused in the World Database of Happiness [..] where the standard deviation is also usedfor nation rankings and time trends.” It seems worth noting that one year after thepublication of Allison and Foster (2004), Kalmijn and Veenhoven (2005, p.359)conclude their article with the following advise: “The standard deviation is the mostcommonly used statistic for measuring inequality of happiness in nations and in testsit performs equally as well as three other statistics of disparity; i.e. the mean absolutedifference, the mean pair distance and the interquartile range. Hence there is no reasonto discontinue the use of the standard deviation.”15 Interestingly, the cited comparison ofstatistics does not encompass any of the measures developed specifically for ordinal datain other fields of research (section 2.2.2) some of which were already available by then.16

Even five years later the happiness researchers Kalmijn (2010) and Kalmijn andArends (2010, p.153) derive a so-called “pseudo-metric” measure that relies on theassumption of equidistance between the categories of the ordinal variable and state,

15 Original text not in italics.16 Instead, they only discuss standard measures of dispersion for cardinal data, some of them well-known

and widely used in research on income inequalities. Among others, the Theil and Gini coefficientsare included in the comparison.

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that “[n]o suitable statistic ordinal index [..] has been proposed yet [..]”(Kalmijn andArends, 2010, p. 155).17

An article regularly cited in support of a simple cardinal interpretation of ordinal vari-ables is van Praag (1991), who aims to assess the “possibilities of translating qualifyingverbal labels into figures, where we do not specify the subject matter.”18 In this context-free setting test persons had to determine the intervals of an ordinal variable using twodistinct approaches. One visual/graphical and one purely numerical assessment. As canbe expected, variance is slightly higher for the graphical solutions. What is even moreimportant here, is that a downward bias seems to exist, leading to a deviation of esti-mated numerical values from the theoretical predictions (equidistance) which is higherfor lower categories than for the upper ones. This seems to be especially the case for thesecond lowest category, even more so in the graphical compared to the purely numericalassignment method. Indeed, for the graphical solution, estimate and prediction for thesecond lowest category differ by more than one standard deviation. In this sense, thisstudy, usually cited in support to a simple cardinal interpretation, can also be interpretedas questioning this same approach. A recent study by Studer (2011) suggests a contin-uous scale to circumvent the cardinality problem. His experimental evidence also castsdoubt on findings based on cardinalization approaches.In favor of the approach taken by the happiness studies cited so far, one could argue,

that – as long as there is no change in orderings – they are simply pragmatic. However,to confirm that results are indeed qualitatively consistent the performance of robustnesschecks as suggested above would be required. Clark et al. (2012) take a step in thisdirection trying to confirm that their result is not an artifact. Based on the standarddeviation as inequality measure, they posit a decline in the dispersion of happiness since1980 in countries with a positive income growth. To gain confidence in this finding, theyalso calculate the IOV (see 2.2.2) introduced by Berry and Mielke Jr. (1992) andshow that standard deviation and IOV are more or less parallel lines in the plot theyprovide. This somehow supports their finding, without however being a strict proof ofrobustness. Also, in order to gain this additional confidence, one either needs to use somemeasure tailored for ordinal data as a benchmark or check whether the Allison-Fosterordering applies. However, this procedure raises the question why one should go back tothe standard measure again.The problem of robustness in applied studies has e.g. been demonstrated by Madden

(2010) in his longitudinal comparison of subjective health assessments for Ireland. Hecomputes the ordinal inequality measure introduced by Abul Naga and Yalcin (2008)(see section 2.2.2) with different parameters for the years 2003-2007 and compares theassociated rankings with the orderings resulting from the (cardinal) Generalised EntropyIndex. The choice between ordinal and cardinal measures turns out to be more criticalfor the rankings than the specific parameters. Hence, not only in theory may simple17 When deriving their pseudo-metric measure Kalmijn and Arends (2010) start out with a simple

dichotomization of ordinal data. Although there are valid measures available for dichotomous data(e.g. the Index of Qualitative Variation), it is obvious that this strategy is associated with a wasteof information. Hence, these approaches are not discussed in more detail.

18 van Praag (1991), p. 72.

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cardinalization be associated with a lack of quantiative precision (which might deemacceptable for reasons of pragmatism) but indeed may be prone to a serious risk ofproducing qualitatively inconsistent results.To sum up, it seems that in happiness research, the insight into the need of specific

measures has not completely been appreciated, or at least, not been considered to bepractically relevant. Only very recently Madden (2011) and Dutta and Foster (2013)have used some of the inequality measures, that were developed specifically for use withordinal data (see section 2.2.2 for a discussion of measures and their use in appliedstudies). From a Philosophy of Science perspective it is interesting that these applicationsin the field of happiness research were not carried out by established happiness researchesbut by researchers known for their works in other fields, like Health Economics.

Sophisticated Cardinal Interpretation To overcome the assumption of equidistance un-derlying simple cardinalization, various alternatives have been suggested. Many of themwere developed or applied in the field of Health Economics, in which research on healthinequalities and its interdependence with income is still a question of vibrant debates.Basically all of the different approaches developed since the mid 1990s assume that un-derlying the ordinal variables is a latent, continuous variable which can be estimatedfrom the ordinal data and then be used to calculate a conventional inequality measure.In the context of happiness research such an approach is described in Veenhoven

(2009) who aims to estimate numerical values for the ordinal variable “happiness” byemploying web-based visualization methods resembling those described in van Praag(1991) (see above): Test subjects had to move a slider to those positions that they felt tobe corresponding to the demarcation between two categories of verbal evaluation. In thisway a possibly nonlinear scale is experimentally generated. Based on these scales it ispossible to study whether test subjects from different countries apply a similar underlyingscale, when answering questions in happiness surveys. In fact the results presented byVeenhoven (2009) confirm the theoretical supposition of noninformative mean values:the means of happiness ratings for England and the Netherlands differ by 6.3 percenton a four category scale, whereas they only differ by 3.2 percent on the scale derivedby the method just described.19 With respect to measuring inequality and in line withthe general idea already sketched above, Kalmijn and Arends (2010) suggest to usethese experimentally generated metric results as boundary estimates of a cardinalizedhappiness variable and apply standard dispersion measures on the mid-interval values.This however is problematic for reasons to be discussed below.Another approach has been suggested by Wagstaff and Van Doorslaer (1994),

who assume the latent variable to be distributed lognormal in order to reflect the skewnesspresent in many measures of health. They find values of the latent variable by dividingthe area under the lognormal curve according to the relative frequencies observed forthe categories of the ordinal variable. Groot (2000) describes another alternative basedon an ordered probit model to estimate the cutpoints. Similarly, in happiness researchStevenson and Wolfers (2008) employ a generalized ordered probit model to assess19 The author himself however interprets this result not as questioning the cardinalization approach.

