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A New Approach to Measuring
Socio-Spatial Economic
Segregation
Sean F. Reardon
Stanford University
Glenn Firebaugh
Pennsylvania State University
David O’Sullivan
University of Auckland
Stephen Matthews
Pennsylvania State University
Presentation prepared by Jacques Silber (Bar-Ilan University)
1
Goal of the paper
• To develop an approach to measuring
spatial economic segregation
• To develop different measures of
segregation along an ordinal dimension
since income data are often reported using
ordered categories
• To show how these measures can be
adapted to take into account the spatial or
social proximity of individuals
2
I.
Existing Measures of
Economic Segregation
1) Category-Based Measures of Economic
Segregation
•
•
3
Most common way of measuring income
segregation has been to divide the population into
two categories, based on some chosen income
threshold
Segregation between the two groups is then
computed using some segregation measure (e.g.
the dissimilarity index) that assumes the existence
of only two groups
Category-Based Measures of
Economic Segregation
Problems :
- Dichotomizing the income distribution amounts to
discarding a lot of information
- The results of such an approach may depend on the
choice of threshold
Possible Solutions :
- Compute the two-group segregation for all possible
pairs of income categories, and then construct some
summary measure of the pairwise indices
- Compute segregation among multiple income category
groups using for example the Theil information theory
index of segregation
4
Category-Based Measures of
Economic Segregation
But still unsolved issues:
• The Theil index measures segregation
among a set of unordered groups (such as
racial groups) and hence is insensitive to
the ordinal nature of income segregation
• Such an approach remains sensitive to the
number and location of the thresholds
used to define income categories
5
I.
Existing Measures of
Economic Segregation
2) Variation-Ratio Based Measures of
Economic Segregation
•
•
6
An alternative approach defines segregation as
a ratio of the between neighborhoods variation
in income to the total population variation in
income
As measures of dispersion one can use the
variance of incomes, the so-called TheilBourguignon population-weighted
decomposable inequality index, etc…
Variation-Ratio Based Measures of
Economic Segregation
• Advantages of such an approach :
Can use complete information on the
income distribution
Does not rely on arbitrary thresholds
These measures are often invariant to
certain types of changes in the income
distribution
7
I.
Existing Measures of
Economic Segregation
3) Spatial Autocorrelation Measures of
Economic Segregation
• Most proposed measures of income
segregation are aspatial in the sense that they
do not account for the spatial proximity of
individuals/households.
• A third approach would therefore consider that
segregation should measure the extent to
which households near one another have more
similar incomes than those that are farther from
one another
• This approach is the least well-developed
8
II.
Measuring Segregation by
an Ordinal Category
• Following Reardon and Firebaugh (2002) the
idea is to assume that segregation is the
proportion of the total variation in a population
that is due to differences in population
composition of different organizational units
(e.g., schools or census tracts)
• Assume a measure of variation v. The
segregation measure S(v) will then be
expressed as :
9
Measuring segregation by an
ordinal category
S(v) = j=1 to J (tj /T) (1 – (vj / v))
where :
the subindex j refers to the organizational unit
(e.g. region),
tj to the population in unit j,
T to the total population,
vj to the variation in unit j and
v to the variation in the total population
10
Measuring segregation by an
ordinal category
 Measuring ordinal variation
• Consider an ordinal variable x that can take on
any of K ordered categories 1, 2, …, K
• Ordinal variation will be assumed to be maximal
(e.g. equal 1) when half the population has x=1
and half has x=K
• Ordinal variation will be minimal (e.g. equal to 0)
when all observations have x=k for some k1,
2,…, K
• Measuring ordinal variation then amounts to
measuring how close the distribution of x is to
these minimum and maximum variation states
11
Measuring segregation by an
ordinal category
• Consider the set C of [K-1] cumulative proportions ck ,
with C=(c1 , c2 ,…, cK-1), where ck is the cumulative
proportion of the sample with values of X in category k or
below (cK = 1 by definition).
