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Discussion Paper Series
CPD 02/15
Cultural Diversity – A Matter of Measurement
Peter Nijkamp and Jacques Poot
Centre for Research and Analysis of Migration
Department of Economics, University College London
Drayton House, 30 Gordon Street, London WC1H 0AX
w w w .c re a m -mi grati on. org
CULTURAL DIVERSITY –
A MATTER OF MEASUREMENT*
Peter Nijkamp
Jacques Poot
Department of Spatial Economics
VU University
Amsterdam, The Netherlands
[email protected]
National Institute of Demographic and Economic Analysis
University of Waikato
Hamilton, New Zealand
[email protected]
Abstract
Cultural diversity – in various forms – has in recent years turned into a prominent and
relevant research and policy issue. There is an avalanche of studies across many
disciplines that measure and analyse cultural diversity and its impacts. Based on
different perspectives and features of the available data, a great variety of diversity
indicators have emerged. The present paper aims to highlight some critical issues
involved in applying such measures of cultural diversity. A selection of commonly used
or recently advocated measures are reviewed. Measures of population diversity can be
calculated at different spatial scales and used to analyse spatio-temporal heterogeneity.
Additionally, there is a growing interest in measuring spatial dependence, particularly
in the form of segregation or clusters. We conclude that there will be in the future
considerable scope for adopting multidimensional and cultural distance-weighted
measures of diversity. Such measures will be increasingly calculated by means of rich
geo-referenced longitudinal micro data. However, adopted measures must be better
motivated by behavioural theories. Further research on the determinants and impacts of
observed measures of diversity is also likely to be fruitful, particularly in a dynamical
setting.
JEL Classification: C00, D63, J15, R23, Z13
Keywords: diversity, dissimilarity measurement, ethnicity, culture, segregation,
polarization, fractionalization
*
This is a draft of Chapter 2 forthcoming in J. Bakens, P. Nijkamp and J. Poot (eds) The Economics
of Cultural Diversity. (Cheltenham UK: Edward Elgar, 2015). An earlier version of this paper was
presented at the Workshop E Pluribus Prosperitas: The Economics of Cultural Diversity, Tinbergen
Institute, Amsterdam, April 15-17, 2013 and at a seminar at the Netherlands Interdisciplinary
Demographic Institute (NIDI) in The Hague on October 29, 2014. We would like to thank Annekatrin
Niebuhr, Daniel Arribas-Bel and workshop/seminar participants for comments. Ceren Ozgen, Alina
Todiras and Emma van Eijndhoven provided helpful input in developing this paper. The research
reported in the paper was initially conducted as part of the 2009-2013 Migrant Diversity and Regional
Disparity in Europe (MIDI-REDIE) project, funded by the NORFACE–Migration research programme,
www.norface-migration.org. Jacques Poot also acknowledges a grant from the Netherlands Institute for
Advanced Study in the Humanities and Social Sciences (NIAS) and grant UOWX1404 provided by the
New Zealand Ministry of Business, Innovation & Employment (MBIE) for the research programme
Capturing the Diversity Dividend of Aotearoa New Zealand (CaDDANZ).
1
INTRODUCTION
Migration and cultural diversity are intertwined phenomena. Following several decades
of growing numbers of immigrants in the developed world, the populations of host
countries have become more diverse: culturally, socio-economically, but also spatially.
This transformation has been particularly prominent in Europe. During the last three
decades, the foreign-born population in Europe has increased more than in any other
part of the world.
While migrants from some backgrounds adopt the dominant culture of the host
society quickly, others maintain the culture of their home country and pass this on to
their children and subsequent generations. Aspects of foreign cultures are also adopted
by the host population and people may feel individually attached to several cultures.
Moreover, cultures are never static but evolve and adjust in migrants’ countries of
origin and countries of destination.
The issue of cultural diversity has in recent years prompted a wealth of scientific
research, both conceptually and empirically. In our ‘age of migration’ we observe
interesting distinct patterns of living, working, communicating, bonding and other
behavior among and between different cultural groups. The question whether the socioeconomic and spatial outcomes of cultural diversity are positive or negative has led to
a rapidly growing literature. This literature is reviewed briefly in the next section. To
date, there have been many contrasting findings. These depend inter alia on: the
geography and spatial scale of analysis in the countries concerned; time scale;
classification of the groups considered; socio-economic conditions; institutions; and the
selected definition of diversity. This prompts the questions whether, firstly, there is a
best way to measure cultural diversity – non-spatially or spatially – and, secondly,
whether measurement of the impacts of diversity is sensitive to the choice of a diversity
indicator. These questions are the focus of the present chapter.
A vast collection of measures/indicators of diversity of populations has been
proposed in the literature. While such measures can all be applied in principle to cultural
diversity, many have their origin in other areas of socio-economic research or even in
other disciplines. To introduce the need for a systemic perspective on measuring
cultural diversity, we just mention here six – rather randomly selected – studies on
diversity, which use different measures of diversity among population groups. Firstly,
Åslund and Skans (2009) develop an index of systematic segregation in which they
1
account for covariates in a nonparametric way. Secondly, Constant and Zimmermann
(2008) define an ethnosizer index which aims to measure the intensity of a person’s
ethnic identity. Thirdly, Fearon (2003) uses an index of ethnic fractionalization and an
index of cultural fractionalization (with structural distances between languages being a
proxy for cultural distance in the latter measure). Fourthly, Olfert and Partridge (2011)
investigate the creative classes in cultural communities and employ a modified globalregion-of-birth fractionalization index. Fifthly, Ottaviano and Peri (2006) measure the
effect on wages and rents in US cities by means of the standard fractionalization index.
Finally, Wong and Shaw (2011) use an exposure measure of ethnic segregation but take
an activity space approach rather than the conventional residential perspective.
It is evident from the above examples that there is a wealth of indicators for
measuring non-spatial and spatial diversity or heterogeneity (we will use the terms
interchangeably) of populations. Following a short review of the literature on cultural
diversity and its socio-economic impacts in the next section, we outline some general
principles regarding measurement of diversity in Section 3. This is followed by a
discussion of common measures of population heterogeneity in Section 4. The
penultimate section is concerned with the combination of spatial and population
heterogeneity. We conclude with some retrospective and prospective remarks in
Section 6.
2
THE SIGNIFICANCE OF CULTURAL DIVERSITY
Socio-cultural diversity embodies a host of cultural, ethnic, religious, political and
demographic factors. These are to be distinguished from economic characteristics
(wealth, ability, education, occupation, etc.). Even without socio-cultural diversity, the
presence of people with different socio-economic attributes and heterogeneous
preferences will lead through competitive market forces and through societal
institutions to a non-random ‘sorting’ of people. This sorting can take place spatially
within cities (as already explained by urban land rent theories of Alonso, 1964, and
Muth, 1971; but see also: Borjas, 1998; Ihlanfelt and Scafidi, 2002; Johnston et al.,
2007), but also more broadly geographically across regions (e.g., Roback, 1982;
Moretti, 2011). Finally, sorting can be by skills and occupations across and within firms
(e.g., Heckman et al., 1998).
2
In the economist’s classic view of the world, the influence of intra- and intergroup
relationships on economic processes has been largely assumed away. However, in
recent years economists have started to pay increasingly attention to phenomena such
as social capital (Putnam, 2000) and interdependence of preferences (Brock and
Durlauf, 2001). In the broader social science literature, attention has already been paid
much longer to the question whether and how interactions between distinct population
groups have an influence on their behavior. Distinct population groups may be the result
of specific social values, religious convictions, cultural attitudes, lifestyles, languages
or traditions. Residential sorting – and in a stricter sense, residential segregation –
reflects the economic and socio-cultural diversity among subgroups of the population.
This observation is well articulated in the seminal studies of Sakoda (1971) and
Schelling (1971) that focused the attention on the intensity of social bonds or linkages
within a given household’s neighborhood in a city. Households will then try to move to
an urban district where the share of their friends and relatives is above a certain
threshold. These studies have prompted an avalanche of spatial segregation research in
cities (Bruch and Mare, 2006; Fossett, 2006a, 2006b; Benenson, 1998; Boal, 1978;
Speare, 1974). Unfortunately, a solid quantitative test of these ideas that lends itself to
generalization and comparative study is still missing. As a consequence, there is much
speculation about the assumed drivers and effects of socio-cultural diversity in urban
neighborhoods. Especially the definition of a clearly and unambiguously demarcated
urban space that forms a justified unit of analysis for comparative study is lacking.