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inequality in the US. One advantage of ordered probit models, but also interval regres-sion (see below), is seen in the potential to address reporting bias – using additionalsocioeconomic variables allows to check if it is plausible that observations in the samecategory reflect the same true situation (e.g. health status) which does not necessarilyneed to be the case.20 It seems rather difficult however, to validate that the cardinalizedvariable really contains more information about the true ratings than the original ordinalvariable. Hence, validity is likely to be a major problem of these approaches.

van Doorslaer and Jones (2003) provide a review of some common approachesand were among the first to assess their internal validity. The adjective internal refersto the circumstance of the authors making use of the fact that their dataset containsboth an ordinal health variable (Self Assessed Health, henceforth SAH) and a standardcardinal variable on health utility (McMaster Health Utility Index, henceforth HUI). Thelatter is assumed to be an appropriate measure of individuals actual health and is usedto calculate the benchmark (income-related) inequality index – (i).21 The actual HUI iscompared with four latent variable approaches, which can thus be assessed in terms oftheir validity:

(i) “an OLS regression using actual HUI as the dependent variable [..],

(ii) an OLS regression on mean HUI per category of SAH,

(iii) an ordered probit regression using SAH as dependent variable,

(iv) an interval regression using SAH as dependent variable and HUI-based thresholdlevels”22

Note: the idea behind (iv) is, that in datasets which (typically) do not contain individ-ual level information on both SAH and HUI there might be some external data availablefor the HUI of the relevant population which in turn can be used to estimate thresholds.It turns out that interval regression (iv) performs best. Compared to the OLS regres-

sion on mean HUI per category (ii) the advantage lies not so much in the estimate ofthe mean, but in the ability to preserve more of the inter-individual variability in health.Also, predicted range and percentiles are closer to the predicted actual ones. Orderedprobit turns out to not provide good approximations.23

For Costa-Font and Cowell (2013) the positive finding of van Doorslaer andJones (2003) (with respect to interval regression as method of cardinalization) does notmean that these approaches are without problems. Regarding interval regression and

20 Apouey (2007).21 As a measure of relative income-related health inequality van Doorslaer and Jones (2003) use the

health concentration index.22 van Doorslaer and Jones (2003), p. 74 f.23 Another interesting finding presented in van Doorslaer and Jones (2003) is, that the scale of SAH

seems to be nonlinear, which again casts doubt on simple cardinalization approaches of the kinddiscussed in section 2.2.1. More specifically, health differences between categories of SAH seem todecline with increasing SAH categories.

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imputation methods, van Doorslaer and Jones (2003) state that “there is an insuffi-cient discussion of the economic rationale for these methods or the practical implicationsof using one method rather than another.” The Meta-Analyis carried out by Costa-Font and Hernández-Quevedo (2013) suggests that this lack of foundation needsto be taken rather seriously. Not really susprisingly, they find that the main reason forheterogeneity in results on health inequalities is the method of cardinalization which hadbeen applied in the first place. This, again, underlines that caution is advisable for thesekinds of approaches.

2.2.2 Measures Tailored for Ordinal Data

Prior to the seminal paper by Allison and Foster (2004) Blair and Lacy (2000,p.253) already suggested “that conventional approaches that treat ordinal data as inter-val/ratio [..] are problematic for the analysis of dispersion.” Yet, before Allison andFoster (2004), researchers typically developed parametric measures without referringto dominance arguments. This may explain the modesty expressed by the preferred clas-sification of the early measures as dispersion (rather than inequality) measures. One suchearly example is the Index of Ordinal Variation (IOV) defined by Berry and MielkeJr. (1992) which can be shown to belong to a class of parametric inequality measuresfor ordinal data defined by Blair and Lacy (2000).24 The following discussion does notaim to give a complete bibliography of these kinds of approaches. Instead, the remainderof this section aims at presenting the measures used in the empirical application of thispaper.Before turning to the specific measures, it should be noted that all ordinal measures

share the common feature of building on probability distributions instead of values as-signed to categories.25 As has been pointed out earlier, the crucial insight is that theheight of the CDF is scale invariant. Consequently all measures rely solely on the (rela-tive) CDF, thus accepting the fact that it contains all distributional information.26

Absolute Value Measures and Measures Referring to the Median Blair and Lacy(2000) add to this insight by arguing that perfect equality can not serve as a referencedistribution in the ordinal case. Cardinal measures of inequality and dispersion in oneway or another capture distance from a uniquely defined distribution of perfect equality– the mean. In the ordinal case, however, there are as many perfectly equal distributionsas there are categories so that choosing among these distributions becomes arbitrary.27

Hence, Blair and Lacy (2000) turn things upside down by suggesting a class of mea-sures based on distance to perfect inequality. Indeed, perfect inequality is unique in theordinal case with half the probability mass located in the lowest and highest category,respectively. This strategy however causes a problem of distinguishing inequality from

24 The IOV is equivalent to Blair-Lacy’s (1 − l2).25 See e.g. Abul Naga and Yalcin (2008) or Kobus (2014).26 Note that the first k−1 categories carry all information of the relative CDF (Blair and Lacy, 2000).27 Obviously, the reason is that with categorical data, perfect equality means that all observations are

in the same category.

12

polarization (reflecting the idea of a disappearing middle) in the ordinal case (Zheng,2008; Kobus, 2014).All measures in the family defined by Blair and Lacy (2000) build on the ideas just

described. The fact, that in the perfectly unequal distribution, the cumulative relativefrequency of the first k−1 categories is exactly one half is used to measure distance frommaximal dispersion. By referring to squared Euclidian distance, maximal dispersion canbe defined as d2 = ∑k−1i=1 (Xi −

12)

2. This can also be normalized by the maximum possiblevalue d2max =

(k−1)4 . Following Blair and Lacy (2000), the percentage of maximum

dispersion attainable for an ordinal variable with k categories can thus be defined as

IBL = 1 −d2

d2max, (1)

which is equivalent to the IOV previously defined by Berry and Mielke Jr. (1992).With regard to the properties of the class of measures based on d2, Blair and Lacy

(2000) hesitate to call them inequality measures, preferring the more humble interpre-tation of representing dispersion measures. This prudence can be explained by the factthat their measure satisfies the criterion of scale invariance. However, before Allisonand Foster (2004) they could not be sure whether the Pigou-Dalton transfer principlewas also met. Meeting this principle is however critical in distinguishing measures ofdispersion and of inequality.Following the lines of argument in Abul Naga and Yalcin (2008), it can actually

be shown that the Blair and Lacy (2000) IBL also satisfies the Allison-Foster ordering– which has not yet been done to the best knowledge of the author of this paper. Ina rather ad hoc way, this can be shown by defining some distributions satisfying themedian-preserving spread criterion and then applying the IBL to see whether the ap-propriate ranking is generated. Possible (cumulative relative) distributions satisfyingthe median-preserving spread principle are defined in Abul Naga and Yalcin (2008):Π = (.05, .1, .2, .95,1), Θ = (.05, .1, .2, .85,1), Π̂ = (0,0,0,1,1) and Π̌ = (.5, .5, .5, .5,1).Applying IBL to these distributions yields the results expected for a measure satisfyingthe median-preserving spread principle:IBL(Π̂) = 0 < IBL(Π) = .345 < .425 = IBL(Θ) <

1 = IBL(Π̌).Abul Naga and Yalcin (2008) and Apouey (2007) independently develop classes

of measures using axiomatic approaches. One common axiom to be satisfied by themeasures in both approaches is the median-preserving spread principle.Abul Naga andYalcin (2008) define a class of measures, based on the (endogenous) median category ofthe distribution, which generally implies that only distributions with the same mediancan sensibly be compared. To overcome this restriction, Abul Naga and Yalcin (2010)introduce the idea of equivalence classes, to derive measures, which satisfy the Allison-Foster ordering without being constrained to the case of distributions with identicalmedians. Another type measure defined in Abul Naga and Yalcin (2008) and similarlyin Apouey (2007) are the so-called “absolute value measures” which do not rely onthe endogenous median. Instead they refer to the perfectly unequal distribution as thepoint of reference, just like the IBL. In fact, Abul Naga and Yalcin (2010) classify

13

the absolute value measure by Apouey (2007) to be median-independent. Due to thesimilarity of the IBL and the measure by Apouey (2007) it seems plausible to classifythe IBL as belonging to the median-independent measure as well, though this is notexplicitly proven here.Thus, the IBL satisfies the most critical features for an ordinal inequality measure:

scale invariance, median-preserving spread principle and (as an absolute value measure)median-independence. Therefore, in the empirical part, the IBL measure will be used asa representative index for the families of measures discussed so far. This approach canbe justified by the result in Kobus (2014). She transfers the classic Atkinson result intothe ordinal framework by showing that the Allison-Foster ordering is the largest relation,with which all measures satisfying the median-preserving-spread criterion are compatible.In other words, when distributions can be ranked based on the Allison-Foster ordering, allmeasures satisfying the median-preserving-spread criterion will yield the same ranking.Hence it suffices to use any one of these measures.In the field of policy research Van der Eijk (2001) suggested a measure, which closely

resembles the idea of equivalence classes of Abul Naga and Yalcin (2010). Usingpatterns of predefined 0-1 triples and weights allows the decomposition of an empiricalfrequency distribution into layers consisting of 0-1-tuples. Accounting for the applicationin political question Van der Eijk (2001) refers to inequality/polarization as (political)dissent, whereas perfect equality is interpreted as consent. His agreement measure A isdefined as follows:

A =

(k − 2) ⋅ TU − (k − 1) ⋅ TDU

(k − 2) ⋅ (TU + TDU)

⋅ (1 −(s − 1)

(k − 1)) , (2)

where s (k) is the number of nonempty (total) categories. TU (TDU) denote thenumber of triples of categories conforming to (deviating from) unimodality. Applyingthe same crude check as in the case of IBL shows, that A does not satisfy the median-preserving spread principle. However, Ruedin (2014) implemented a normalized versionof A in his R-package agrmt, which shall be denoted IvdE .28 This statistic is zero in thecase of consent, 0.5 in the case of a uniform distribution and 1 in the case of perfectinequality/polarization. Applying the check from above yields that the IvdE satisfies themedian-preserving spread principle.Finally, the measure defined by in the field of information science shall be mentioned.

Although claiming to account for the ordinal nature of data, Tastle and Wierman(2007) use the mean as a point of reference. Therefore, the measure shall not be discussedin more detail here.Two of the measures described in this section have already been used in applied works

in happiness research. In his study on boom and well-being in Ireland (see also section2.1) Madden (2011) computes the measures suggested by Abul Naga and Yalcin(2008) and Apouey (2007) with the parameterization recommended by the developersof these indexes. As can be expected, since the Allison-Foster ordering applies to the

28 The command to compute this statistic is polarization.

14

majority of pairs of annual distributions, the measures generally confirm the findingsdescribed above. The boom was accompanied by decreasing inequality in the domainsof life satisfaction under consideration. But the results also show that rankings by theparametric measures are not unambigious when dominance does not apply. For example,in the domain work satisfaction no dominance can be found for the years 1999, 2000 and2001. While the Apouey-measure yields an inequality ranking 2001 < 1999 < 2000, theorder is 1999 < 2001 < 2000 for the Abu Naga/Yalcin-measure.

Digression: Ontological Considerations and Methodological Implications In the dis-cussion above, it has been mentioned that methodologically, a distinction between mea-sures of dispersion and measures of inequality can be based on whether they accountfor (rank) preserving transfers of the Pigou-Dalton type or not. The problem with thisnotion in the ordinal framework has already been mentioned: measures of inequalitycan hardly be distinguished from those of polarization. However, Kobus (2014, p.3)mixes up ontological and methodological aspects when arguing that a meaningful inter-pretation of the Pigou-Dalton Transfer is difficult to imagine, because “e.g., what wouldtransferring health from a healthy to a less healthy individual mean?” An interpersonaltransfer of health (or happiness) may in fact seem to be an odd idea, but it is not rootedin the ordinal nature of the data – a simple reference to the possibility of income beingmeasured on an ordinal scale shall suffice for illustration. The real methodological issueis, that “the impact of the Pigou-Dalton transfer may remain indeterminate, dependingon whether the transfer is sufficient to move individuals to different categories” (Kobus,2014, p.3). This argument can also be found in Blair and Lacy (2000) and Cowelland Flachaire (2012).Nevertheless, there is some truth to the first argument. But it is an ontological rather

than a methodological issue. In fact, it seems important to distinguish the ontologicalfrom the methodological dimension, though there are clearly strong interdependencies,especially in applied works. From an ontological perspective, it indeed seems problematicin the case of happiness (or health etc.) to refer to the idea of interpersonal transfersas standard inequality measures do. For Kalmijn and Veenhoven (2005, p.366ff), thepossibility of a transfer necessitates a “capacity” interpretation of the underlying variable.Referring to physics examples, a capacity variable like volume is characterized by the fea-sibility of summation. In contrast, it does not make sense to add up “intensity” variableslike temperature.29 Following Kalmijn and Veenhoven (2005), distributions of capac-ity variables may be interpreted as distributions of numbers and quantities alike. Theformer interpretation is purely frequentist, just summarizing the number of observationsin certain parts/intervals of the distributions so that it applies for both – for capacityand intensity variables. In contrast, a quantity interpretation refers to a fraction of theaggregate amount of the underlying variable (earned by certain subpopulations) and isthus sensible in the case of capacity variables only. Income in this terminology clearlyqualifies for being a capacity variable – it can be aggregated to a total income of a pop-ulation, which may then be (re-)distributed. In contrast, Kalmijn and Veenhoven

29 See Kalmijn and Veenhoven (2005) for a more detailed explanation.

15

(2005) argue that happiness should be interpreted as an intensity variable. Indeed, theidea of an aggregate total amount of happiness for a population that can be redistributedseems odd.Obviously, these ontological aspects directly carry over to the choice of adequate

methodologies. For Kalmijn and Veenhoven (2005) it follows from these consider-ations that standard inequality measures should be disregarded in happiness studies be-cause they generally rely on a quantity interpretation of underlying distributions. Theyexplicitly criticize the Gini index building on the Lorenz-curve which is special in that ithas a number distribution on the abscissa and a quantity distribution on the ordinate.30

No doubt, disregarding approaches relying on the idea of transfers is a stringent call atfirst sight.Before taking a closer look at this issue, another related aspect shall be mentioned.

Apart from the intensity vs. capacity variables issue, the problem of indeterminacy of atransfer to move subjects between categories remains to be resolved. Here, methodologi-cal and ontological considerations are directly intertwined. The Allison-Foster ordering isone way of addressing the methodological issue, but whether it really captures the onto-logical concerns may be doubted. However, a less direct interpretation of transfers mightmitigate the problem. A more humble interpretation could e.g. be that inequality mea-sures compare shifts in distributions of intensity variables over time. Compared to theinterpersonal notion suggested by Kalmijn and Veenhoven (2005) such an interpre-tation seems to be compatible with the Allison-Foster ordering. Happiness distributionsare likely to change over time, very likely caused by changes in underlying variables.While these changes need not be a zero-sum game as the idea of transfers suggests, adisregarding of measures relying on mean- (or median)-preserving spread principles doesnot seem necessary for ontological reasons.Finally, another mainly methodological issue shall be mentioned. Ordinal data, like

happiness, are typically measured on a scale which is bound from above and below (e.g.from zero to ten). This is in contrast to cardinal variables, like income, which possessnatural zero points and, typically, open scales. In fact, the boundedness of the scale mightbe the reason for the downward bias that was among others documented by van Praag(1991) in his study on the cardinalization of ordinal scales (section 2.2.1). To date itseems that this boundedness of scale issue has not been considered in its implications formethodology and interpretation of results, so that future research on this topic is likelyto be valuable.