• The maximal variation corresponds to
C = (½, ½, ½,…, ½) which is the case where half the
population has the lowest possible value and half has
the highest possible value of x.
Note also that there are K possible distributions of x such
that there is no variation in x, which implies that
cj = 0 for j<k and cj = 1 for jk.
12
Measuring segregation by an
ordinal category
•
Two possible measures of ordinal
variation
1)
Entropy based measure
E0 = (1/(K-1))k=1 to K-1[ck log2 (1/ck)+(1-ck) log2 (1/(1- ck))]
2)
Measure based on the concept of diversity
I0 = (1/(K-1))  k=1 to K-1 4 ck (1- ck)
13
Corresponding Measures of
Segregation
1. Ordinal information theory index (an ordinal
generalization of the categorical
information theory index H of Theil)
H0 =  j=1 to J (tj/T) (1 – (Eoj/E0))
2. Ordinal variation ratio index (an ordinal
generalization of the categorical variance
ratio index, based on the concept of
diversity)
R0 =  j=1 to J (tj/T) (1 – (Ioj/I0))
14
Ordinal segregation as an average
of pairwise segregation
• Both H0 and R0 can be written as weighted averages of a
set of K-1 pairwise segregation indices:
H0 = (1/(K-1)) (1/E0)  k=1 to K-1 Ek Hk
R0 = (1/(K-1)) (1/I0)  k=1 to K-1 Ik Rk
where the subscript k indicates variation or segregation
computed between the two groups defined by the kth
threshold (i.e., Hk and Rk are segregation levels
measured between a group consisting of all those with
incomes in category k or below and a group consisting of
all those in category k+1 or above)
15
Measuring segregation by an
ordinal category
Graphical Representation
• Figure 1 shows cumulative household income percentile density
curves for each of the 176 census tracts in San Francisco County, in
2000
• Figure 2 shows the corresponding curves for the 613 tracts in
Wayne County
• In both figures, the x-axis indicates both the local (i.e., San
Francisco or Wayne County) income percentiles and the 15 income
thresholds used in the 2000 census
• The income distribution in San Francisco is generally “higher” than
in Wayne County (25% of households in San Francisco reported
incomes greater than $100,000, compared to 12% of Wayne
County)
• If there were no income segregation in either county, each tract’s
cumulative household income percentile density curve would fall
exactly on the 45-degree line
16
Measuring segregation by an
ordinal category
• If income segregation had been complete, each tract’s
curve would be a vertical line at some income level,
indicating that within each tract all households have the
same income
• Thus, income segregation can be measured by the
average deviation of the tract cumulative household
income percentile density curves from their regional
average (which is, by definition, the 45-degree line)
• By this measure, Wayne County appears more
segregated by income than San Francisco, since the
variation of the tract cumulative density curves around
the 45-degree line is greater in Wayne County
17
Measuring segregation by an
ordinal category
Figure 1
Cumulative Household Income Percentile Densities,
176 San Francisco Census Tracts, 2000
100
80
60
40
20
0
Threshold (Dollars and Income Distribution Percentile)
18
,000
,000
100
$200
$150
,000
90
$125
,000
80
$100
00
70
$75,0
00
60
$60,0
00
00
00
50
$50,0
$45,0
00
00
00
40
$40,0
$35,0
$30,0
00
30
$25,0
00
20
$20,0
$10,0
00
10
$15,0
0
Measuring segregation by an
ordinal category
Figure 2
Cumulative Household Income Percentile Densities,
613 Wayne County (Detroit) Census Tracts, 2000
100
80
60
40
20
0
Threshold (Dollars and Income Distribution Percentile)
19
,000
100
$125
,000
$150
$200,000
,000
90
$100
00
80
$75,0
00
70
$60,0
00
$50,0
00
60
$45,0
00
$40,0
00
50
$35,0
00
40
$30,0
00
$25,0
00
30
$20,0
$15,0
$10,0
20
00
10
00
0
Measuring segregation by an
ordinal category
• Figures 3 and 4 show the pairwise household
income segregation levels computed at each of
the 15 Census 2000 thresholds for San