It is noteworthy that even leading researchers in the field have come up with rather
contrasting conclusions on the impacts of diversity. We will illustrate this by briefly
summarizing two important studies. First, Page (2007) highlights in his book The
Difference the beneficial effects of diversity in a social environment, such as schools,
firms and organizations. Diversity of membership of such social constellations appears
to improve their functioning. Page seeks to explain the emergence of diversity benefits
mainly in terms of the ability of organized social structures to employ a variety of
cognitive tools based on training, experience and genes. His work and his positive
conclusions on diversity are supported by a wealth of empirical material, pertaining to
different countries and different social groupings. A thorough review of his study can
be found in Ioannides (2010).
An opposite conclusion can be found in a 2006 article entitled E Pluribus Unum by
Putnam, in which he argues that (ethnic) diversity leads to an erosion of social capital.
3
Diversity undermines communication between people and hence affects social
solidarity, not only between distinct groups but also among group members.
Consequently, diversity will negatively impact on social norms (like reciprocity, trust,
worthiness, etc.). Based on a series of empirical studies he arrives at the conclusion that
diversity has always a negative effect and undermines stable social networks, leading
in the end to increased social isolation (the ‘hunkering down’ hypothesis).
Putnam (2000) highlighted the distinction between bonding and bridging in the
development of social capital in migrant communities. The interaction of workers of
different cultural backgrounds with the host population can increase productivity due
to knowledge spillovers or other forms of positive externalities (see e.g. the sociocultural mixed embeddedness hypothesis by Kloosterman and Rath, 2001). This result
is only an advantage up to a certain degree. When the variety of ethnic or cultural
backgrounds is too diverse, fractionalization may imply excessive transaction costs for
communication (the ‘Babylon effect’; see Florax et al., 2005) and may therefore lower
productivity.
On the other hand, cultural diversity among migrants may impact positively on host
societies by enhancing the quality of life in cities and consumer choice options. A
tolerant native population may value a multicultural society because it increases the
range of available goods and services. Although cultural economics is a growing field
(Blaug, 2001), the potential benefits of migration-induced cultural diversity on the arts
– and culture in general – has hardly been investigated from an economic perspective.
Admittedly, diversity may be perceived as an unattractive feature if, e.g., natives
recognize it as a distortion of national or local identity. They may discriminate against
other ethnic groups and fear that social conflicts between different foreign nationalities
may be imported into their own neighborhoods. Because of the spatial selectivity of
migration, the impacts of the aforementioned mechanisms are likely to be amplified at
the regional and urban level. Diversity may also influence subjective wellbeing of
individuals in positive or negative ways (e.g., in terms of perceptions of neighborhood
safety).
Given the growing importance of linkages between migrant source and destination
countries through global diaspora, the perspectives of migrant source countries are also
of great relevance. Migrants and culturally defined communities form various networks
that have a range of impacts, such as on international trade and on the spatial clustering
of migrant groups. In the presence of growing benefits of urban agglomeration, which
4
the world is witnessing at present, the fact that migrants are often drawn from peripheral
areas and drawn to large agglomerations may reinforce differences in socio-economic
development between these areas (see Ozgen et al. 2010 for meta-analytic evidence).
Research on the socio-economic and spatial impact of cultural diversity is nowadays
intensifying (see for overviews: Alesina and La Ferrara, 2005; Olfert and Partridge,
2011; Ratna et al., 2009). Economic studies of socio-cultural diversity tend to
emphasize the positive externalities emerging from the presence and interaction of
diverse population groups. However, growing socio-cultural diversity of the population
may affect the host economy through many different channels. The net impact depends
on the relative strength of these effects. A seminal study by Ottaviano and Peri (2006)
demonstrated that US-born citizens experienced both a wage increase and a rising rental
price of their house in metropolitan areas in which the share of foreign-born people
increased. They conclude that there is a significant positive effect of cultural diversity
on the productivity of the native born.
A promising start has also been made with the analysis of the interrelationships
between cultural diversity and economic performance in Europe (Bellini et al., 2008;
Suedekum et al., 2014; Ozgen et al., 2012). While most research finds that there is a
positive causal link from migrant diversity to regional productivity and innovation, the
relationship is complex and may be specific to certain industries, types of firm, or types
of workers (for a review, see Ozgen et al., 2014). In addition, there has been a series of
new studies – largely on the basis of micro data – that have aimed to assess the
disaggregate benefits of cultural diversity, for instance, on the degree of firm’s
innovativeness (e.g., Ozgen et al., 2013), on entrepreneurship (e.g., Audretsch et al.,
2010; Sahin et al., 2010), on plant productivity (e.g., Haas and Lucht, 2013; Trax et al.,
2012), on labor demand (e.g., Brunow and Blien, 2014), on income (e.g., Brunow and
Brenzel, 2012) but also on social issues such as trust (e.g., Gundelach and Freitag,
2014), social cohesion (e.g., Sturgis et al., 2014) and subjective wellbeing (e.g., Longhi,
2014).
The above sketched contrasting viewpoints and findings suggest that, on one hand,
diversity increases contacts between diverse people and thereby improves mutual
understanding and trust and, on the other hand, diversity will magnify conflicting
interests between groups and thereby create the basis for a disintegration of society. It
is clear that both sets of phenomena are potentially concurrent manifestations of reality
5
with the relative strengths of each being context specific and probably also dependent
on the operationalization of the concept of diversity.
The literature on diversity has an abundance of related concepts. Some of these are
non-spatial and refer to the (multi-dimensional) composition of a given population.
Others are concerned with the varying presence of population sub-groups across space.
Two common concepts of the latter are segregation and (spatial) concentration.
Concentration is usually interpreted as referring specifically to geography or space, i.e.
the extent to which a population defined by a certain attribute is unevenly dispersed
across spatial units. Segregation refers to the degree to which two or more groups
defined by one attribute (age, sex, ethnicity) are distributed unequally across another
attribute (location, occupation, wealth, etc.). Clearly, a relatively low segregation
implies that a difference between the groups is not discernible for the given
classification. For example, if there was no gender segregation across occupations,
knowing a person’s occupation would not be informative of the person’s gender (this
idea of reducing attributes to those that have predictive power in comparing individuals
is the foundation of rough set theory, see Pawlak, 1997).
A wealth of conceptual and empirical studies has been undertaken to understand
and map out segregation and concentration (see, e.g., Friedrichs, 1998; van Kempen,
2005; Musterd and van Kempen, 2009; Phillips, 1998). In many cases, a high level of
segregation and a high spatial concentration of certain socio-cultural or ethnic groups
are regarded as undesirable situations that may reinforce a disadvantaged socioeconomic position of certain population groups. This has prompted a debate on policies
to reduce segregation in cities, for instance, by introducing a residential quota system,
by fostering urban economic policies that would benefit less privileged people, or by
implementing a system of affirmative action which would give priority to
disadvantaged ethnic or socio-cultural groups in the local labor or housing market (see
e.g., Feinberg, 1985; Moro and Norman, 2003).
Additionally, the policy debate in many European countries has focused on
improving the skill composition of the immigrant flow. In some countries attempts are
made to attract high-skilled temporary and permanent immigrants to specific industries
suffering from labor shortages (e.g., in the IT industry). Cultural diversity of migrants
has to date played only a minor role in this policy debate.
Hence, a fundamental question is whether, and to what extent, migrant diversity –
based on different origins and socio-economic capabilities – may be seen as a
6
potentially positive contributor to economic welfare, and to what extent it influences
socio-economic disparity. Additionally, the way in which residential choice and
mobility of migrants impacts on socio-economic outcomes at specific locations is also
of great interest.
All empirical studies must operationalize concepts of cultural diversity and spatial
segregation of population sub-groups. Many use the same measures but rarely assess
the properties and suitability of such measures in the specific context. Following a brief
discussion in Section 3 of concepts, terminology and notation that can be applied to all
measures of cultural diversity and segregation discussed in this chapter, we first review
measures of heterogeneity of (sub) populations in Section 4 and then diversity of places
in Section 5.
3
GENERAL PRINCIPLES
Diversity is a concept that is nowadays used in a broad array of social science research.
In an edited volume by Knotter et al. (2011), the following application fields are
mentioned and extensively discussed: social anthropology, sociology, criminology,
ecology, linguistics, architecture, urban planning, geography, economics, management
and organization, psychology, law, and public policy/political science. There are
evidently different meanings and terminology (e.g. heterogeneity, diversity,
dissimilarity) attached to the concept of diversity, but a critical question is: how to
measure it? In socio-economic research on diversity the fractionalization index (a
measure of diversity among people) and the segregation index (a measure of diversity
among places) have certainly become the ‘market leaders’. The former refers to the
probability of two persons meeting who are not of the same type (Alesina et al., 2003).
The latter refers to the proportion of people of a particular type who would have to be
redistributed to make their spatial distribution (or with respect to some other type of
classification, such as occupation – see Duncan and Duncan, 1955) identical to that of
the rest of the population.
