(Segregation) Index Wars: Progress or
Regression?
Martin Watts
Centre of Full Employment and Equity
University of Newcastle
NSW
Australia
Paper prepared for the New Directions in Welfare
Congress, OECD, Paris, 6 -8 July 2011
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Introduction
 Segregation: groups of individuals differ in access to occupations,
social networks, schools etc (Reardon & Firebaugh, 2002).
 Reliance directly or indirectly on paid work for economic welfare
so school & occupational segregation particularly important.
 Academic disciplines (demography, economics & sociology)
active. 50 year index war since seminal article (D&D, 1955).
 War punctuated by comprehensive reviews (James & Taeuber,
1985; White, 1985; Watts, 1998; Reardon & Firebaugh, 2002a)
but, despite warnings, skirmishes have continued post 1998.
 Research should rigorously document pattern of segregation &
changes over time/across space, via index. Then develop & test
hypotheses about trends or x-country differences in segregation.
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Introduction
 Different methodologies:
axiomatic – but math tractability problem (Hutchens, 2001,
2004; Chakravarty & Silber, 2006; Alonso-Villar & Del Río,
2010; Frankel & Volij, 2011)
ad hoc: indexes satisfying desirable properties used for
particular research study (Flückiger & Silber, 1999; Reardon &
Firebaugh, 2002a,b; Watts 2003; Mora & Ruiz-Castillo, 2003,
2005, 2008, 2009a,b,c, Charles & Grusky, 2004; Deutsch &
Silber, 2005).
 Also disagreements persist over desirability of particular
properties
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Outline of Paper
1) Reflect on recent debates about segregation properties:
index decomposability;
construction of local indexes; &
Margin Independence.
2) Calculate Karmel/MacLachlan & Mutual Information Indexes of
gender segregation (43 occupations) in Australia 1996-2010.
Centre of Full Employment and Equity
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Segregation Index Criteria
Scale Invariance
Scale Interpretability
Symmetry
Independence
Weak Transfers
Strong Transfers
Org Division Property
Composition Invariance
Group Division Property
Occupational Equivalence
Transpose Invariance
Occupations Invariance
Additive Decomposability
IS
IDG
SQR
H
G
A
M
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2
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2
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Strong Organisation Decomposability
Strong Group Decomposability
Local Decomposability
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2
2
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2
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Notes: IS denotes multi-group KM index; IDG is multi-group analog of Index of Dissimilarity;
SQR is Hutchens’ two group square root index; H is Theil Entropy Index; G is Gini Index; A is
Charles & Grusky (two group) Index; & M is Mutual Information Index. ‘2’ indicates that
property only hold in 2 group case. Bolded properties are forms of Internal Decomposability.
Source: Reardon & Firebaugh (2002a:56), Frankel & Volij (2011:15) & author’s own analysis.
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Segregation Indexes: Linear
 Multi-group KM Index (IS) is related to multi-group ID (IDG):
G
G
N
g 1
g 1
n 1
IS  (2 I ) IDG   p g IS g   p g  p.n p gn / p g . p.n )  1
N
G
N
  pn  [Tgn  p.nTg . ] /T.n   pn IS n
(1)
pgn = Tgn/T = share of all individuals (T) in group g & organisation
n; pg. = Tg./T = share
of T in group g; p.n = T.n/T is share of T in
G
occupation n & I   pg . (1  pg . ) (Frankel & Volij, 2011:11).
n 1
g 1
n 1
g 1
 IS = weighted local indexes based on both organisations & groups
(Alonso-Villar & Del Río, 2010) due to Transpose Invariance;
but normalising factor I removed, so no Scale Interpretability.
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Segregation Indexes: Non-Linear
 Mutual Information Index (M):
N
N
G
n 1
g 1
N
G
M   [ p.n M ]   p.n  p g n log( p g n / p g . )   p gn log( p gn /( p g . p.n )
n
n 1
G
N
n 1 g 1
G
  p g .  pn g log( pn g / p.n )   [ p g . M g ]
g 1
n 1
(3)
g 1
pg n  pgn / p.n is conditional probability. Conversely p
ng
 M = reduction in uncertainty about individual’s group after
discovering organisation to which individual belongs (0 if no
segregation), & via transpose invariance, reduction in uncertainty
about individual’s organisation …..