Francisco and Wayne County, respectively
• In addition, each figure illustrates the relative
weight (dashed lines) that the pairwise
segregation computed at each threshold is given
in the calculation of the ordinal segregation
measures (which are shown by the thin
horizontal lines in each figure)
20
Measuring segregation by an
ordinal category
Figure 3
Pairwise Household Income Segregation Measures, by Threshold,
176 San Francisco Census Tracts, 2000
Household Income Segregation
0.20
1.5
Relative Weight (density)
0.15
0.10
0.05
Pairwise H
Pairwise R
E weight
I weight
Ordinal H
Ordinal R
1.0
0.5
0.00
0.0
,000
,000
100
$200
$150
,000
Threshold (Dollars and Income Distribution Percentile)
21
90
$125
,000
80
$100
00
70
$75,0
00
60
$60,0
00
00
00
50
$50,0
$45,0
00
00
00
40
$40,0
$35,0
$30,0
00
30
$25,0
00
20
$20,0
$10,0
00
10
$15,0
0
Measuring segregation by an
ordinal category
Figure 4
Pairwise Household Income Segregation Measures, by Threshold,
613 Wayne County (Detroit) Census Tracts, 2000
Household Income Segregation
0.20
1.5
Relative Weight (density)
0.15
0.10
0.05
Pairwise H
Pairwise R
E weight
I weight
Ordinal H
Ordinal R
1.0
0.5
0.00
0.0
Threshold (Dollars and Income Distribution Percentile)
22
100
$125
,0
$150 00
$200,000
,000
,000
90
$100
00
80
$75,0
00
70
$60,0
00
00
$50,0
00
60
$45,0
$40,0
00
50
$35,0
00
40
$30,0
$25,0
00
30
00
00
$15,0
$10,0
20
$20,0
10
00
0
Measuring segregation by an
ordinal category
• Segregation, (whether using an approach based on entropy or
one based on diversity) is relatively flat across most of the
middle of the income percentile distribution in both places, but
increases or decreases sharply at the extremes of the
distribution, depending on which measure is used
• As expected, measured segregation at each income
percentile is generally higher in Wayne County than in San
Francisco, regardless of which measure is used
• N.B. The ordinal segregation measures for San Francisco and
Wayne County in Figures 3 and 4 are not exactly comparable
to one another because income thresholds do not fall at the
same percentiles of the distributions (there are differences in
the overall income distributions in the two counties)
• Moreover the measures clearly depend on the choice of thresholds
23
Measuring segregation by an
ordinal category
• Therefore if pk is near 0 or 1, the entropy related index contains little
information about the segregation experienced by an individual,
since it distinguishes among individuals only at one extreme of the
income distribution. Conversely, if pk is near 0.5, then the entropy
related index will contain a maximal amount of information, since the
distinction between the two groups takes place at the median of the
distribution.
• The same phenomenon occurs with a diversity based segregation
index. For a given threshold k, the probability that two randomlyselected individuals from the population will have incomes on
opposite sides of threshold k is 2 pk(1- pk).
• Because such a probability is maximal when pk=0.5 and minimal
when pk=0 or pk=1, a greater weight will be given to the case where
segregation between groups is defined by the median of the income
distribution than to the cases where a distinction is made between
an extreme income group and the remainder.
24
Measuring segregation by an
ordinal category
 Incorporating spatial proximity into
measures of income segregation
• Spatial rank-order information theory index and
spatial rank-order ratio index
• The expressions are similar to those given
previously for the rank-order segregation indices
but take into account spatial proximity, as
explained in the paper by Reardon and Sullivan
on measures of spatial segregation
25
IV. Empirical Examples
• Table 1 reports estimated household income
segregation levels for 6 metropolitan areas. The first
column in each panel of the table reports the ordinal
income segregation measures H0 and R0.