At the most fundamental level, all such measures are concerned with a population
(census or sample) of individuals who are classified across a range of characteristics,
also referred to as attributes. In order to assign people to groups, the attributes are
assumed to be qualitative/categorical variables. They may be ordinal (e.g. education)
or not (e.g. ethnicity). Attributes drawn from continuous distributions can be converted
7
into categorical variables by means of some process of discretization (e.g. from exact
age to age groups). Attributes are measured at different points in time, hence the date
and time of observation is itself an attribute of the person being observed. Given the
importance of geographical space, the attribute of location is often considered
separately from the others. Geographical space is partitioned into areas and individuals
are located in one of these areas.
Once groups have been defined in terms of one or more attributes, the population
can be partitioned into groups such that within a group all individuals have identical
observed attributes. Even though the description of groups may be detailed (i.e. many
attributes are considered), some unobserved heterogeneity may remain, which can of
course influence statistical analyses if it is non-randomly related to the observed
heterogeneity.
Traditionally, cultural diversity research focused only on one aspect and a
corresponding classification: culture, ethnicity, race, country of birth, language,
citizenship, religion, etc. However, it is clear that socio-economic impacts may depend
on attributes drawn from multiple classifications. Additionally, in our increasingly
mobile world with a growing complexity of migration types (e.g. Poot et al., 2008), the
number of relevant attributes is increasing and may include migrant visa status, spatial
selectivity and international networks. It is therefore not surprising that Vertovec (2007)
coined the term ‘super-diversity’ to describe this new reality. On the other hand, when
either the number of classifications is high and/or the categorization in each
classification is rather fine, many of the potential combinations are not observed in
practice. This is referred to as the ‘sparse matrix’ or ‘empty cells’ problem. Another
interesting issue when considering many attributes is the extent which the measure of
overall diversity is simply the sum of diversity measures for each of the attributes. Such
additivity would generally require at least independence across classifications.
This general setting is identical to that of the multi-way cross tabulation of
categorical data (also called a contingency table), which fundamentally refers to a
multivariate discrete distribution (with location in our context being one of these
variables). Many methods have been developed for analyzing such data, with the core
objective usually being the association and interaction (or independence) between
attributes (e.g., Bishop et al., 1975). Information theory has turned out to be helpful for
analysis of such tables (e.g., Gokhale and Kullback, 1978) and most statistical software
nowadays contains appropriate procedures, such as log-linear or general linear models.
8
For a recent review of this literature, see Fienberg (2011). In this context, measures of
diversity can be interpreted as the ‘moments’ of the multivariate distribution. Such
measures could be concerned, for example, with the dominance of the modal group,
bimodality (polarization), dispersion or evenness, skewness, etc.
For any given classification, the ‘distance’ between possible outcomes may be
important to describe the ‘intensity’ of diversity. While for cardinal variables this is
straightforward (the distance between age groups is the difference in mean ages), for
ordinal (e.g. qualifications) and non-ordinal (e.g. language) classifications it may
depend on multiple characteristics of each category. For example, a group of people
equally distributed across their four native languages French, Italian, Spanish and
Portuguese may be considered less diverse than an equally sized group equally
distributed across French, Japanese, Russian and Chinese. Distance-weighted measures
of diversity will be described later. It is useful to note that the recent emergence of ‘big
data’ (e.g., McAfee and Brynjolfsson, 2012), such as on interactions between
individuals measured through email and mobile phone traffic, or face-to-face visits
monitored through transportation data, may provide new sources of information that
signal socio-cultural distance between groups.
We will now introduce some notation that will be maintained throughout the
remainder of the chapter. We assume that we focus on a population of size Nt observed
at time t (t = 1, 2, …, T), in which each individual has a set of attributes (characteristics)
C and is located in area a (a = 1, 2, …, A). A group g (g = 1, 2, …, G) is defined by
people with identical attributes. The number of people who belong to group g in area a
is denoted by Pgat. A subscript  denotes aggregation over that index. Hence Nt  Pt =
∑ீ௚ୀଵ ∑஺௔ୀଵ ܲ௚௔௧.
We consider two types of distance: geographical distance and social-cultural
distance. The geographical distance may be measured in various ways, as in spatial
econometrics (e.g. Anselin, 1988). Socio-cultural distance between groups may also be
conceptualized in different ways (e.g., Karakayali, 2009; Desmet et al., 2009). The
geographical distance between areas r and s (defined, for example, by the distance
between population centroids) is denoted by rs and social-cultural distance between
groups i and j by ij. In both cases we normalize these distances to be on the interval
(0, 1] when r  s and i  j, but aa = 0 for all a and gg = 0 for all g. These distance
measures satisfy rs = sr and ij = ji for all r, s, i, j; and also rs  rq + qs and ij  ik
9
+kj. Consider for example linguistic distance: ij would be close to zero if i refers to
Dutch and j to Flemish, whereas ij would be close to one if i refers to Dutch and j to
Japanese.
Because group membership generally refers to multiple classifications, distances
between groups are multidimensional too and depend on the similarity between the
categories of the classifications (e.g., Bossert et al., 2003).
Even within one
classification, such as language, distance can be determined by the combination of a
number of criteria (Greenberg, 1956; McMahon and McMahon, 2005). The idea of
quantifying cultural (power) distance was introduced by Hofstede (2001). Various
subjective and objective methodologies may be used to define groups, and measure
distances between groups, but this remains challenging (Shenkar, 2001). However, for
the purpose of describing measures of diversity, the concepts and notation introduced
in this section is adequate.
Once groups have been defined, individuals may of course transition from one
group to another and from one location to another between successive points in time.
This calls for a longitudinal analysis in which irjs(t,t+d) refers to the transition
probability that an individual belongs to group i in area r at time t and to group j in area
s at time t + d, with d > 0. All those who enter our observations at time 0 with given
attributes i* and in area r* may be referred to as a cohort. It is clear that 0  irjs(t,t+d)
 1 and ir(t,t+d) = 1 for all d>0 when one group may exclusively capture attrition. If
irjs(t,t+1)
is independent of t, the transition process becomes a Markov chain.
It is plausible that transition will have some kind of ‘gravity’ property, as in
migration and trade modelling (e.g. Anderson, 2011): irjs(t,t+d) is likely to be
negatively correlated with both socio-cultural distance ij and geographical distance
rs, while the number of transitions is likely to be positively related to Pirt and to Pjst. In
fact, we could gauge the extent of similarity between groups by observing actual
transitions (‘revealed preference’) or by means of surveys (‘subjective valuation’). Of
particular interest for cultural diversity are language acquisition, cross-cultural
marriage and changing cultural identity across entry cohorts (first generation migrants)
and birth cohorts (higher generations).1
1
It is clear that the groups and transitions must be defined carefully. A migrant who becomes fluent
in a host country language and who would then refer to that language as her/his ‘first’ language would
become a member of a group that is bilingual rather than monolingual, as such a person would not, or at
least not fully, ‘unlearn’ their native language.
10
4
COMMON MEASURES OF POPULATION HETEROGENEITY
Measuring diversity is a fundamental aspect of the work of ecologists and biologists.
Some of the measures that are available in the literature on biodiversity have been
applied in the social sciences too; often using identical mathematical formulae but a
different terminology and interpretation. Maignan et al. (2003) show that while
economists are usually interested in measures of inequality with respect to quantitative
phenomena, such and income or wealth, and ecologists are mostly concerned with
relative and absolute abundance of different qualitative types such as species and subspecies, there is potential for a greater transfer of measures across disciplines. There is
also a need for a better understanding of how different diversity measures are related to
each other.
In this context, a family of measures developed by Patil and Taillie (1982) is very
helpful (see also Maignan et al., 2003). Consider a group g with relative frequency of
occurrence Pg/P. (we drop here the area and time subscripts, a and t respectively, for
simplicity because area and time are assumed fixed in this section). Whether we
consider this group ‘rare’ could be signaled by a function that takes this group share to
a certain power. Hence ‘rarity’ of group g may be given by
R() = [1  (Pg /P )]/
(1)
in which we assume that  ≥ 1. The value of  determines whether we emphasize
‘abundance’ or ‘evenness’ (a small  yields a measure of how representative the group
is in the population, a large  signals the likelihood that the group contributes to an even
distribution across categories; examples follow below). Many common non-spatial
measures of diversity are weighted averages of ‘rarity’ with weights given by the group
shares (Pg /P):
ఉ
D()= ∑ீ௚ୀଵ(ܲ௚ /ܲ· )ܴ(b) = ∑ீ௚ୀଵ(ܲ௚ /ܲ· )[1 -൫ܲ௚ ⁄ܲ· ൯ ]/b
(2)
We can roughly interpret  as generating the ‘moments’ of the categorical
distribution. The simplest measure is obtained when  = -1. It is easy to see that
11
D( -1) = ‫ܩ‬- 1 . This is almost identical to the richness or abundance of the diversity
of the population, AB = G, which is simply the number of distinct types.2 Of course we
would expect that the larger the population we consider, the more likely it becomes that
even rare groups are present. It is therefore useful to also consider indices of relative
abundance such as the Margalev diversity index MA (Margalef, 1958):
MA =
஽(ିଵ)
(3)
௟௡௉·
or the Menhinick index ME (e.g., Whittaker, 1977):
ME =
஺஻
(4)
ඥ௉·
Both measures have been developed in the context of ecological diversity where
sampling is common. These relative abundance measures are also useful in socioeconomic contexts where indices are compared across populations of quite different
sizes.