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Segregation Indexes: Non-Linear
 Frankel & Volij (2011): M satisfies: Scale Invariance, Symmetry,
Independence, School & Group Division Property; Strong, Weak
Transfers; Local, Strong Organisation & Group Decomposability.
 M is Transpose Invariant (Mora & Ruiz-Castillo, 2009a,b).
Normalisation, via E or E*, for Scale Interpretability, rejected –
similar to IS.
 Properties shown in red also characterise IS.
 Neither M nor IS are CI or OI – so not margin free.
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Contemporary Debates - Decomposability
 ‘Horses for courses’ approach. So internal decomposability may
be less useful than construction of local indexes & use of Deming
& Stephen (1940) algorithm to neutralise Margin Dependence.
 If meaningful clustering of groups or organisations, then index
should satisfy Strong Decomposability (SD), such as M Index.
 School segregation: relevant benchmark of socio-economic
composition is local catchment area (LCA), so between & within
decomposition based on LCAs appropriate but not Margin Free.
 Using M, Frankel & Volij (2011:18) geographically decompose
total US school US segregation by race into 4 components.
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Contemporary Debates - Decomposability
 BUT Inter-city comparisons of segregation meaningless due to
margin dependence & unequal number of schools.
 Time series comparisons more robust but still vulnerable to MD.
From policy perspective, time series comparisons more useful.
 Index exhibiting Internal Decomposability is misleading if
investigating rates of integration of ‘homogeneous’ occupational
clusters, e.g. Managerial, Professional, Sales & Service, Skilled
Blue Collar & Unskilled (cf. Reardon & Firebaugh, 2002b:90).
 Skilled Blue Collar (SBC) occupations low female employment
shares (Watts, 2003). Within cluster segregation very low.
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
Contemporary Debates - Decomposability
 OTOH, between cluster component picks up sum of disparities
between overall & cluster based female employment shares.
 Incorrect benchmark, so internal decomposition yields no
insight about segregation trends across clusters to inform policy.
 IS & M: correct local decomposition, but Margin Dependent.
 Two approaches to achieving Margin Free measurement.
i) Employ MF index, such as A.
ii) External decomposition of index with desirable properties.
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Contemporary Debates – Margin Dependence
 A is Margin Free (Charles & Grusky, 2004) but no
Organisational Equivalance – all organisations same
weight.
0.5
J
J


A  exp (1 / J ) (ln( F j / M j )  (1 / J ) ln( F j / M j )) 2 
j 1
j 1


 Jerby et al (2005a): index class exhibits singularity; undefined
for zero (fe)male population in organisation j, non –linearity.
 Simple ad hoc solutions fail because of both sensitivity &
absence of Organisational Equivalence (Jerby et al, 2005a) .
 Debate whether zeros are sample or structural in traditional
societies, where cultural norms restrict organisation
(occupation) access to some group(s).
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Contemporary Debates – Margin Dependence
• Jerby et al (2006) segregation defined by margins, so margin
attributes endogenous to segregation; cf. purging MD.
• Jerby et al (2006) MF index is valid for ‘students of segregation
interested in capturing segregation through prism of occupational
categories’. Occupations as ontological entity.
• Segregation measurement should capture individuals’ distribution
across occupations, but TI indexes allow both interpretations.
 Jerby et al: segregation as outcome of ‘selection process with a
double barrier’. 1st barrier: entry into LM & 2nd barrier to
occupations.
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Contemporary Debates – Margin Dependence
 Define ISR (index for segregation regime):
J
J
j 1
j 1
ISR   Ai ri  R   (T j / T ) ( Fj  M j ) /(T j / 2)  ( F  M ) /(T / 2)  KM / 2
 2nd term = LM entrance barrier, 0 if F=M, equal representation.
 Barriers to individual occupations, revealed (& measured) as
variation of occupational measure around aggregate measure.
 Jerby et al (2005b): depict 2 dimensions, ISR & R on x axis.
 Inclusion/exclusion mechanisms (cultural forces, G policy, eg
tax policy), impact on women’s LFP & occupational attainment.
 BUT not clear conceptual & empirical separation of women’s
representation in employment & occupational attainment.
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Contemporary Debates – Margin Dependence
 2nd approach to obtain MF measure: external decomposition of
index to reveal changes (differences) in segregation, after impacts
resulting from changes in group shares & organisational
distribution (margin dependence) have been excised.