• The subsequent columns report the rank-order
income segregation measures estimated based on
polynomial approximations of orders M=2 through
M=10.
• The rank-order measures are remarkably stable,
regardless of the order of polynomial used. This is
largely because the functions H(p) and R(p) are
relatively smooth functions
26
Empirical Examples
Table 1: Estimated Household Income Segregation Levels, Selected Metropolitan
Areas, 2000
Panel A: Rank-Order Information Theory Index
Metropolitan
Area
Atlanta
Denver
Minneapolis
New York
Pittsburgh
San Jose
HO
HR
HR
HR
HR
HR
HR
HR
HR
HR
0.1360
0.1652
0.1385
0.1475
0.1088
0.0974
(M=2)
0.1365
0.1670
0.1366
0.1478
0.1062
0.1020
(M=3)
0.1362
0.1664
0.1360
0.1462
0.1053
0.1020
(M=4)
0.1362
0.1662
0.1357
0.1472
0.1055
0.1028
(M=5)
0.1362
0.1662
0.1357
0.1465
0.1049
0.1028
(M=6)
0.1364
0.1664
0.1360
0.1476
0.1055
0.1031
(M=7)
0.1364
0.1664
0.1360
0.1475
0.1054
0.1030
(M=8)
0.1364
0.1664
0.1360
0.1466
0.1051
0.1029
(M=9)
0.1364
0.1664
0.1360
0.1467
0.1052
0.1029
(M=10)
0.1365
0.1666
0.1360
0.1469
0.1052
0.1029
Panel B: Rank-Order Relative Diversity Index
Metropolitan
Area
Atlanta
Denver
Minneapolis
New York
Pittsburgh
San Jose
27
RO
RR
RR
RR
RR
RR
RR
RR
RR
RR
0.1472
0.1764
0.1465
0.1620
0.1161
0.1029
(M=2)
0.1523
0.1846
0.1504
0.1613
0.1159
0.1141
(M=3)
0.1528
0.1849
0.1503
0.1613
0.1159
0.1143
(M=4)
0.1529
0.1849
0.1504
0.1611
0.1158
0.1140
(M=5)
0.1529
0.1849
0.1504
0.1612
0.1160
0.1140
(M=6)
0.1528
0.1848
0.1503
0.1609
0.1159
0.1141
(M=7)
0.1528
0.1848
0.1503
0.1610
0.1160
0.1141
(M=8)
0.1528
0.1848
0.1503
0.1608
0.1160
0.1140
(M=9)
0.1528
0.1848
0.1503
0.1609
0.1160
0.1140
(M=10)
0.1528
0.1849
0.1503
0.1609
0.1160
0.1138
Empirical Examples
• Figure 5 illustrates the values of Hk at each of the 15
thresholds for the New York metropolitan area, as well as
the fitted polynomials of order M=2, 3,…, 10. Note that
for polynomials of order 4 or higher, the curves fit the
points extremely well through most of the range
• Table 1 indicates that the ordinal measures H0 and R0
are often a reasonably good estimate of the rank-order
measures HR and RR, though not always. In San Jose,
for example, R0 is 10 percent lower than the values of
RR. In Pittsburgh, in contrast, H0 is 3% larger than the
values of HR. Moreover, Atlanta appears more
segregated than Minneapolis on the basis of H0, but less
segregated on the basis of HR
28
Empirical Examples
Figure 5
Fitted Household Income Segregation Threshold Profiles
New York Metropolitan Area, 2000
Polynomial Order
0.35
Segregation (H)
0.30
M=2
M=3