Next, we consider the case  = 0. Because R() = [1  (Pg /P )]/, it follows,
using L'Hôpital's rule, that R(0) = ln (Pg /P ) and we call the index D(0) the ShannonWeaver index SW:
SW = D(0) = - ∑ீ௚ୀଵ(ܲ௚ /ܲ· )ln(ܲ௚ /ܲ· )
(5)
This measure is identical to the Shannon index from information theory and is also
referred to as the Shannon-Wiener or entropy index (e.g. Theil, 1972). This index can
only be calculated when each group has at least one member. The index varies between
zero (when there is only one type) and a maximum of ln G when all groups have an
equal number of members. In order to easily compare populations that have coarse
(small G) or fine (large G) classifications, we can also introduce a relative Shannon
index, also referred to as the Shannon evenness index, which is D(0)/ln[1+D(-1)], i.e.
which divides D(0) by ln G. Hence,
2
Each of the 29 measures of diversity defined in this chapter will be denoted by a two letter
acronym.
12
SE = D(0)/D(-1) = -{∑ீ௚ୀଵ(ܲ௚ /ܲ· )ln(ܲ௚ /ܲ· )}/ ln‫ܩ‬
(6)
When we consider D(1), we arrive at the most commonly used measure of
dispersion across categories of a classification, called the fractionalization index (also
referred to as the Gini-Simpson index, see e.g. Alesina et al. 2003 and Desmet et al.,
2009), which is given by
௉೒ ଶ
‫(ܦ = ܴܨ‬1) = ∑ீ௚ୀଵ(ܲ௚ /ܲ· )ൣ1 − ൫ܲ௚ ⁄ܲ· ൯൧= 1 − ∑ீ௚ୀଵ ቀ௉ ቁ
·
(7)
In economics, this index is most often used to measure (a lack of) concentration of
market power among firms, in which case Pg refers, for example, to employment or
output of firm g and 1−FR is commonly known as the Herfindahl index (HD). This
index ranges between 0 (when all individuals belong to only one group) and 1−1/G
(when all groups have P /G individuals). Again, to compare fine and coarse
classifications, FR can be standardized into a modified fractionalization index which
varies between zero and one (see e.g. Olfert and Partridge, 2011, for an application):
ீ
ீ
௉೒ ଶ
‫ିீ = ܨ ܯ‬ଵ ‫(ܦ‬1) = ீିଵ ൤1 − ∑ீ௚ୀଵ ቀ௉ ቁ ൨
·
(8)
Equally common in the literature is the closely related Simpson diversity index SI.
This measures the probability that two randomly selected (without replacement)
individuals belong to two different groups. Since this is one minus the probability that
they belong to the same group, we get
ܵ‫ =ܫ‬1 −
∑ಸ
೒స భ ௉೒ ൫௉೒ -ଵ൯
(9)
௉· (௉· -ଵ)
The last term on the right hand side is also referred to as the Simpson dominance
index (DO). Greater diversity/lesser dominance implies a greater SI (smaller DO). If
each group is present but has only one member, P = AB = G and SI = 1 (DO = 0). If
everyone belongs to the same group, SI = 0 (DO = 1).
13
It should be noted that diversity indices such as FR and SI fail to satisfy the
decomposition condition. This condition is satisfied when in a multiple classification
(groups are defined by two or more attributes) the total diversity measure is equal to
the sum of the diversity measures for each of the classifications, assuming that the two
attributes are independently distributed across the respective categories. It is easy to see
that the SW index does satisfy this condition.
Another problem with the fractionalization index FR and related diversity indices
is that when a population consists of one large group (the ‘majority’) and various small
groups, these diversity indices are highly correlated with the share of the large group
in the population. Diversity may then be more effectively measured by calculating FR
among the minorities only. Alesina et al. (2013) show that the fractionalization index
FR=D(1), with g =1 referring to the majority population, can be decomposed into
௉
௉
௉
௉
௉
௉ ଶ
௉
௉
௉
೒
೒
೒
೒
‫ீ∑ = ܴܨ‬௚ୀଵ ௉ ቂ1 − ௉ ቃ= 2 ௉భ ቂ1 − ௉భቃ− ቂ1 − ௉భቃ ∑ீ௚ୀଶ ௉ ି௉ ቀ1 − ௉ ି௉ ቁ
·
·
·
·
·
·
భ
·
భ
(10)
in which 2 ௉భ ቂ1 − ௉భቃ can be interpreted as ‘between majority and minorities’
·
·
diversity and the second component as ‘among the minorities’ diversity. If
௉భ
௉·
is close
to one, the latter part is approximately zero and FR becomes approximately a decreasing
quadratic function of
௉భ
௉·
, the share of native born. This explains the positive correlation
which is observed by many researchers between the share of migrants and FR (e.g.,
Ozgen et al. 2013).
Increasing  further to  = 2, we obtain an evenness index:
ଶ
EI = D(2) = 0.5 ∑ீ௚ୀଵ(ܲ௚ /ܲ· )[1 -൫ܲ௚ ⁄ܲ· ൯ ]
(11)
which approaches zero when virtually everyone belongs to one group and 0.5 when
all groups have equal shares. The literature usually focuses on the opposite of evenness,
which is referred to as polarization. An obvious measure in terms of D(2) would be
simply
ଷ
0.5- D(2) = 0.5 ∑ீ௚ୀଵ൫ܲ௚ /ܲ· ൯
(12)
14
(similar to the third moment that measures skewness in a continuous distribution).
However, a more commonly used way of measuring polarization has been introduced
by Reynal-Querol (2002):
ଶ
RQ = 1− 4 ∑ீ௚ୀଵ൫0.5 − ܲ௚ /ܲ· ൯ (ܲ௚ /ܲ· )
(13)
which can also be expressed as
௉
ଶ
௉
೒
೒
RQ = 4 ∑ீ௚ୀଵ ቀ௉ ቁ (1 − ௉ )
·
(14)
·
Such polarization measures are in practice often generalized to take account of
social distances between groups (see also later in this section) and have been applied,
for example, to the impact of ethnic divisions on conflict (Esteban et al., 2012).
Rather than just summarizing the ‘moments’ of the population distribution across
the classification as we have done above, we can of course depict the unevenness of
dispersion graphically by means of the Lorenz curve. This curve is obtained by ranking
the groups from the rarest to the most common, and then plotting the population share
of the k smallest groups (k = 1,2, …, G), which is ∑௞௚ୀଵ ܲ௚ /ܲ· , on the vertical axis
against k/G on the horizontal axis.
An example is given in Figure 1. The extent of diversity and polarization can be
gauged from the shape of the Lorenz curve. When all groups are equally represented,
i.e. there is maximum diversity, the Lorenz curve is the diagonal B. A common
summary measure of inequality between groups (i.e. a lack of diversity in the present
context) is the Gini index (e.g. Rousseau et al., 1999). This index is calculated by
measuring two times the area between curve A and line B.
Figure 1 about here
The Gini coefficient can be easily calculated by sorting the G population shares
௉ ௉మ
௉
, … , ಸ ቁ
௉· ௉·
௉·
ቀ భ,
௉෨ ௉෨
௉෨
from the smallest to the largest. Let ቀ௉భ , ௉మ , … , ௉ಸ ቁdenote the ranked
sequence, with
·
௉෨ೝ
௉·
·
·
the rth smallest fraction. Define Yr as the cumulative sum of the
15
fractions up to r; and Y0  0. Given this notation, the Gini coefficient of population
diversity can be calculated for the population in any given area (or for all areas
combined) as
ଵ
‫ = ܣܩ‬1 − ீ ∑ீ௥ୀଵ(ܻ௥ + ܻ௥ିଵ)
(15)
The Gini coefficient ranges from 0 to 1. Its value is 0 when each group is of equal
size (maximum diversity) and 1 when the entire population belongs to one group only
(Y1 = 0). One benefit of the Gini coefficient is that it is a population scale-independent
measure. Another benefit is that it permits a visual comparison of changes in
unevenness over time or across areas. It should be noted that a given value of the Gini
coefficient can represent quite different distributions. Moreover, the value of the index
is influenced by granularity of the group classification, i.e. how many groups are being
considered.