Rank-based Measures and the Notion of Status Cowell and Flachaire (2012)address some of the ontological and methodological issues by introducing the notion ofstatus, while also incorporating most of the ideas already discussed. Status can gener-ally be defined as a function of individual utility, the distribution of utilities and thepopulation size. By taking the detour of status instead of directly working with utility,the authors graphically demonstrate that the problem of cardinalization is avoided. As

30 They also explicitly criticize the Theil index, which also assumes some aggregate amount that canbe redistributed.

16

a first step towards a parametric inequality measure for ordinal data, they introduce themerger principle which already imposes some structure on the index function. Accordingto the merger principle, status of individual i is not affected by merging categories of thedistribution which i does not belong to. This generally induces an additive structure.Cowell and Flachaire (2012) distinguish between upward- and downward-lookingstatus. In the former (latter) case individual i′s status si depends on the proportion ofpeople with the same or higher status than i. A sensible choice between upward- anddownward-looking status depends on the meaning of high or low values on the underly-ing ordinal scale. When high values on the ordinal scale indicate a more (less) desirablesituation, then upward-(downward)-looking status is chosen. Thus, status is simply are-interpretation of the cumulative relative frequency distributions.Based on their axiomatic approach, Cowell and Flachaire (2012, p.9ff) derive a

general form of the inequality measure IαCF :

IαCF (s; e) =1

α(α − 1)[

1

n

n

∑

i=1

(

sie)

α

− c(e)] , (3)

where α is a sensitivity parameter for weighting different parts of the distribution,31 eis a reference point and c(e) is a constant conditional on e. Not surprisingly, inequalityis interpreted as total distance from equality. Here, as for ordinal measures in general,distance is established in “terms of proportions rather than [..] absolute differences”(Cowell and Flachaire (2012, p.11)). The question is however, which reference point(or distribution) should be chosen, as the general formulation leaves various possibilities.Cowell and Flachaire (2012) discuss the mean, the median and maximum status aspotential points of reference. When choosing the median as reference point, the approachseems largely equivalent to the idea of the median-preserving spread. In this sense, theunderlying axioms may be interpreted as a generalization of the median-preserving spreadintroduced by Allison and Foster (2004).Whereas the mean has been criticized as reference point by all authors developing

ordinal measures, Cowell and Flachaire (2012) also discard the median as a pointof reference. The reason being that it is not necessarily well-defined for ordinal data.32

The authors take a stance for maximum status (e = 1) as a reference point, because inthe case of perfect equality, both upward- and downward-looking status take on a valueof unity. Hence the maximum value is always well-defined and ensures a well-behavedindex in the 0-1 interval (Costa-Font and Cowell, 2013). Choosing the maximumas the point of reference the approach basically reproduces the ordinal variable with thenotion of status. This is why IαCF (s; e) is classified as a (mere) rank-based measure inthis article.Besides the reference point, a specific form of IαCF (s; e) for applied work obviously de-

pends on the choice of α. Cowell and Flachaire (2012) discuss various combinationsof parametrization and ensure that individual relative status to be an affine transfor-

31 For low values of α the index is more sensible to low status values.32 See page 16 of Cowell and Flachaire (2012) for examples.

17

mation of utility. In this context, invoking Eichhorn‘s Theorem yields (Cowell andFlachaire, 2012, p.19) the following function:

IαCF (s; 1) = {

1α(α−1)

[1n ∑

ni=1 s

αi − 1] , if α ≠ 0 ∧ α < 1

−1n ∑

ni=1 log(si) , if α = 0

(4)

To the authors, it does not seem to be overly restrictive to require values of α < 1.33 Ascan be expected, the index thus specified does not satisfy the median-preserving spreadprinciple.

Finally, the ranks-based Gini coefficient, recently defined by Giudici and Raffinetti(2011) is presented. This measure also exclusively resorts to the ranks implied by theordinal distribution. Accordingly, the approach is somehow related to that of Cowelland Flachaire (2012) just described. However, underlying the ranks-based Gini isneither an axiomatic approach nor an elaborated theoretical concept, as is the case withthe notion of status in Cowell and Flachaire (2012). Parallels to the standard Giniindex are however remarkable, the most important analogy being that the ranks-basedGini is based on a redefined version of the Lorenz-curve. The coordinates of this Lorenzcurve are (X(.)) and (Q(r)), which can be calculated by applying the following simpleformula (Giudici and Raffinetti, 2011, p.7):

(X(.),Q(r)) =⎛

⎝

∑ij=1 nj

∑kj=1 nj

,∑ij=1 rjnj

∑kj=1 rjnj

⎞

⎠

. (5)

Hence, the values for the abscissa are simply the cumulative relative frequencies. Theordinate is given by the cumulative relative ranks, which are calculated by sorting theindividual observations of the underlying variable and then assigning ranks, which arethen aggregated to a cumulative distribution. Note that the assignment of ranks is alsonecessary in Cowell and Flachaire (2012), but the aggregation of these rank tablesis done differently, owing to the underlying concept of status.Employing the trapezoids rule, the definition of the ranks-based Gini is given by (Giu-

dici and Raffinetti, 2011, p.9):

IGR = 1 −k

∑

r=1

(X(r − 1) +X(r))(F (r) − F (r − 1)) (6)

Just like the standard Gini, the ranks-based Gini takes on values between zero andone. It is zero in the case of equality (all observations in one category) thus satisfyingthe normalization axiom (Giudici and Raffinetti, 2011). The ranks-based Gini (justas the IαCF (s; 1)) is not confined to comparisons of distributions with equal median.However, for obvious reasons it might be more informative to also consider the level ofthe distribution in empirical works.

33 In fact the same restriction yields the Atkinson-family indexes from the class of generalized entropymeasures.

18

Summing up, the discussion in this section has shown, that there are good reasons, forusing either dominance criteria or measures specifically tailored for the use with ordinaldata to assess inequality in subjective data measured on ordinal scales. A sensible ap-proach seems to be a combination of both. This appears to be adequate, since measuresof different families may come up with different orderings if dominance does not apply.Moreover, sensitivity analyses can help to gain a better picture of the data, depending onhow certain parts of the underlying distributions are weighted with these measures. Sofar, the Allison-Foster ordering is the only approach with a notion of second order domi-nance applicable for ordinal data. Because of its relatedness to the Lorenz dominance, ithas been integrated in different families of parametric measures by purpose, or as in thecase of Blair and Lacy (2000) rather accidentally. Besides these methodological as-pects, it has been argued that ontological considerations are also relevant in the curse forsensible inequality measures in the ordinal case. Indeed, some doubts remain whether theAllison-Foster ordering is an ontologically sound concept for happiness data. However, apossible conciliation has been suggested by reinterpreting the idea of transfers which isat the heart of the Allison-Foster ordering. Other approaches, especially Cowell andFlachaire (2012) refrain from the idea of transfers altogether without having to sacri-fice the possibility to measure inequality – instead of dispersion only. Indeed, it seemsthat these status- or rank-based-approaches are compatible with intensity variables andthus less critical in ontological terms. This probably holds more for the IαCF (s; 1) thanfor the IGR. The latter index is not axiomatically derived and hence, properties remainslargely unclear. In general, more research is necessary on these issues.

3 Data and Results

The empirical part of this paper builds on cross-sectional data of the European SocialSurvey (ESS). This survey is carried out with the purpose to provide microdata for mon-itoring social change in the participating European countries and its close neighbors.34

Some general remarks on this dataset are given in section 3.1, before section 3.2 turns tothe discussion of the main empirical findings.

3.1 The European Social Survey

The ESS survey is carried out every two years, the latest wave (of six in total) covers 30countries in the year 2012. Only 17 out of 36 countries in total participated in all sixwaves. However, at the time of preparing the dataset for this article, data for only 15countries were available.35 Since one aim of this article is to gain insights on developmentsin inequality over time and across countries, only those 15 countries were included, forwhich data of all waves are available: Belgium, Denmark, Finland, Germany, Hungary,Ireland, Netherlands, Norway, Poland, Portugal, Slovenia, Spain, Sweden, Switzerland,

34 Neighboring countries included in the dataset are e.g. Russia, Ukraine, Iceland, Turkey or Israel.35 For France (wave 6) and Austria (wave 5 and 6) the most recent data have not yet been released by

the time the dataset for this article was prepared.