M=4
M=5
M=6
M=7
M=8
M=9
M=10
0.25
0.20
0.15
0.10
0.05
0.00
Threshold (Dollars and Income Distribution Percentile)
29
,000
,000
100
$200
,000
$150
$125
,000
90
$100
00
80
$75,0
00
70
$60,0
00
00
00
60
$50,0
$45,0
00
50
$40,0
$35,0
00
00
40
$30,0
$25,0
00
30
$20,0
$10,0
00
20
$15,0
10
00
0
Empirical Examples
Illustration based on spatial rank order
information theory index
• The authors use a biweight kernel proximity function with
radii varying from 500m to 4000m. The information
theory based index will then indicate how much less
income variation there is in a radius of 500m (or 4000m)
than in the metropolitan area as a whole
• Table 2 reports values, for each of the 6 metropolitan
areas, of the spatial rank order information theory index
and the spatial relative diversity index, each computed at
radii of 500m, 1000m, 2000m, and 4000m
30
Empirical Examples
Table 2: Estimated Household Spatial Income Segregation Levels, by Spatial Scale,
Selected Metropolitan Areas, 2000
Panel A: Rank-Order Information Theory Index
Aspatial HR
Metropolitan
Area
(Block groups) (500m)
Atlanta
0.1364
0.1457
Denver
0.1664
0.1685
Minneapolis
0.1360
0.1422
New York
0.1466
0.1236
Pittsburgh
0.1051
0.1098
San Jose
0.1029
0.1020
Spatial HR
(1000m) (2000m)
0.1355
0.1177
0.1435
0.1143
0.1229
0.0980
0.1076
0.0906
0.0939
0.0726
0.0821
0.0605
(4000m)
0.0951
0.0905
0.0733
0.0698
0.0517
0.0431
Granularity Ratio
HR(4000m)/ HR(500m)
0.6528
0.5368
0.5157
0.5646
0.4704
0.4226
Panel B: Rank-Order Relative Diversity Index
Aspatial RR
Metropolitan
Area
(Block groups) (500m)
Atlanta
0.1528
0.1630
Denver
0.1848
0.1878
Minneapolis
0.1503
0.1577
New York
0.1608
0.1378
Pittsburgh
0.1160
0.1215
San Jose
0.1140
0.1140
Spatial RR
(1000m) (2000m)
0.1524
0.1332
0.1614
0.1296
0.1375
0.1106
0.1206
0.1017
0.1046
0.0814
0.0928
0.0688
(4000m)
0.1082
0.1034
0.0835
0.0782
0.0583
0.0490
Granularity Ratio
RR(4000m)/ RR(500m)
0.6639
0.5508
0.5296
0.5678
0.4800
0.4299
Note: All values of HR and RR are estimated using an 8th-order fitted polynomial approximation. Spatial indices are
computed using a biweight kernel proximity function with bandwidths 500m, 1000m, 2000m, and 4000m.
31
Empirical Examples
• Table 2 indicates that income segregation
declines with scale, though at different rates
across metropolitan areas. In Atlanta, for
example, income segregation computed using
4000m-radius local environments is two-thirds of
segregation computed using a 500m radius.
• We examine the patterns of income segregation
in more detail for each of the metropolitan areas
in Figures 6-11.