A related index is the Hoover index HO (also referred to as the Robin Hood index)
which calculates the proportion of the population of each group that would have to be
redistributed in order to achieve an even distribution, with each group having P / G
members. The Hoover index is given by
௉
ଵ
೒
HO = 0.5 ∑ீ௚ୀଵ ቚ௉ − ீ ቚ
·
(16)
The Hoover index is a special case of the Duncan and Duncan dissimilarity index
DI in which the distribution of individuals across a classification is compared for two
populations (Duncan and Duncan, 1955). Section 4 discusses this index when the
classification refers to spatial areas.
All diversity measures discussed above do not take the cultural distance between
groups into account. Effectively they assume that cultural distance is at a maximum for
every combination of different types (i,j), i.e. ij = 1. Because many of the formulae
above are weighted ‘rarity’ measures, it is straightforward to generalize these to modify
the weighting based on how culturally dissimilar groups are. This is nicely
demonstrated by Desmet et al. (2009) on the basis of a family of measures of social
effective antagonism, introduced by Esteban and Ray (1994). Following these authors,
the aggregate level of social effective antagonism can be defined as
16
௉
௉
ଵାఊ
೒
K(, ) = ∑ீ௚ୀଵ ∑ீ௛ୀଵ ௉ ቀ௉೓ ቁ
·
·
ߪ௚௛
(17)
in which  is the cultural distance matrix and  is a parameter that ‘tunes’ the
measure with respect to abundance or evenness. Desmet et al. (2009) note that if  = 0
and max is a matrix with gh = 1 for all g  h and gg = 0 for all g, then K(0, max) is the
fractionalization index FR of Eq. (7). For intermediate cases, with 0 < gh < 1 for some
g  h, DF is the cultural distance-based fractionalization index which effectively
measures the population-weighted average cultural distance:3
௉ ௉೓
೒
DF = K(0, ) = ∑ீ௚ୀଵ ∑ீ௛ୀଵ ௉
· ௉·
ߪ௚௛
(18)
Similarly, K(1, max) can be shown to be identical to the Reynal-Querol polarization
index RQ given in Eqs. (13) and (14), while the polarization equivalent in the general
case becomes the one of Esteban and Ray (1994):
௉
௉
ଶ
೒
ER = K(1, ) = ∑ீ௚ୀଵ ∑ீ௛ୀଵ ௉ ቀ௉೓ ቁ ߪ௚௛
·
·
(19)
Rather than pre-allocating individuals to groups in terms of several characteristics,
the analysis of cultural-distance weighted diversity can also be conducted at the micro
level of individual data, see Bossert et al. (2011). All measures described in this section
can be calculated at any spatial scale and the values of any given diversity index can be
compared for areas at any given spatial scale. Alternatively, we could fix the group and
consider how diverse or polarized a given spatial distribution of a given population
group g is by calculating measures that simply swap area and group subscripts, and run
sums over A areas rather than G groups in the above formulae. However, in practice we
often want to focus on the diversity of among groups and places simultaneously. The
next section therefore reviews a range of global and local measures of cultural diversity
among places.
3
Desmet et al. (2009) also discuss a special case in which there is a large cultural distance between
the ‘host population’, which tends to be the majority, and minority groups – but not between the minority
groups themselves.
17
5
DIFFERENCES IN POPULATION HETEROGENEITY BETWEEN PLACES
In this section we introduce several statistics that can be used to capture spatial patterns
of diversity. In all equations that follow we will again use a common notation. Some of
the spatial measures are referred to as global measures in that they provide a summary
measure of the spatial pattern across all areas, whereas others are local measures in that
they are calculated for each area.4 Let Pgat again refer to the population of group g
(=1,2,...,G) in area a (=1,2,...,A) at time t (=1,2,…,T). A subscript  refers again to the
sum over that particular subscript (t is removed when the analysis is purely crosssectional). Each area a has a set of neighborhood areas/spaces S (to be defined in detail
later) that are indexed by sa and numbered from 1 to Sa.
Massey and Denton (1988) refer to the extent to which the spatial dispersion of a
particular population group is different from the spatial dispersion of the general
population as segregation. They classify 20 commonly used measures of segregation
in terms of five distinct features (evenness, exposure, concentration, centralization and
clustering) of the spatial distribution of minorities that such measures can potentially
convey. They calculate all 20 measures with data on the location of Hispanics, blacks,
Asians and non-Hispanic whites (the latter were defined as the majority population) in
US metropolitan areas. They use factor analysis to identify the distinct types of
information these measures convey and the measures that had the greatest factor
loadings. Their conclusion is that their five posited features of segregation can all be,
at least to some extent, identified in the data and that some of the 20 segregation
measures capture these features better than others. Hence for each feature they identify
a measure that signals this feature best in the data (based on the factor analysis and
other information).
Two of the preferred five measures do not take the geography of population
distribution explicitly into account. One of these, the dissimilarity index, informs on
evenness. The other is concerned with exposure of a minority group, in terms of
potential contact, either with members of the own group (the isolation index) or
4
Assuming the availability of geo-referenced data, all of the diversity measures reviewed in the
previous section can be calculated for any given area. However, in this section we are interested in how
such a diversity measure of an area compares with those of other areas. Local spatial measures of
diversity can then refer to, for example, the average diversity of areas surrounding any given area.
18
members of the majority group (the interaction index). These indices will be described
mathematically below.
Three features of segregation that take geography explicitly into account are
concentration, centralization and clustering. Concentration measures are concerned
with population density and quantify the relative amount of physical space occupied by
a given group. Massey and Denton (1988) define a relative concentration index that
describes this feature of spatial dispersion in the best way. Centralization refers to the
extent to which a given group is located near the center of a city or region, e.g. the
Central Business District or the largest city respectively. An absolute centralization
index is Massey and Denton’s preferred measure of centralization. Finally, clustering
refers to the extent to which the distribution of a given group is in a ‘contiguous and
closely packed’ way, thereby creating enclaves. For this feature Massey and Denton
(1988) recommend a spatial proximity index. However, the huge growth in spatiallyreferenced data and Geographic Information Systems (GIS) since the 1980s has led to
many new developments in the spatial statistics and spatial econometrics literatures that
can also applied to measuring diversity. The most common measures of this type will
be reviewed later in this section.
One of the most common global, i.e. ‘averaged’ across areas, spatial diversity
measures cited in the literature is the dissimilarity index, which – as just noted above –
is advocated by Massey and Denton (1988) as the best measure of spatial (un)evenness
(when not geo-referenced). The index is a measure of displacement – the proportion of
people in group one which would have to relocate in order to make their distribution
identical to that of group two (Duncan and Duncan, 1955). When the dissimilarity index
is computed for one group (a minority) and the remainder of the population combined,
it is known as the segregation index. The group segregation index for group g across
area units a (=1,2,…,A) is5
ଵ
௉
೒ೌ
DIg= ∑஺௔ୀଵ ฬ −
ଶ
௉೒·
൫௉·ೌ ି௉೒ೌ ൯
ฬ
(20)
൫௉·· ି௉೒· ൯
5
Not that for simplicity we did not use the subscript a for each of the diversity measures in the
previous section, even though they can be calculated for each area a. In this section we will consider
some measures that are specific to a group g and to an area a, while others are ‘averaged’ across groups
or across areas. Subscripts for the diversity measures are therefore useful from hereon.
19
Note that, while being a global index, the group segregation index provides very
limited information on clustering patterns and will only reveal an average situation for
the group. It does not take account of the location of group clusters, a phenomenon
known as the checkerboard problem (Brown and Chung, 2006). The checkerboard
problem recognizes that there may be one big cluster of a group, or many small
communities scattered around the total area, but no way of knowing which one is
present from a global index calculation. The Moran’s I index of spatial correlation
provides a global spatial measure that informs on which of these patterns is more
plausible (see below).
The segregation index DIg can be interpreted as the fraction of the group g’s
population that would have to be redistributed in order for the spatial distribution of
group g to become the same as that of the rest of the population. Such redistribution
could lead to unrealistic changes in the population of various areas. In some contexts it
is more meaningful to calculate a modified index that measures the fraction of group g
and the fraction of the rest of the population that would need to be both redistributed to
achieve identical spatial distributions, under the condition that the area populations
remain constant. Such an index was earlier applied by Van Mourik et al. (1989) to the
case of occupational segregation, where the standard segregation index measures the
percentage of women who would need to change occupation to equalize the male and
female distribution of the labor force across occupations. Any such redistribution is
likely to imply unrealistic changes in the total number of people in various occupations.