19

United Kingdom. Statistics have been computed for all of these countries, but only aselection of results is presented here.36

ESS is administrated at the Centre for Comparative Social Surveys at City UniversityLondon and financially supported by the European Commission, the European ResearchFund and National Funds. The ESS administration provides guidelines for carrying outthe surveys. However, sampling and partly also the specification of the questionnaire(e.g. Income Categories) is administrated by the partners in Participating Countries.These are generally the National Statistics Offices. A guideline with respect to samplesize is 1,500 observations per country and wave (for countries with less then two millioninhabitants the minimum is 800 observations). Thus unidimensional analyses on thenational level can build on rather large samples.The dataset contains a variety of variables on sociodemographics, social capital and

trust, attitudes and beliefs and migration backgrounds. Most importantly in the contextof this article are the questions on subjective well-being. Questions on subjective eval-uations cover income and job satisfaction, satisfaction with democracy, education andhealth services in the country of residence, to name only some. Also included are ques-tions on subjective well-being which need to be answered on a 0-10 Likert-scale (Survey,2012, p.8 and 12):

• Taking all things together, how happy would you say you are?

• All things considered, how satisfied are you with your life as a whole nowadays?

The first question (happiness) is typically associated with a notion of happiness asan affective concept, whereas the second question (life satisfaction) is a construct witha rather cognitive connotation (Sirgy, 2001, p.30f).37 Most of the studies cited so faremploy the second type of question. Hence, the results below are also based on the lifesatisfaction variable. It should however be noted, that the distributions are rather similarand highly correlated. Nevertheless, they are likely to measure different things and arealso differently influenced by underlying factors – at least relatively. The other variablesfrom the ESS employed in this study are country and year identifiers as well as designweights.38

3.2 Results

A first valuable impression about inequality in life satisfaction may be gained by inspect-ing Figure 1. For the selection of countries, histograms for the years 2002-2012 clearlyshow that the (weighted) relative frequencies in life satisfaction differ markedly. Notso much, but still noticeably over time and very clearly between countries. Except forHungary (HU) and less strikingly for Portugal (PT), distributions in all countries areleft skewed. Denmark (DK) is well-known for ranking among the happiest nations in the

36 Results for countries not covered in the discussion in this section can be provided upon request.37 In a recent paper Studer and Winkelmann (2014) find that time of response actually matters for

reported happiness: slower responses typically lead to lower reported happiness.38 Since the data will not be pooled in the analysis, there is no need to use population weights.

20

world. Almost 57 Percent of people rate their lives with 8 or higher. In the Netherlands(NL), the distribution is much more peaked, with almost 40 Percent of Dutch peoplerating their lives with 7. However, compared to the Danish people, relatively few Dutchrate their lives with 8 or higher (about 26 Percent).The distributional patterns observed in Figure 1 are likely to be influenced by a myriad

of factors. Economic, but also cultural aspects are likely to play a prominent role.39 Theyhowever cannot be detailed here. Instead, the question to be addressed in the remainderof this paper is whether methodological tools, discussed in the previous section, can:

• reflect the intuitive perception of inequality in histograms?

• yield consistent results for the data at hand?

Figure 1:Histograms of Life Satisfaction by Country and Year.

NL DK ES GB IE DE FR PT HU

0%10%20%30%40%

0%10%20%30%40%

0%10%20%30%40%

0%10%20%30%40%

0%10%20%30%40%

0%10%20%30%40%

20022004

20062008

20102012

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Extremely dissatisfied

1 2 3 4 5 6 7 8 9 Extremely satisfied

Life Satsisfaction on Likert−Scale

Fre

quen

cy

Note: Green Line indicates median category.

Own Illustration using ESS-data.

3.2.1 Findings based on Dominance Criteria

As has been argued, dominance criteria allow a robust evaluation of distributions, irre-spective of underlying scales. With ESS data, the issue of scale may be documented bycomparing e.g. Denmark and the Netherlands again. In all of the years covered by the

39 Senik (2011) explains the relatively low level of happiness in France with cultural factors.

21

ESS data set, average life satisfaction measured by the mean (with an underlying linearscale) is higher in Denmark. However, a quick check for first order dominance reveals thatfor 2010 and 2012 first order dominance does not apply. Hence, scale actually matters forranking these two countries in terms of levels of happiness. This has obvious implicationsfor inequality orderings as well. Moreover, even when first order dominance applies, thereliability of cardinal inequality measures is not clear at all because arithmetic meansmay vary relatively to each other with different underlying scales (see section 2.1).The problem with dominance is, however, that it does not always allow an ordering.

For the countries and years displayed in Figure 1, inequality comparisons are obviouslyapplicable with respect to the two dimensions – time and country. In the time dimension,dominance can be assessed for pairs of years (for the period 2002-2012) in each country.This is a 6x6 Matrix for 9 countries, so that 270 (off-diagonal) pairs need to be compared.First order dominance applies 39 times. An Allison-Foster Ordering is possible only once:2008 is less unequal than 2002 in the Netherlands. Considering the country dimensions,dominance may be evaluated for pairs of countries in every single year. Out of these432 (off-diagonal) pairs, 151 are characterized by first order dominance. The Allison-Foster ordering again is hardly found. It applies for only two of the pairs: in 2010, thedistribution of life satisfaction is less unequal in Spain (ES) than in Germany (DE) andGreat Britain (GB). For the complete sample of 15 countries, these proportions hardlychange.40

Thus, for the data at hand, dominance criteria in fact allow only a small proportion ofdistributions to be ordered, especially with respect to the Allison-Forster principle. Thisis in contrast to other studies applying dominance criteria with ordinal data. In theiroriginal paper, Allison and Foster (2004) compare health ratings in 25 US states. Outof these 600 pairs, first order dominance applies for 199 pairs which is in the range ofthe above findings for ESS data. However, the Allison-Foster ordering holds for 55 pairs.Also in the context of health ratings Abul Naga and Yalcin (2008) find the Allison-Foster ordering to be satisfied for 12 out of 42 pairs Swiss Cantons. In the context ofhappiness research, dominance criteria have been applied by Dutta and Foster (2013).Here, Allison-Foster dominance occurs even more frequently than first order dominance.One last study that shall be mentioned, is Madden (2011). For the different domains ofsatisfaction (work, finance etc.) considered in this paper, there are considerable variationsin proportions between first and second order dominance.The main reason for the differences between findings in these studies and those based

on the ESS data presented above is simple. While the Likert scale in the ESS ranges from0 to 10, the data employed in the other studies are measured on scales with considerablyfewer categories. Happiness ratings in Dutta and Foster (2013) are measured on ascale with 3 categories only, health ratings in Allison and Foster (2004) and AbulNaga and Yalcin (2008) on a 1-5 scale and the data employed by Madden (2008)are measured on scale ranging from 1-6. This obviously makes an important difference,

40 In the comparison by countries, out of 1260 pairs, 352 are characterized by first order dominance,and 4 pairs can be ordered with the Allison Foster principle. For the comparison by years, first orderdominance (Allison-Foster) applies for 57 (1) out of 450 pairs.

22

especially for the Allison-Foster ordering. As can be seen in Figure 1, medians differ notonly between countries but, when considered over time, also within them. This precludesan Allison-Foster ordering of these distributions from the outset.In this light, a 0-10 scale appears to cause a dilemma. It suggests to capture more

information than scales with fewer categories, but at the same time, the probability ofactually finding dominance is smaller. An approach of merging adjacent categories tocircumvent this problem is, however, not a sensible solution as this would imply to discardinformation. An equally bad solution is to argue that a 0-10 Likert scale is fine enoughto be interpreted as cardinal. As has been pointed out, the problem with Likert scales,in general, is that equidistance cannot be taken for granted. This is probably due tothe boundedness from above and below, which is another problem in itself. Hence, anarbitrarily dense scale is not a solution. A more adequate conclusion is rather to refrainfrom measuring subjective data on discrete Likert scales altogether. Alternatives havee.g. been suggested by Calcagnì and Lombardi (2014) or Studer (2011). However,for the time being, most survey data is still measured on Likert scales. So, one has tocope with the associated shortcomings. A reasonable strategy laid out in this paper is tocombine dominance criteria with specific inequality measures to gain the best possiblepicture of the data.

