1.1 Introduction
Multilateral comparisons of collections or groups of things are ubiquitous. League tables of national (or regional) education levels, health outcomes, income levels, average self-reported happiness levels, poverty and inequality measures and sports team performance abound. In other spheres, asset returns distributions of portfolio managers are compared; school based student educational attainments are contrasted; and treatment effects, the result of controlled experiments, are compared, ordered and contrasted. Degrees of societal generational income persistence are correlated with the levels of relative inequality in those societies. In other paradigms, the persistence of class membership over generations or the dependence of outcomes on different sets of circumstances is of interest. All of these pursuits relate to ranking, ordering or assessing the extent of differences between, groups or collections of things according to some implicit or explicit underlying criterion function. This book is about this generic activity as seen through the lens of wellbeing measurement, largely because researchers in that particular sphere have given a great deal of thought to the matter; however, it should be stressed that these ideas have application in many spheres beyond that of wellbeing.
Invariably group comparisons are based upon measures, usually summary statistics, of varying degrees of sophistry1 that summarize and represent an aspect (or aspects) of interest that prevail in each of the groups under comparison. As such, they provide a complete ordering of the groups in that any one group is measurably worse, better or the same as any other group in the comparison set. The choice of statistic is based on an implicit or explicit criterion function underlying the process. It should be noted that the generality of the criterion function can often be a source of confusion because the particular configuration of groups being compared inherently engender ambiguity or conflict between alternative comparison instruments appropriate for a particular criterion function. Indeed, this particular problem has led to an alternative comparison methodology offering an unambiguous but only partial ordering of groups in the sense sometimes one group is revealed as not definitively better, worse or the same as another group. Interestingly it transpires that the partial ordering methodology can inform the complete ordering methodology as to the potential for ambiguity in the ranking process. These issues will be explored in the book.
When comparing groups, a basis for comparison has to be established at the outset. Are the group aspects being compared reflective of some quality of “goodness” or “badness” of the group? If so, does it matter how that “goodness” or “badness” is shared within a group? Are there mitigating features which mean that individual impoverishment or overabundance of the quality needs to be compensated for? When studying the effectiveness of a collection of treatments is the spread of outcomes of a particular treatment a matter for concern? Does the relative size of the groups matter? These questions have to be addressed by the investigator in articulating the criterion function and contemplating potential comparison approaches if a complete ordering of the groups is to be attempted, formulating the sort of index or indicator that is to be used in the process. A field of economics that has paid some attention to the appropriate criterion function is welfare economics and social choice theory; however, its deliberations are relevant and applicable in fields far beyond comparison of group wellbeing.
In truth, in wellbeing measurement, these summary statistics are attempts at reflecting aspects of something that is inherently unmeasurable; the criterion function is in effect a measure of the aggregate wellbeing of a group or its “common good”. However, formulating an idea of what should be embedded in such a function can inform the choice of summary statistic (or whether some other approach should be taken). Indeed, outside the realm of wellbeing measurement, it makes much sense for an investigator to articulate the criterion function underlying the comparison exercise since it will lend clarity to the choice of comparison instrument and methodology. So, before embarking upon a study of the measurement and comparison exercise, the nature of a “welfare”, “common good” or criterion function which underlays the many different notions of the wellbeing measurement exercise (together with some of the objections that economists have made to such a comparison activity) will be explored later in this introductory chapter.
1.2 An Outline of What Follows
Since much of the following relies upon statistical constructs (i.e. probability distributions) which describe of the manner in which the aspect of interest is allocated across groups, Chap. 2 provides some necessary background for the various applied statistical components that later chapters rely upon. It outlines the very basic concepts regarding the nature, properties and use of probability distributions which treat the variables of interest as random and describe their allocation across the groups under comparison. Generally, for practical comparison purposes, distributions are not known a priori and have to be estimated, and so the chapter moves on to distinguish between parametric versus non-parametric investigative approaches to estimating distributions. When there is no information about a parametric distributional structure, or at least an unwillingness on the part of the investigator to make assumptions about parametric structure, probability distributions can be estimated using kernel techniques, the basic essentials of which are also outlined. The book is focused on multilateral comparisons; however, from a wellbeing comparison perspective, the main approach is to test for stochastic dominance relations which, together with overlap and Transvariation methods, are bilateral comparison techniques. The basic essentials of these bilateral comparison techniques are described in this chapter with a view of extending them to multilateral comparisons in later chapters. This family of techniques can sometimes fall foul of the test inconsistency problem which, together with some of the solutions to it, are described in the final section.
The most common practice is to formulate an index, frequently on an axiomatic basis, which provides a complete ordering for the ranking process and this, together with some of the pitfalls of this activity, is what is discussed in Chap. 3. After an introduction which deals with some of the difficulties associated with formulating and employing indices, Sects. 3.3 and 3.4 discuss some basic level of wellbeing and unit free inequality indices which leads to a discussion of inequality adjusted wellbeing level measures in Sect. 3.5. Notions of segmentation and polarization are distinguished in Sect. 3.6 with multivariate extensions of these being considered in Sect. 3.7. Poverty measurement and equality of opportunity and mobility indices are discussed in Sects. 3.8 and 3.9. Having alluded to the ambiguity problem in the introduction it is illustrated in the final section of the chapter.
An alternative comparison strategy is outlined in Chap. 4. Stochastic dominance criteria are used to study the orientation of group distributions to see if, given the nature of the chosen criterion function class, the juxtaposition of the group distributions will admit an unambiguous ordering. Sometimes they will, sometimes they won’t, in effect the ordering is only partial; however, in the event that an unambiguous ordering is not admitted, a similar exercise can be pursued in the light of a more restrictive criterion function class. This chapter outlines some basic dominance criteria and the way in which successive orders of dominance restrict the nature of the criterion function. Based upon these concepts, Chap. 4 then considers techniques for revealing the extent of ambiguity surrounding the comparison exercise, develops some ideas for establishing unambiguous groupings and formulates a general family of ordering indices that reflect restrictions implied on the criterion function implied by the level of stochastic ordering. The relationships between stochastic dominance and inequality and poverty orderings are covered in later sections as is the special case of polarization. The problem of ambiguity and how to measure the extent to which it prevails are discussed and ...