The latter may be assumed to be demand determined. Van Mourik et al.’s modified
segregation index calculates the percentage of both women and men that would need to
change occupation to have an equal distribution across genders and unchanged totals in
each occupation. The equivalent situation in spatial segregation is to consider, for
example, relocating migrant and native born households in a social housing program
such that their spatial distribution is equalized and the housing stock in each area
remains unchanged. The modified segregation index is
ଵ
௉
௉
ܸ‫ ܯ‬௚ = ଶ௉ ∑஺௔ୀଵ ቀቚܲ௚௔ − ܲ௚· ௉·ೌ ቚ+ ቚ൫ܲ·௔ − ܲ௚௔ ൯− ൫ܲ·· − ܲ௚· ൯௉·ೌ ቚቁ
··
··
··
20
(21)
The modified segregation index turns out to be equal to the conventional
segregation index times a factor that depends on the fraction of the total population that
is in group g (for the proof, see Van Mourik et al. 1989):
ܲ
ܲ
ܸ‫ ܯ‬௚ = 2 ܲ݃· ቀ1 − ܲ݃·ቁ‫݃ܫܦ‬
··
(22)
··
The isolation index is Massey and Denton’s (1988) preferred measure of the degree
of potential exposure of individuals to members of their own group (isolation) or
another group (interaction). The isolation index captures the extent to which members
of a population group are disproportionately located in the same areas, i.e. they are more
clustered. Consider first the weighted average fraction of the population across all areas
௉
೒ೌ
that belongs to group g, ∑஺௔ୀଵ w௚௔ ௉ , with w௚௔ =
·ೌ
௉೒ೌ
௉೒·
and therefore ∑஺௔ୀଵ w௚௔ = 1 for
all g (we can also measure the weighted average exposure of group g to group h:
∑஺௔ୀଵ w௚௔
௉೓ೌ
௉·ೌ
). The isolation index IIg simply normalizes this average fraction in the
following way (Cutler et al. 1999):
‫ܫܫ‬௚ =
ು ೒ೌ
w೒ೌ
ು ·ೌ ௉೒ ·
ି ൘௉
೒ ·൘
··
ು ··
௉೒·
ଵି ൘௉
··
∑ಲ
ೌస భ ು
(23)
This measure captures the degree to which group members live in areas in which
they are over-represented. An isolation index value of 0 indicates that the group is
distributed in proportion to the total population, while a value of 1 can be interpreted
as total isolation whereby all of the group locate in one or several particular areas a,
and no-one of the rest of the population locate in those areas. Exposure can be a useful
concept in studying acculturation. Exposure measures are usually based on information
linked to a person’s usual residential address, but exposure at work can be calculated
when workplace addresses are known.
Another way of examining an uneven distribution of a group across a number of
areas is by means of the Lorenz curve and the Gini coefficient, which were already
introduced in the previous section in terms of gauging the diversity of groups in a
particular area (see again Figure 1). In the spatial context a Lorenz curve is constructed
by first calculating a group’s fraction of the population of each area and then ranking
areas from the one with the smallest fraction to the one with the largest fraction. Next,
21
consider for each ranked area the points representing the cumulative percentage of total
population up to that area on the horizontal axis and the cumulative percentage of the
group’s population up to that area on the vertical axis. If a group is spatially distributed
identically to the total population, the Lorenz curve would coincide with the 45 degree
line. Half the total area between the observed Lorenz curve and the 45 degree line is the
Gini coefficient of a group’s spatial segregation.
As in the case of diversity among groups, the Gini coefficient can also be easily
௉
௉
௉
calculated for spatial diversity. First, sort the population shares ቀ௉೒భ , ௉೒మ , … , ௉೒ಲ ቁ of
·భ
·మ
th
·ಲ
group g in the various regions from the smallest to the largest. For the r smallest
fraction, let Xr be the cumulative sum of the corresponding shares of the regions’
෩
ܲ
populations in the total population, i.e. ܺ௥ = ∑௥௜ୀଵ ܲ·݅. Similarly, Ygr is the cumulative
··
෩
ܲ
sum of the corresponding shares of group g’s population, ܻ௚௥ = ∑௥௜ୀଵ ܲ݃݅ and X0  0, Yg0
݃·
 0. Given this notation, the Gini coefficient of segregation of group g can be calculated
as
‫ܩܩ‬௚ = 1 − ∑ீ௥ୀଵ(ܺ௥ − ܺ௥ିଵ)൫ܻ௚௥ + ܻ௚,௥ିଵ൯
(24)
The final two global non-GIS segregation or clustering measures that we consider
are the closely-related Ellison and Glaeser (1997) and Maurel and Sédillot (1999)
concentration indices, denoted EGg and MSg respectively. Both are derived as the
correlation between location decisions made by members of a particular group, which
can be positive or negative. The measures were originally derived to capture the
geographic concentration of industries that take into account differences across
industries in the firm size structure. We report here formulae proposed by Maré et al.
(2012).6 A value of close to zero for either of these indices would indicate a lack of
spatial segregation. The two indices differ only slightly. The EGg index has a more
positive value for groups that are concentrated in areas with higher shares of the overall
population.
Using the same notation as before, the EGg index is given by
6
Our equations differ slightly from the original formulations to reflect the focus on people rather
than firms. Unlike firms, which differ in size, all people carry equal weight. Hence the final term in both
the numerator and denominator of the formulae, which is in the firm case a Herfindahl index of firm
concentration, simply becomes 1/Pg , as in the unweighted index of Maurel and Sédillot (1999).
22
‫ܩܧ‬௚ =
ು ೒ೌ ು ·ೌ మ
ቊ∑ಲ
ೌసభ ( ು ೒ · షು ·· ) ቋ
భ
ି
ು ·ೌ మ
ು ೒·
(భష∑ಲ
(
)
)
ೌస భ ು ··
భ
(ଵି )
ು ೒ ·
(25)
while the MSg index is given by
‫ܵ ܯ‬௚ =
ು ೒ೌ మ
ು ·ೌ మ
) ష ∑ಲ
ೌస భ ( ು ·· ) ቋ
ು ೒·
భ
ି
ು ·ೌ మ
ು ೒·
(భష∑ಲ
ೌస భ ( ು ·· ) )
భ
(ଵି )
ು ೒ ·
ቊ∑ಲ
ೌస భ (
(26)
Ellison and Glaeser (1997) suggest that in order to determine a benchmark for their
measure of industry concentration, the index could be calculated for an industry which
would not be considered concentrated. In our present context of spatial segregation, the
spatial distributions of males and females may be used as a point of comparison. They
are expected to be very similar. Hence, for both genders EGg (and MSg) will have low
values which can be used as a benchmark for gauging spatial segregation of other
groups.
With respect to local diversity measures it should be noted that all of the non-spatial
measures defined in Section 3 can be calculated for all areas for which data are
available. Hence each index given in the previous section can be calculated for any area
a or any amalgamation of areas. For example, the fractionalization index for each area
a becomes
௉೒ೌ ଶ
‫ܴܨ‬௔ = 1 − ∑ீ௚ୀଵ ቀ௉ ቁ
(27)
·ೌ
The values of ‫ܴܨ‬௔ can be mapped with GIS software to gauge the spatial patterns.
Corresponding measures of spatial statistics such as spatial autocorrelation can also be
calculated.
A simple measure of local concentration or clustering of group g in area a is the
location quotient, also referred to as the local concentration ratio:
23
ܲ݃ܽ
൘
ܲ݃ܽ
൘
ܲ݃·
‫ܳܮ‬௚௔ = ܲ
= ܲ
݃·൘
·ܽ൘
ܲ··
ܲ··
ܲ·ܽ
(28)
‫ܳܮ‬௚௔ measures whether group g’s share of the population in area a is larger or
smaller than group g’s share of the total population (implying ‫ܳܮ‬௚௔ > 1 and ‫ܳܮ‬௚௔ < 1
respectively). Alternatively, ‫ܳܮ‬௚௔ measures whether the proportion of g’s population
that is located in a is larger or smaller than the proportion of the total population that is
located in a.
All measures of clustering reviewed above do not account for the topological
relationship of neighborhoods or areas to one another. Hence high spatial concentration
may or may not coincide with spatial clustering, which is the checkerboard problem
referred to earlier. This is illustrated in Figure Consider a minority distributed across
areas with three population shares: high (black square), medium (grey square) and low
(white square). The segregation measures discussed above will have higher values for
the left checkerboards than for the right ones, but identical values for the upper and
corresponding lower checkerboards.