3.2.2 Findings from Applying Inequality Measures

In the discussion of inequality measures (section 2) it has been pointed out that happi-ness researchers still recommend the use of the Standard Deviation (SD). Therefore, andbecause many of the studies on happiness inequality cited above, the SD and the stan-dardized Coefficient of Variation (COV) are also included in the following comparison ofspecific measures. Results based on these two cardinal measures are plotted in Figure2, together with the values of four indexes developed specifically for inequality measure-ment with ordinal data. The IBL and the IvdE were shown to satisfy the Allison-Fosterordering without being restricted to distributions with equal median. This is why theyare plotted on top of each other. For the same reason of comparability, the IαCF (s; e) andthe ranks-based Gini (IGR) are plotted on top of each other. They are based on similarideas of rankings without being median-based or satisfying the Allison-Foster ordering.For being applicable to specific data, the parameters of the IαCF (s; e) need to be speci-

fied. Here, I0upCF shall indicate that the same specification as in Cowell and Flachaire(2012) has been chosen: parameter α is set to zero and the concept of upward-lookingstatus is applied. The latter seems natural, since higher ratings on the 0-10 Likert scaleare desirable. Setting α = 0 means, that no tail of the distribution is given special em-phasis. These choices are of course critical, and sensitivity analyses would be advisable.This would however go beyond the scope of this paper and is consequently left for futureresearch. To justify the choice of combining upward-looking status and α = 0 refer-ence to the finding of Costa-Font and Cowell (2013) shall suffice: in the context ofinequalities in SAH changes in α are less critical in terms of potential re-rankings forupward-looking status, than for downward-looking-status. The IBL and the IvdE are

23

available as functions in the R-package provided by Ruedin (2014). The routines forcomputing I0upCF and IGR had to be programmed for this study.41

Figure 2:Evolution of Inequality in European Countries.

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The results for the different measures can be evaluated in terms of time consistencyand country consistency. The former would imply that lines in Figure 2 do not cross(are almost parallel), so that different years in a specific country would be ranked in thesame way for different measures. The latter would be characterized by the same verticalsorting of the different countries for a specific year, irrespective of the measure applied.This is obviously not the case here. Nevertheless, some insights may be gained aboutinequality between countries and the development of inequality over time.On first inspection, the plots resembling each other the most are the SD and the IBL.

The COV looks somehow similar. A closer look at these three plots, reveals however,that for a significant number of years, country rankings differ depending on the measureunder consideration. In 2002 e.g., the most unequal country according to the SD is France(FR). Though much closer, this is also the case with the IBL in 2002 but not in 2012(where FR is slightly more equal than Hungary (HU) according to the IBL). In contrast,the COV clearly ranks HU to be the most unequal country in terms of life satisfactionover the whole period 2002-2012.

41 These routines were also programmed in R. It is planned to make them available as an R-package.Until then, the functions are made available upon request.

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Thus, cardinal measures do not come up with consistent results in terms of rankingcountries. The same can be said for the measures based on the Allison-Foster ordering.This, however, is not surprising given that the Allison-Foster dominance can hardly befound in the data. Nevertheless, plots for the IBL and IvdE closely resemble each other.When rankings differ, the measures are often very close. Interestingly, plots for thesemeasures rather closely resemble the plots for cardinal measures. Both types of measuresdo not come up with consistent country rankings. However, considering time trajectoriesfor any specific country, the results seem to be more consistent. Inequality re-rankingsof years appear to happen less frequently than re-rankings of countries. This somehowsupports the general finding of many happiness researchers of declining inequality in lifesatisfaction during the last 40-50 years or so.So far, the panels on the right in Figure 2 have not been discussed. Visual inspection

suggests that results for the I0upCF and the IGR differ from their neighboring panels tothe left, but also between each other. The I0upCF is consistent with the other measures inranking HU as (one of the) most unequal countries and DK (among) the most equal ones.But overall, country re-rankings seem to be more common here. Also, the magnitude ofcountry re-rankings seems to be larger. PT is ranked the second most unequal countryby the measures in the left and middle panel in only a few years. In contrast, PT,ranks second in the I0upCF -ranking over the whole period 2002-2012. Time-trajectories arequalitatively similar to the panels to the left.The IGR almost completely differs from all other panels. The most unequal country

according to the Gini is NL, the most equal ones are DE and FR. Especially the rankingof NL to be the most unequal country in terms of life satisfaction does not seem veryplausible when inspecting the histogram (Figure 1) more closely. As has been noted,almost 40 Percent of ratings are in category 7. Recalling that the standard (cardinal)Gini is especially sensitive to changes in the middle of the distribution, there might besome analogous implicit weighting with the IGR.As a first result of this discussion, it can be noted that results for the different indexes

– with the exception of the IGR – are resembling each other. Nevertheless, re-rankingsfrequently occur: more often for countries than for years in a time trajectory of a specificcountry. If re-rankings happen, these are typically close calls. With respect to the IGRfuture research should clarify if there is an implicit weighting that potentially causesimplausible results. Another peculiarity with the IGR is the fact that the results are allin a very narrow interval which might be another reason for implausible re-rankings.This directly leads to the question of significance of the results. In particular, the

questions are, whether between-country differences are in fact significant and whethermovements over time are actually mapping a change in specific countries instead ofdisplaying only noise. To shed light on these issues, bootstrapped confidence intervalshave been calculated following a strategy that shall briefly be described here: Fromthe original data 2,000 subsamples of 1/3 of the number of original cases are drawnwithout replacement from separate subsets (one set for each combination of country

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and year).42,43 These boot samples were then used to compute the different inequalitymeasures. The quantiles of these results were then used to get the 95 Percent confidenceintervals.

Figure 3:Bootstrapped Confidence Intervals for Inequality Measures.

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For the sake of clarity, confidence intervals are displayed in Figure 3 as shaded areasfor only four countries (HU, DE, ES and IE). When confidence intervals for one countryare non-overlapping between years, this means that there is a 90.25 Percent probability,that index values for those years with non-overlapping intervals do not only differ bychance.Hence, when using the SD, inequality in HU and DE significantly differ in all years

except 2004. For the IGR inequality in these countries however differs in 2010 and 2012.For all other measures, inequality in HU and DE significantly differs over the wholeperiod. There is no pair of countries, for which inequality significantly differs over thewhole period irrespective of the measure employed. Disregarding results of the IGR, theonly pair of countries that significantly differ in all years and for all measures is IE andHU.

42 The decision to draw without replacement was made due to the fact that the subsets make up ratherlarge subsamples of around 2,000 observations from which to draw.

43 Karoly and Schröder (2014) suggest a Jackknifing-Approach for calculating confidence intervalsof inequality measures in the cardinal case.

26

With respect to changes over time, there is no country in which inequality has notsignificantly changed between any pair of years. However, for HU no such change hasoccurred when inequality is measured by the SD, the IBL, the IvdE and the I0upCF , so thatchanges in the index values may in fact be only random. However, when measured bythe COV and the IGR inequality in life satisfaction significantly changes in HU for onepair of years, respectively. According to the COV, inequality seems to have declined inHU between 2008 and 2010. For the IGR a rather ‘long-term’ decline can be diagnosedbetween 2002 and 2012 (though changes between other pairs are insignificant). For theother three countries displayed in Figure 3, changes in inequality over time are significantfor all measures. This may indicate some effects of the EU debt crises which seems to beaccompanied by increasing inequality in Ireland (IE) and ES, whereas the distributionof life satisfaction has become more equal in DE. This description holds for all measures,except the IGR, which leads to the opposite diagnosis.Recalling the result in Madden (2011) of a tendency of convergence in different do-