32
Empirical Examples
Figure 6
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, Atlanta Metropolitan Area, 2000
0.35
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.25
0.6
H4000m/H500m Ratio
Spatial Segregation (H)
0.30
0.8
Block Group
0.20
0.15
0.10
0.4
0.2
0.05
0.00
0.0
Threshold (Dollars and Income Distribution Percentile)
33
$200
,000
,000
100
$150
,000
90
$125
,000
80
$100
00
70
$75,0
00
60
$60,0
00
00
$50,0
00
50
$45,0
00
00
40
$40,0
$35,0
00
00
30
$30,0
$25,0
00
20
$20,0
$10,0
00
10
$15,0
0
Empirical Examples
Figure 7
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, Denver Metropolitan Area, 2000
0.35
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.25
0.6
H4000m/H500m Ratio
Spatial Segregation (H)
0.30
0.8
Block Group
0.20
0.15
0.10
0.4
0.2
0.05
0.00
0.0
Threshold (Dollars and Income Distribution Percentile)
34
100
$125
,000
$150
,000
$200
,000
90
,000
80
$100
00
70
$75,0
00
60
$60,0
00
$50,0
00
50
$45,0
00
00
40
$40,0
$35,0
00
00
00
30
$30,0
$25,0
00
20
$20,0
$10,0
00
10
$15,0
0
Empirical Examples
Figure 8
Figure 9
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, Minneapolis Metropolitan Area, 2000
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.30
H4000m/H500m Ratio
0.25
0.20
0.15
0.10
0.6
0.8
Block Group
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.25
0.20
0.4
0.15
0.2
0.10
0.6
0.4
0.2
0.05
Threshold (Dollars and Income Distribution Percentile)
0.00
0.0
,000
,000
,000
100
$200
$150
,000
Threshold (Dollars and Income Distribution Percentile)
90
$125
$100
00
80
$75,0
00
70
$60,0
00
00
00
60
$50,0
$45,0
00
50
$40,0
$35,0
00
40
$30,0
00
30
00
20
$25,0
10
00
0
$20,0
100
,000
$150
,00
$200 0
,000
90
$125
,000
80
$100
00
70
$75,0
00
60
$60,0
00
00
00
50
$50,0
$45,0
00
00
40
$40,0
$35,0
00
00
30
$30,0
$25,0
00
20
$20,0
$15,0
$10,0
00
10
00
0.0
0
$15,0
0.00
$10,0
0.05
35
0.35
H4000m/H500m Ratio
0.30
0.8
Block Group
Spatial Segregation (H)
0.35
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, New York Metropolitan Area, 2000
Empirical Examples
Figure 10
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, Pittsburgh Metropolitan Area, 2000
0.35
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.25
0.6
H4000m/H500m Ratio
Spatial Segregation (H)
0.30
0.8
Block Group
0.20
0.15
0.10
0.4
0.2
0.05
0.00
0.0
Threshold (Dollars and Income Distribution Percentile)
36
100
,000
$125
$150,000
$200,000
,000
90
$100
00
80
$75,0
00
$60,0
00
00
70
$50,0
$45,0
00
60
$40,0
00
00
50
$35,0
$30,0
00
40
$25,0
00
30
$20,0
$15,0
$10,0
20
00
10
00
0
Empirical Examples
Figure 11
Fitted Household Income Segregation Threshold Profiles,
by Spatial Scale, San Jose Metropolitan Area, 2000
0.35
Spatial Segregation (H)
0.30
0.8
Block Group
500m Radius
1000m Radius
2000m Radius
4000m Radius
H4000m/H500m Ratio
0.6
H4000m/H500m Ratio
0.25
0.20
0.15
0.10
0.4
0.2
0.05
0.00
0.0
,000
$200
$150
$125
Threshold (Dollars and Income Distribution Percentile)
37
90
,000
80
,000
70
,000
60
$100
50
00
40
$75,0
30
00
20
$60,0
10
$10,0
00
$15,0
00
$20,0
00
$25,0
00
$30,0
00
$35,0
00
$40,0
00
$45,0
00
$50,0
00
0
100
Empirical Examples
• In each figure, we plot the pairwise spatial
information theory index computed at the
threshold defined by the various income
percentiles, estimated at the four radii.
• Figure 9, for example, illustrates the case of
New York. It appears that the segregation
gradient at the high end of the income
distribution is quite steep. This indicates that the
highest-income households are substantially
more segregated from other households than
are the lowest-income households.
38
DISCUSSION
• This is a very interesting paper that suggests
new ways of measuring segregation by income
• I will not discuss the section of the paper that
deals (at the end) with spatial segregation
measurement because this is a vast field to
which Reardon has made very important
contributions
• I have to confess however that I am only starting
to read this literature (to which until now mostly
sociologists and geographers have contributed).