Moran’s I is a global measure of clustering that yields greater values for the upper
checkerboards than for the lower checkerboards. The index is a measure of spatial
autocorrelation which essentially determines whether or not spatial dispersion is
random. Hence Moran’s I complements the global measures introduced above as the
latter provide an aggregate measure of dispersion of a group across areas without a
spatial reference frame, whereas Moran’s I calculates the degree of clustering that takes
into account neighborhoods. In this context, the neighborhood of an area is defined as
the set of areas that are within a pre-defined distance of the area considered.
Figure 2 about here
Moran’s I index of the degree of global clustering of group g is defined as:
‫ܫ ܯ‬௚ =
ܲ݃ܽ 1
ܲ݃‫ ܽݏ‬1
ܽ ‫ݓ‬
- ‫ܣ‬ቇቆ∑ܵ
- ‫ܣ‬ቇቇ
ܽ‫ܽݏ‬ቆ ܲ
ܲ
‫ݏ‬
=1
݃
·
݃·
ܽ
஺
∑௔ୀଵ
ܲ݃ܽ 1 2
∑ಲ
- ‫ܣ‬ቇ
ೌస భቆܲ
݃·
ቆ
24
(29)
where a refers to areas as before and sa to the areas in the neighborhood of a (there
are Sa such areas). The spatial proximity weights ‫ݓ‬௔௦ೌ can be defined in various ways.
For example, they can be defined as the reciprocal of the distance between the
population centroid of area a and the centroids of surrounding areas sa,
d௔௦ೌ . Alternatively they can be defined simply by adjacency/contiguity (with a
‘proximity’ of ‘1’ assigned to pairs of adjacent regions and ‘0’ to non-adjacent ones).7
In all cases, the weights are row-standardized, such that ∑௦ೌ ‫ݓ‬௔௦ೌ = 1 for all a. One set
of weights that automatically satisfies row-standardization is a set of population-based
weights, whereby the weight of neighborhood area sa of area a is assumed to be equal
ௌ
ೌ
to ܲ·௦ೌ / ∑௝ୀଵ
ܲ·௝ if the selected area and a neighborhood area are adjacent, or if the
neighborhood area is within some distance band of the selected area, and zero if
otherwise.
MIg can be easily visualized as the slope of a regression line of the spatially
weighted value of the fraction of the group’s population in any area in the neighborhood
of a selected area on the value of the fraction of the group’s population in the selected
area itself. A plot of the points through which the regression line can be drawn is
referred to as a Moran scatterplot. MIg lies between -1 and 1 and provides a measure of
how similar an area’s share of the total population of group g is to the populationweighted share in surrounding areas. A negative Moran’s I may be indicative of isolated
enclaves in which areas with a large share of a group are surrounded by areas in which
a low share of the same group can be found. This is referred to as negative spatial
autocorrelation.8 A value of Moran’s I close to 1 is indicative of highly significant
positive spatial autocorrelation that in our context can be interpreted as evidence of
extensive segregation that straddles many areas.
Moran’s I as calculated in Eq. (29) provides only an average indication of clustering
of a given group g. There may be a range of different spatial patterns occurring for
different groups, despite similar spatial autocorrelation results. To investigate the
spatial patterns of concentration, Getis and Ord’s (1992) G* local measure of
7
The emergence of ‘big data’ on actual face-to-face encounters or online interactions through email
or social media provides new and promising sources of measuring proximity between individuals and
groups across areas. This can lead to new ways of measuring geographic weights matrices.
8
The question of whether the observed spatial pattern is different from a random spatial allocation
of population (i.e. Moran’s I is statistically significant) is more complex. Two sets of standard errors can
be calculated under the assumptions of standardisation and normality (Cliff and Ord, 1981; Pisatio,
2001).
25
∗
concentration can be calculated for every area a and every group g. ‫ܩ‬௚௔
is a calculation
which identifies areas of neighborhood clustering that are significantly different from
the average situation in the total study area (Johnston et al., 2009). Using a row∗
standardized spatial weights matrix W*, ‫ܩ‬௚௔
can be calculated as:
∗
‫ܩ‬௚௔
=
ೄ
ೌ
∗ (
∑ೞೌసభ ௪ ೌೞ
ೌ
ܾܲ݃ 2
൬ ൰
ܲ·ܾ
ඪ൮∑ಲ
್సభ
‫ܣ‬
ು ೒ೞೌ
ು ·ೞೌ
൲-ெ ೒మ ඨ
-ெ ೒ )
(30)
మ
ೄ
ቀಲ ∑ೞೌసభ ೢ ∗ೞೌ -భቁ
ೌ
(ಲషభ)
തതതതതതത
where ‫ ܯ‬௚ = ത
ܲത
௚௔ /ܲ.௔ refers to the mean group share of group g across all areas a
(see, e.g., Maré et al., 2012). Here sa includes the area a itself in the spatial weights that
are in Eq. (30) indicated by an asterisk. The index values are normally distributed z
∗
scores under the null hypothesis of no spatial clustering. A value of ‫ܩ‬௚௔
for an area that
is greater than 1.96 indicates that there is less than a 2.5 percent chance that the high
degree of concentration of group g that is observed in and around the area a would be
∗
observed under random location decisions (and, similarly, a value of ‫ܩ‬௚௔
for an area
that is less than 1.96 indicates that there is less than a 2.5 percent chance that the
extreme absence of group g in and near a would be observed under random location
decisions). G* values can be displayed on a map for each group to show specific
neighborhoods where groups are over and under-represented.
Various other local measures can be calculated as well. Anselin (1995) notes that
Moran’s I is the aggregate across all areas of what he refers to as a local indicator of
spatial association (LISA). Given Eq. (29), we define the local indicator LIga by
‫ܫܮ‬௚௔ =
ܲ݃ܽ 1
ܲ݃‫ ܽݏ‬1
ܽ ‫ݓ‬
− ቇቆ∑ܵ
ቆ
− ቇቇ
ܲ݃· ‫ܣ‬
‫=ܽݏ‬1 ܽ‫ܣ ·݃ܲ ܽݏ‬
ܲ݃ܽ 1 2
∑ಲ
ቆ
− ቇ
ೌసభ ܲ
݃· ‫ܣ‬
ቆ
(31)
and, hence, ‫ܫ ܯ‬௚ = ∑஺௔ୀଵ ‫ܫܮ‬௚௔ . As is the case with other local measures, ‫ܫܮ‬௚௔ can be
plotted on a map; for example to identify ‘hot spots’: areas where high values of the
index are strongly clustered (e.g. Anselin, 1995).
26
All spatial measures discussed in this section either refer to spatial segregation of a
specific group g (either dispersed or clustered across areas) or refer to the local diversity
in area a (either homogeneous/uniform or heterogeneous/diverse). Effectively this
involves calculating indices for a column or row respectively of the A x G two-way
cross tabulation of observed occurrences. This is depicted in Figure 3.
Figure 3 about here
Additionally, we may want to consider all cell frequencies in the table
simultaneously and account, or not, for the geography that defines the various areas.
For example, we could calculate the isolation index IIg for each group g in a
metropolitan area and, next, calculate a group size-weighted average of the IIg indices
to obtain an indicator of the overall extent to which groups are isolated or spatially
mixed across the metropolitan area. But there are of course many other ways of
combining segregation measures for individual groups. Reardon and Firebaugh (2002)
suggest six multigroup segregation indices: a dissimilarity index, a Gini index, an
information theory index, a squared coefficient of variation index, a relative diversity
index and a normalized exposure index. Reardon and Firebaugh (2002) list seven
criteria for evaluating multigroup measures of segregation.9 They conclude that the
information theory-based index, which was originally proposed by Theil and Finezza
(1971) and Theil (1972), is the most satisfactory index in terms of their criteria. This
index is the only one that satisfies the principle of transfers (when a person is moved
from an area i, where its share of the population is larger than in an area j, to that area
j, then segregation is reduced). Moreover, this is the only multigroup index that can be
decomposed into a sum of between- and within-group components.10 The Theil index
TH is given by the following formula:
9
These criteria are: (1) organizational equivalence; (2) size invariance; (3) transfers; (4) exchanges;
(5) composition invariance; (6) additive organizational decomposability; and (7) additive group
decomposability. Space constraints preclude us from elaborating on these here. See Reardon and
Firebaugh (2002) for details.
10
For example, a decomposable index of segregation of all ethnic groups should be the sum of the
index of between-supergroups segregation (e.g., Africans, Asians, Europeans, etc.) and the aggregate of
ethnic segregation measures for ethnicities within supergroups (e.g., Chinese, Korean, Japanese, etc.
among the Asians).