mains of life satisfaction during the economic boom in Ireland, it seems plausible thatinequality may increase in times of recession or crisis. Of course, one can easily imaginetheoretical reasons why crises may also be accompanied by convergence: One such argu-ment could be that in times of crises, people show more solidarity which in turn leadsthem to rate their lives more uniformly. Which kind of causality is actually true cannotbe answered here. But the fact that 5 out of 6 measures yield the same qualitative resultof increasing inequality in IE and ES together with decreasing inequality in DE, is insupport of inverting the findings of Madden (2011). Especially the timing appears plau-sible, too. IE was the first European country hit by the crisis (around 2008), and indeed,inequality rises first. But since this rise can already be seen fo the years before 2008 inFigure 3, there maybe other factors than the crises explaining this increase. For ES, thehypothesis of a crisis effect seems more plausible. Here, the marked increase happensbetween 2010 and 2012 nicely coinciding with the occurrence of the Spanish debt crisis.Interestingly in the case of DE, inequality has hardly changed between 2002 and 2006 –a period of massive changes in social policy often accused to having lead to a divisionof society. The decrease in German inequality in life satisfaction in recent years mayindicate that it is not absolute booms, being needed for convergence, but that relativeperformance matters more in terms of life satisfaction.Summing up, the analysis has shown, that dominance, especially of the Allison-Foster

type, is rather restrictive when applied to data measured on a 0-10 Likert scale. Never-theless, the dominance findings reiterate the problems with (simple) cardinalization ashas been pointed out with the example of Denmark and the Netherlands. The findingsfor specific measures may be summarized as follows: Except for the ranks-based Gini Co-efficient, the results for the parametric measures differ more when used in cross-countrycomparisons, compared to a perspective over time. In the time dimension, re-rankings areless frequently occurring than in the cross-country dimension. This is somehow surpris-ing, given that histograms differ more between countries than within countries over time.What is not surprising, is the fact of ambiguous findings depending on the meausureemployed. Since dominance is hardly found in the data, the specific parameterization

27

is crucial for rankings. Despite these caveats it seems, nevertheless, fair to claim thatinequality has increased in Spain and Ireland, while Germany has experienced a conver-gence in terms of life satisfaction accompanying the relatively good economic performanceduring the EU debt crisis. Only when the results of the IGR are taken for granted, wouldthis description of the data be misleading. But properties of the ranks-based Gini arelargely unknown, so that more research is indicated to gain further confidence in thisinterpretation of the data.

4 Conclusion

More than 10 years ago, Blair and Lacy (2000, p.252) argued on “the insufficientlyrecognized substantive importance of dispersion in ordinal data.” The discussion insection 2 has shown that this critique still remains valid for some fields of research, suchas happiness studies. However, methodological progress has been made in other fields,allowing to better address the nature of ordinal data which are ubiquitously availablein social survey data nowadays. The integrated approach of discussing methodologicalissues together with empirical results on life satisfaction has proved to be fruitful inyielding important insights. One such insight concerns appropriate research designs forstudies employing ordinal data. A combination of dominance criteria and (parametric)inequality measures from different families seems promising to gain a richer picture ofthe data.From a Philosophy of Science perspective, an important insight of this discussion is

that different fields in the social sciences apply rather distinct methods, without appre-ciating works in other fields. Simple cardinalization is still the predominant approach inhappiness research which casts doubt on empirical findings in this discipline. This doubtseems appropriate not only for the theoretical arguments, but also for contradictionspresent in empirical findings. The problem of possibly reversed orderings when ordinalscales are interpreted as being cardinal has been prominently emphasized by Allisonand Foster (2004). They propose a partial ordering for the ordinal case having thepotential of coming up with robust inequality rankings. This property has spurred thedevelopment of corresponding parametric measures in the last 10 years or so, especiallyin health economics. But the Allison-Foster ordering turns out to be problematic, intheory and practice (see below). In theory, Cowell and Flachaire (2012) argue thatthe median is not a sensible point of reference when introducing an alternative inequal-ity measure for ordinal data. Moreover, the Allison-Foster ordering is questionable inontological terms (Kalmijn and Veenhoven, 2005).In recent years, a myriad of measures has been suggested for measuring inequality with

ordinal data. However, authors often do not relate their proposals to existing measures,making it hard for the applied researcher to distinguish substitutes from complements.This article tried to classify some of the measures. In this context, it could be shown thatthe measures by Blair and Lacy (2000) and Van der Eijk (2001) indeed satisfy theAllison-Foster ordering. A feature of these measures which has not been documented inthe literature until now. Together with their property of median independence, this was

28

the reason to choose them to represent the family of Allison-Foster type measures in theempirical part pf this paper. Two other families of measures (cardinal, ranks-based) werealso used in the empirical part to gain a picture of the data as complete as possible. Also,this procedure allows to assess the quality of cardinal measures. Beforehand, dominancecriteria were applied to get an idea of how robust findings of different measures are.The application of dominance criteria revealed another issue with the Allison-Foster

ordering. It turns out that this partial ordering is very restrictive when applied to arather dense ordinal scale as the 0-10 Likert scale employed in the ESS (and many othersurveys). In fact Allison-Foster dominance hardly shows up in the data. Consequently, itis not surprising that results from the family of measures incorporating the Allison-Fosterordering (in this study the IBL and the IvdE) do not completely coincide. However, thegeneral characterization of inequality between countries and within countries over time,can still be termed as being fairly similar for these measures. Interestingly, the patternsof the IBL and the IvdE over time also resemble those of the cardinal measures. Oncloser inspection however, re-rankings can frequently be observed. Different patterns arefound for the ranks-based measures. Here, the results from the IGR almost completelydiffer from all other measures, whereas for the ICF some commonalities with cardinaland median-based measures can be found.Generally, it seems that results for different measures are qualitatively more similar

with respect to the ranking of years for a specific country than for the ranking of countriesin a specific year. With respect to within country developments in the period 2002-2012,it seems fair to claim that inequality has not changed in Hungary, but increased in Spainand Ireland. In Germany, which has performed relatively well economically during theyears of crisis, a convergence in terms of life satisfaction can be observed. This suggeststhat it is not absolute booms, but relative economic performance that matters for lifesatisfaction ratings.These interpretations need to be taken with caution, since no empirical strategy has

been employed to really address the link between booms/crises and life satisfaction. Theyare rather based on plausibility and primarily are of heuristic value. Obviously, there areseveral lines for future research. A promising one is to move forward in classifying differ-ent parametric measures, so that the applied researcher can better discriminate betweensubstitutes and complements when certain dominance criteria apply. Also, properties ofsome measures need to be better understood. This is especially true for the ranks-basedGini which yields findings that rather contradict those of the other measures.Finally it should be pointed out that these methodological and empirical results are

of great relevance for policy. The methodological insights matter because only robustfindings can prevent the political exploitation of scientific results. The empirical resultsare important in that they complement findings on political movements, such as Maarekand Dorsch (2014), who find inequality in income to be affecting political movementsin an asymmetric way during booms and recessions.

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References

Abul Naga, Ramses H. and T. Yalcin (2008): Inequality Measurement for OrderedResponse Health Data. Journal of Health Economics, 27(6): S. 1614 – 1625.

Abul Naga, Ramses H. and T. Yalcin (2010): Median Independent Inequality Order-ings. SIRE Discussion Papers 2010-118, Scottish Institute for Research in Economics(SIRE).

Allison, R.Andrew and J. E. Foster (2004): Measuring Health Inequality UsingQualitative Data. Journal of Health Economics, 23(3): S. 505 – 524.

Apouey, Bénédicte (2007): Measuring Health Polarization with Self-Assessed HealthData. Health Economics, 16(9): S. 875–894.

Apouey, Bénédicte and J. Silber (2013): Inequality and bi-polarization in socioeco-nomic status and health: Ordinal approaches. PSE Working Papers halshs-00850014,HAL.

Berry, Kenneth J. and P. W. Mielke Jr. (1992): Assessment of Variation in OrdinalData. Perceptual and Motor Skills, 74(1): S. 63–66.

Blair, Julian and M. G. Lacy (2000): Statistics of Ordinal Variation. SociologicalMethods and Research, 28(3): S. 251–280.