 Iterative procedure, (Deming & Stephan, 1940), transforms initial
(period 1) G*N segregation array into adjusted segregation array,
with same total populations by organisation & same overall group
shares of total population as later (period 2) array.
 Procedure eliminates MF problem because measure of change
based on comparison of ‘like with like’, namely adjusted array &
period 2 array. Margin Dependence finessed.
 If index exhibits Organisational Equivalence then organisational
weights used, overcoming potential sampled zeros problem.
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Contemporary Debates – Margin Dependence
 If time series data, then procedure can re-specify all time series
observations according to a common (period 2) base.
 Segregation arrays corresponding to W time series observations
enables construction of W*W matrix, IC   IPkm  where k,m =
1,2,…W & diagonal elements are simple gross index values;
 Each column represents consistent (MF) time series with
common base period, but which to choose? Suggest column with
max. average correlation with other columns so representative.
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Australian Gender Segregation: 1996-2010
 Both KM & M exhibit many other desirable properties, as noted
so now illustrate use of DS algorithm for Aus quarterly
employment data by occupation & 43 occupations for 1996-2010.

Figure 1: Australian Gender Segregation, 1996-2010 – M & KM indexes
0.25
0.24
0.23
0.22
0.21
0.2
0.19
0.18
0.17
A96 F97 A97 F98 A98 F99 A99 F00 A00 F01 A01 F02 A02 F03 A03 F04 A04 F05 A05 F06 A06 F07 A07 F08 A08 F09 A09 F10 A10
Mg
IPg
 High correlation between 2 indexes (0.975).
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Australian Gender Segregation: 1996-2010
Figure 2: Segregation by Occupational Group, 1996-2010: M index – Margin Free
110
100
90
80
70
60
50
A96 F97 A97 F98 A98 F99 A99 F00 A00 F01 A01 F02 A02 F03 A03 F04 A04 F05 A05 F06 A06 F07 A07 F08 A08 F09 A09 F10 A10
M_F04
MA_F04
PR_F04
SK_F04
LT_F04
US_F04
Notes: M (shown in White) overall M index; MA denotes Managerial; PR
denotes Professional/Paraprofessional; SS is Clerical, Sales &Service; SK
denotes Blue Collar Skilled and US denotes Unskilled.
 Chosen base period Feb. 2004 high average correlation 99.73%
(overall KM index) & 97.5% (M) with corresponding time series
associated with other base periods.
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Australian Gender Segregation: 1996-2010
Figure 3: Segregation by Occupational Group, 1996-2010: KM index – Margin Free
110
105
100
95
90
85
80
75
70
65
A96 F97 A97 F98 A98 F99 A99 F00 A00 F01 A01 F02 A02 F03 A03 F04 A04 F05 A05 F06 A06 F07 A07 F08 A08 F09 A09 F10 A10
IPCC
IPCMA
IPCPR
IPCSK
IPCLT
IPCUS
Notes: M (shown in White) overall M index; MA denotes Managerial; PR
denotes Professional/Paraprofessional; SS is Clerical, Sales &Service; SK
denotes Blue Collar Skilled and US denotes Unskilled.
 Only Managerial & Professional occupations experienced a rising
share of employment over sample period, although all
occupations experienced employment growth.
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Australian Gender Segregation: 1996-2010
Table 1 Summary Statistics for Gross Indexes & Overall & OG MF Indexes
Margin Free M index: Feb. 2004
Growth
Margin Free IP index: Feb 2004
Mg
IPg
M
MA
PR
SK
LT
US
M
MA
PR
SK
LT
US
-9.2
-4.3
-5.7
-39.2
8.5
0.4
-12.6
1.6
-3.4
-21.5
3.3
0.9
-6.3
1.0
 Evidently disparities between index outcomes (unsurprising) but
broad trends similar.
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Concluding Comments
 Properties of multi-group segregation measures reviewed.
 More use of axiomatic approaches, but limited by mathematical
tractability & constraints of internal decomposability (IntD).
 IntD enables construction of between & within cluster measures,
but IntD not always suitable & sensitive to Margin Dependence.
Local Decomposability (LD) addresses first issue.
 IS & M: good properties, including LD but Margin Dependent.
 Margin Free A, Index: no Organisational Equivalence & affected
by sampled zeros, sensitivity & limited to 2 groups.
 DS decomposition enables MF measures of change in segregation
(typically time series). Illustrated with Australian data.
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End of Talk
End of Talk