I do not feel that I know enough about this topic
to make relevant remarks
39
DISCUSSION
• I feel however much more comfortable
with the other topics covered by this paper
• I found out that sociologists are not aware
of important contributions in economics in
the same way as sociologists reviewing
some of my papers have drawn my
attention to the fact that I was almost
completely ignorant of important
sociological contributions
40
DISCUSSION (1)
1. Let me start with Figures 1 and 2 which
are the basis for the derivation of some
of the segregation measures proposed
by the authors
• The curves drawn in Figures 1 and 2 are
in fact what economists have called
Interdistributional Lorenz Curves
• This concept was first proposed by Butler
and McDonald (1987)
41
DISCUSSION (1)
• Let x be a continuous income variable with a probability density
f(x). The hth partial moment, given a target income , will then be
defined as 0 to  xh f(x) dx.
• The normalized incomplete moment of x for x is then defined
as
(,h,x) = [0 to  xh f(x) dx]/E(xh)
where E(xh) = lim 0 to  xh f(x) dx
We may therefore interpret (,1,x) as the proportion of income
received by individuals with income x smaller than or equal to 
Butler and McDonald have then proposed to plot (,h,x) against
(,h,x) for h=0 or h=1 where  and  are two population
subgroups
Plotting (,h,x) on the horizontal line and (,h,x) on the vertical
line, we will obtain an interdistributional Lorenz curve that will lie
below the 45-degree line if subgroup  is unambiguously
disdvantaged, that is if
(,h,x) (,h,x) for every .
42
DISCUSSION (1)
• Deutsch and Silber (1999) have shown
that these curves allow one to compute
Pietra or Gini indices that measure the
economic advantage of one group over
another, a concept originally proposed by
Dagum (1985) but which goes back to
Gini’s notions of “Transvariazione” and
“Ipertransvariazione”
43
DISCUSSION (1)
• The novelty of the paper by Reardon et al.
is that it does not limit the comparison to
two groups but extends it to all relevant
subgroups (e.g. Census tracts) in a given
population (e.g. metropolitan area), the
horizontal axis referring always to the
overall population (metropolitan area).
44
DISCUSSION (1)
• Moreover the paper suggests using all the binary
comparisons between the various census tracts and the
metropolitan area, that give rise to all these curves, to
derive a measure of the dispersion of these curves which
amounts to computing income based segregation indices
• Bishop, Chow and Zeager (2004) have derived statistical
tests for these interdistributional Lorenz curves and I
believe it would be worthwhile to extend their work and
derive tests that would allow to conclude more firmly, for
example, whether an income based segregation index in
a given metropolitan area is higher or lower than the
corresponding one in another area
45
DISCUSSION (2)
• My second basic remark has to do with
the way the authors compute their income
based segregation indices
• Whether they used information or diversity
based indices, their idea is to derive the
ratio of the between tracts segregation
over the total (metropolitan area)
segregation. Clearly if this ratio is high,
most of the segregation takes place
between tracts
46
DISCUSSION (2)
• It is interesting to note that such a ratio will
always
 rise when the between areas segregation
increases
 decrease when the within areas segregation
increases
• But these are precisely the two basic axioms
used by Esteban and Ray as well as others
when devising what they call a polarization index
47
DISCUSSION (2)
• This comes out actually very clearly when one
recalls that Reardon and his co-authors write
that :
 ordinal variation will be assumed to be maximal
when half the population has x=1 and half has
x=k (recall that there are K ordered categories
1,2 …, K)
 ordinal variation will be minimal when all
observations have x=k for some k1,2,..,K.
48
DISCUSSION (2)
• So here again is a proof that there is room
for much more interaction between
sociologists and economists, at least in the
field of segregation measurement
49
DISCUSSION (3)
• My third remark concerns again this idea of
measuring segregation when categories are
ordered
• I would like to draw the attention of the authors
to the fact that Hutchens has recently completed
a paper where he extends his square root index
to the case of ordered categories (e.g.
occupations ranked by prestige or income). This
square root index is in fact used by Jenkins et al.
in the paper they present in this session.
• I would like to stress that Hutchens derived
axiomatically this broader index that he has
recently suggested
50