Calcagnì, A. and L. Lombardi (2014): Dynamic Fuzzy Rating Tracker (DYFRAT): anovel methodology for modeling real-time dynamic cognitive processes in rating scales.Applied Soft Computing, 24(0): S. 948 – 961.

Clark, Andrew E., S. Flèche and C. Senik (2012): The Great Happiness Mod-eration. SOEPpapers on Multidisciplinary Panel Data Research 468, DIW Berlin,The German Socio-Economic Panel (SOEP). Forthcoming in A.E. Clark and C. Senik(Eds.), Happiness and Economic Growth: Lessons from Developing Countries. Oxford:Oxford University Press.

Clark, Andrew E., S. Flèche and C. Senik (2014): Economic Growth Evens-OutHappiness: Evidence from Six Surveys. PSE Working Papers halshs-00936145, HAL.

Costa-Font, Joan and F. A. Cowell (2013): Measuring Health Inequality with Cat-egorical Data: Some Regional Patterns. CESifo Working Paper Series 4427, CESifoGroup Munich.

Costa-Font, Joan and C. Hernández-Quevedo (2013): Inequalities in self-reportedhealth: a meta-regression analysis. Health working paper series in health policy andeconomics 32, London School of Economics and Political Science, LSE Health andSocial Care.

30

Cowell, Frank A. and E. Flachaire (2012): Inequality with Ordinal Data. TechnicalReport, Centre Interuniversitaire de Recherche en Économie Quantitative (CIREQ).

Davidson, Russell (2006): Stochastic Dominance.

Doorslaer, Eddy van and A. M. Jones (2003): Inequalities in self-reported health:validation of a new approach to measurement . Journal of Health Economics, 22(1): S.61 – 87.

Dutta, Indranil and J. Foster (2013): Inequality of Happiness in the U.S.:1972–2010 . Review of Income and Wealth, 59(3): S. 393–415.

Eijk, Cees Van der (2001): Measuring Agreement in Ordered Rating Scales. Qualityand Quantity, 35(3): S. 325–341.

Frey, Bruno S. (2012): Happiness and Public Policies: Fundamental Issues. CREMAWorking Paper Series 2012-16, Center for Research in Economics, Management andthe Arts (CREMA).

Giudici, Paolo and E. Raffinetti (2011): A Gini Concentration Quality Measurefor Ordinal Variables. Serie Statistica 1/2011, Dipartimento di Economia, Statistica eDiritto, Università degli Studi di Pavia.

Groot, Wim (2000): Adaptation and scale of reference bias in self-assessments of qualityof life. Journal of Health Economics, 19(3): S. 403 – 420.

Kalmijn, Wim (2010): Quantification of Happiness Inequality . PhD thesis, ErasmusUniversity Rotterdam.

Kalmijn, Wim and R. Veenhoven (2005): Measuring Inequality of Happiness in Na-tions: In Search for Proper Statistics. Journal of Happiness Studies, 6(4): S. 357–396.

Kalmijn, W.M. and L. Arends (2010): Measures of Inequality: Application to Hap-piness in Nations. Social Indicators Research: an International and InterdisciplinaryJournal for Quality-of-Life Measurement, 99(1): S. 147–162.

Karoly, Lynn A. and C. Schröder (2014): Fast Methods for Jackknifing InequalityIndices. SOEPpapers on Multidisciplinary Panel Data Research 643, DIW Berlin, TheGerman Socio-Economic Panel (SOEP.

Knapp, Thomas R. (1990): Treating Ordinal Scales as Interval Scales: An Attempt ToResolve the Controversy . Nursing Research, 39(2): S. 121–123.

Kobus, Martyna (2014): Polarization measurement for ordinal data. The Journal ofEconomic Inequality, pp. 1–23.

Maarek, Paul and M. T. Dorsch (2014): Recessions, Inequality, and Democratiza-tion. THEMA Working Papers 2014-19, THEMA (THéorie Economique, Modélisationet Applications), Université de Cergy-Pontoise.

31

Madden, David (2008): Ordinal and Cardinal Measures of Health Inequality: An Em-pirical Comparison. HEDG Working Papers, Health, Econometrics and Data Group,Department of Economics, University of York.

Madden, David (2010): Ordinal and cardinal measures of health inequality: an empir-ical comparison. Health Economics, 19(2): S. 243–250.

Madden, David (2011): The Impact of an Economic Boom on the Level and Distributionof Subjective Well-Being: Ireland, 1994–2001 . Journal of Happiness Studies, 12(4): S.667–679.

Madden, David (2014): Dominance and the Measurement of Inequality . In Cuyler,Anthony J., ed.: Encyclopedia of Health Economics, pp. 204–208. Elsevier, 1 ed.

Piketty, Thomas (2014): Capital in the Twenty-First Century . The Belknap Press ofHarvard University Press.

Praag, Bernard M.S. van (1991): Ordinal and cardinal utility: An integration of thetwo dimensions of the welfare concept . Journal of Econometrics, 50(1–2): S. 69 – 89.

Ruedin, Didier (2014): agrmt: Calculate agreement . R package version 1.35/r67.

Senik, Claudia (2011): The French Unhappiness Puzzle: The Cultural Dimension ofHappiness. IZA Discussion Papers 6175, Institute for the Study of Labor (IZA).

Sirgy, M. Joseph (2001): Handbook of Quality-of-Life Research. An Ethical MarketingPerspective, vol. 8 of Social Indicators Research Series. Springer Science+BusinessMedia, Dordrecht.

Stevenson, Betsey and J. Wolfers (2008): Happiness Inequality in the UnitedStates. Working Paper 14220, National Bureau of Economic Research.

Stevenson, Betsey and J. Wolfers (2013): Subjective Well-Being and Income: IsThere Any Evidence of Satiation? . Working Paper 18992, National Bureau of EconomicResearch.

Studer, Raphael (2011): Does it matter how happiness is measured? Evidence from arandomized controlled experiment . ECON - Working Papers econwp049, Departmentof Economics - University of Zurich.

Studer, Raphael and R. Winkelmann (2014): Reported Happiness, Fast and Slow .Social Indicators Research, 117(3): S. 1055–1067.

Survey, European Social (2012): Source Questionnaire Amendment 01 (Round 6,2012/13). Technical Report, Centre for Comparative Social Surveys, City UniversityLondon.

32

Tastle, William J. and M. J. Wierman (2007): Consensus and dissention: A mea-sure of ordinal dispersion. International Journal of Approximate Reasoning, 45(3): S.531–545. North American Fuzzy Information Processing Society Annual Conference{NAFIPS} ’2005.

Veenhoven, Ruut (2005): Return of Inequality in Modern Society? Test by Dispersionof Life-Satisfaction Across Time and Nations. Journal of Happiness Studies, 6(4): S.457–487.

Veenhoven, Ruut (2009): The International Scale Interval Study: Improving the Com-parability of Responses to Survey Questions About Happiness. In Møller, Valerieand D. Huschka, eds.: Quality of Life and the Millennium Challenge, vol. 35 of SocialIndicators Research Series, pp. 45–58. Springer Netherlands.

Wagstaff, Adam and E. Van Doorslaer (1994): Measuring inequalities in healthin the presence of multiple-category morbidity indicators. Health Economics, 3(4): S.281–291.

Yalonetzky, Gaston (2013): Stochastic Dominance with Ordinal Variables: Condi-tions and a Test . Econometric Reviews, 32(1): S. 126–163.

Zheng, Buhong (2008): Measuring inequality with ordinal data: a note. In Bishop,John and B. Zheng, eds.: Inequality and Opportunity: Papers from the SecondECINEQ Society Meeting , vol. 16 of Research on Economic Inequality , pp. 177–188.Emerald Group Publishing Limited.

Zheng, Buhong (2011): A new approach to measure socioeconomic inequality in health.The Journal of Economic Inequality, 9(4): S. 555–577.

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