27
TH
ು ೒ೌ
ቀ୪୬൫௉೒· ⁄௉·· ൯ି୪୬൫௉೒ೌ ⁄௉·ೌ ൯ቁ
௉
=∑ீ௚ୀଵ ∑஺௔ୀଵ ௉·ೌ ൥ು ·ೌ∑ಸ (௉ /௉ )୪୬(௉ /௉ ) ൩
··
೒ · ··
೒స భ ೒ · ··
(32)
However, multigroup segregation measures such as TH do not account explicitly
for geography. A simple way of calculating an average of geographical clustering could
be obtained by calculating Moran’s I index MIg for every group g in a metropolitan area
and then by assigning a global measure of multi-group clustering to the metropolitan
area by calculating global clustering as a group size-weighted average of these indices.
Alternative measures of spatial segregation are reviewed and evaluated by Reardon
and O’Sullivan (2004). These measures take account of the coordinates of the locations
of individuals in the region and the proximity between individuals. This is an important
extension of the various indices discussed so far which all assumed fixed boundaries of
subareas a = 1, 2, .., A of the region or city. Distance measures, such as used in
calculating the Moran’s I index are sensitive to the drawing of boundaries: two points
that are a fixed physical distance apart may either be considered to be in the same
neighborhood (i.e. the distance is zero) or in different neighborhoods (at some positive
distance), dependent on where the boundary is drawn. This is referred to as the
modifiable areal unit problem (MAUP). This problem and the checkerboard problem
mentioned earlier in the chapter are entirely due to the defining of subareas and vanish
once we adopt a fully disaggregated approach that is based on micro data on individuals
and their locations.
After identifying eight desirable properties of spatial segregation measures,11
Reardon and O’Sullivan (2004) conclude that a spatially weighted equivalent of the
isolation index IIg and a spatially weighted equivalent of the information theory-based
Shannon-Weaver index of diversity SW are the optimal spatial segregation measures.
Specifically, for a given individual at point p, let ߨ
ො௣௚ denote the average fraction of
people of type g in the neighborhood of point p, then
‫ܧ‬෠௣ = - ∑ீ௚ୀଵ ߨ
ො௣௚ ln(ߨ
ො௣௚ )
(33)
11
These properties are: (1) scale interpretability; (2) arbitrary boundary independence; (3) location
equivalence; (4) population density invariance; (5) composition invariance; (6) transfers and exchanges;
(7) additive spatial decomposability; and (8) additive grouping decomposability. See Reardon and
O’Sullivan (2004) for details.
28
defines the entropy of the local environment of p. The spatial information theory
segregation index can then be defined by
SS= 1 − ௉
ଵ
ௌௐ
·· ∫௣∈ோ ߬௣ ‫ܧ‬෠௣ ݀‫݌‬
(34)
in which SW is the Shannon-Weaver diversity index of equation (5), ߬௣ is the
population density at point p, R indicates the entire region or city and the integral
indicates that the index is calculated by summing over all points p. SS is a measure of
how much less diverse the local environments of individuals are on average, compared
with the diversity of the total population of region R. Further elaboration and a
discussion of various other spatial indices can be found in Reardon and O’Sullivan
(2004).
All measures discussed in this chapter thus far assume aggregation of individuals
into groups with certain characteristics and into areas within certain boundaries. This
aggregation approach was often the only feasible methodology in the past because data
from population censuses or surveys were subject to strict confidentiality requirements
to satisfy privacy legislation in many countries. Thus, published information on a
population’s demographic and socio-economic characteristics has been predominantly
in the form of multi-way cross-tabulations with each cell reporting the observed number
of people satisfying the specific classification (and often rounded to the base three to
preserve confidentiality requirements). However, researchers are now increasingly
given access to micro data, with certain personal information (such as name and
address) removed and replaced by a location indicator that is determined by the smallest
spatial scale at which data are made available.
In recent years, many new sources of data are becoming available that provide
very rich information, often combining survey and administrative data, at a very local
level and sometimes even including the coordinates of the individual observation itself
(particularly in GIS systems). Additionally, the information available from such
integrated data infrastructures is often longitudinal: individual units are observed at
various points in time. The emergence of this type of information calls for a new
approach to measuring cultural diversity and segregation at the micro level. In this
context, new operationalizations of socio-cultural and geographical distance are needed
in the context of human interactions. For example, rather than measuring exposure to
29
people from different backgrounds by means of simple shares of different socio-cultural
groups in certain regions, network approaches can be used that account for face-to-face
or virtual interaction between individuals. Alternatively, when location coordinates are
known, the individual’s activity space can be defined by spatial kernels (in which a
group is defined by any individual captured within a certain radius of the individual)
that can be varied in robustness checks. While the formal statistical analysis of spatial
dependence and correlation remains computationally demanding for micro-level data,
there are new promising developments (for an application to diversity among firms, see
e.g. Dubé and Brunelle, 2014 and to multi-group spatial segregation of individuals, see
e.g. Kumar et al., 2013). Parallel to the emergence of new rich micro data and
conceptual developments, new statistical software for measurement of diversity and
segregation is emerging too (e.g. Hong et al., 2014).
6
CONCLUSION
In this chapter we have argued that cultural diversity is emerging as an expanding and
promising area of research in economics. Scientific inquiry into the socio-economic
impacts of cultural diversity necessitates a thorough understanding of how to measure
diversity in terms of differences between groups, or individuals, and differences
between areas, or locations. In the past, economists have often adopted measures from
a range of disciplines, such as ecology, biology, sociology, etc. While such measures
can be easily calculated in any socio-economic application, their appropriateness is
rarely critically assessed. In this context we reviewed in this chapter 29 different
measures of diversity among people and places, which are among the most common in
the literature. We have emphasized their interpretation and the relationships between
them. We have not attempted a more formal mathematical or statistical evaluation of
their properties, which can be found in e.g. Andreoli and Zoli (2012), Reardon and
Firebaugh (2002), Reardon and O’Sullivan (2004) and White (1986).
Such evaluations show that for many measures the results are likely to depend on
the extent of disaggregation of space and group attributes (i.e. the granularity of the
data), the selection of relevant attributes and the population scale. An important
warning in this context is that aggregate measures often suffer from the fallacy of
composition. For example, as is clear from the literature on inequality, simply
comparing countries in terms differences in their Gini coefficients provides very little
30
information about the underlying income distribution and the often large differences
across areas within countries. The same would be true for simply comparing cultural
diversity across countries by a single index of fractionalization. It is therefore essential
that sound micro, meso or macro conceptual frameworks and theories are adopted that
provide guidance as to which cultural diversity measure is the most appropriate in the
specific context.
For diversity among people, many papers have adopted the fractionalization index
but it is clear that this captures just one aspect of cultural diversity and the concepts,
and impacts, of other features of the data such as weighted abundance, entropy and
polarization should also be considered, dependent on the context. For place diversity,
researchers appear to have been most comfortable with measuring unevenness by
means of the standard dissimilarity index. Here it is clear that measures of modified
segregation, isolation, concentration, (cross) Moran’s I and GIS-generated local
measures such as location quotients, the local Moran and Getis and Ord’s G* can
provide complementing insights. In all of this, the definition of a ‘group’ is clearly
moving away from a simple majority-minority dichotomy to the consideration of many
groups defined by attributes across a range of domains.
The emerge of ‘big data’ and other exciting new data developments, such as
integrated administrative and survey data infrastructures, suggest promising new
approaches at the micro-level to measure cultural diversity in potential and actual
contact/interaction between individuals. Much of the new data are temporal, and often
longitudinal, which permits the testing of heterogeneity across cohorts, as well as life
course approaches; aspects that have been neglected too much in the past given that
most cultural diversity analyses to date have tended to be simply cross-sectional
analyses. The longitudinal information also permits much more emphasis in the future
on acculturation processes such as those that result from intermarriage and the
absorption of the host country culture by first and subsequent generations of migrants.
Many studies that calculate diversity measures are purely descriptive in nature but
there is still much scope for further multivariate (spatial) econometric models of the
socio-economic impacts of cultural diversity. Varying the measures of diversity in such
econometric models may inform on the aspects of diversity that matter most.
Ultimately, cultural diversity is in most applications an endogenous phenomenon.
The endogeneity of constructed diversity measures is then as much of interest as their
impacts, but accounting for reverse causality remains challenging. Randomized
31
experiments of diversity impacts in economic contexts are rare (but see e.g. Boisjoly et
al., 2006). Given the growing cross-border mobility and the associated emergence of
super-diversity, we may expect a further widening and deepening of new research on
the measurement of cultural diversity and its socio-economic impacts.
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Cumulative Proportion of Groups
Figure 1 The Gini index and Lorenz curve
Low Segregation
Low Spatial Correlation
High Spatial Correlation
High Segregation
Figure 2 Comparing segregation and spatial correlation
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Area Axis
Dispersed
Type Axis
Uniform
Diverse
Clustered
Figure 3 Combining Area and Group Diversity
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