Transcript Modelling Segregation Using Multilevel Models: FSM in England 2001-6 Session 2: Modelling Social Segregation Monday 30th June 2008

Modelling Segregation Using Multilevel
Models: FSM in England 2001-6
Session 2: Modelling Social Segregation
Monday 30th June 2008
Outline
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Motivation: the importance of segregation
Research questions
Data: FSM obtained from PLASC
Traditional index approaches
Problems with an index approach
Model-based approach
Linking the model-based approach to indexes
Applying the model-based approach
Extensions of the model-based approach
The Composition of Schools in England (June 2008)
Motivation: are we become a segregated society?
EG in relation to schools
Virtuous and Vicious circles
Following 1988 Education Reform Act with emphasis on
choice, league tables, competition expectation of
INCREASED segregation
Attractive
to High
status
parents
Apparent
high
performance
Unattractive to
High status
parents
Apparent
poor
performance
Apparent
worsening
Apparent
improved
performance
performance
Choice
increased polarization in terms of ability
Choice
increased polarization in terms of socioeconomic background; poverty; ethnicity etc
Research Questions
FSM eligibility: Only statutory available
information on economic disadvantage
• Has school FSM segregation increased?
• Has LA segregation increased?
• Has segregation been differential between
different types of LA’s
• Which currently are the most segregated
LA’s in England?
FSM: the data
• Source: Pupil Level Annual School Census
• Outcome: Proportion of intake Eligible for FSM
• Intake:
Year 7 of the national curriculum in 2001-2006,
Action
LA’s
Schools
Cohorts Pupils
Complete data from PLASC; 2001-2006
148
5.615
26,178
3,587,459
Omit ‘special’ schools
148
4.088
20.952
3,536,152
Omit cohorts with less than 20 pupils
148
3,636
20,429
3,535,056
Omit schools without a new intake at
aged 11 (ie middle schools) LA loss, eg
IOW, Poole
144
3,076
17,695
3271,010
Omit cohorts with implausible
year-on-year differences (see next slide)
144
3,076
17,637
3,261,372
Greater than 25% departure from 6 year
median
FSM: Eligibility criteria
FSM: Only statutory available information on economic
disadvantage
The current eligibility criteria are that parents do not have to pay for
school lunches if they receive any of the following:
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Income Support
Income-based Jobseeker's Allowance
Support under Part VI of the Immigration and Asylum Act 1999
Child Tax Credit, provided they are not entitled to Working Tax
Credit and have an annual income (as assessed by HM Revenue &
Customs) that does not exceed £14,155
• the Guarantee element of State Pension Credit.
• Children who receive Income Support or income-based Job
Seeker's Allowance in their own right
Measuring segregation: traditional Indexbased approaches EG D index
Segregation or diversity indexes have a long history (e.g. Wright
1937) and there are a lot of them!
Duncan and Duncan’s (1955) D: ones of the most popular
Characteristic segregation curve
100
Variable
D-Index of 0.3
Evenness
Cumulative % of FSM
80
60
40
20
0
0
20
40
60
80
Cumulative % of Non-FSM
100
fsmi is number of pupils in school i
eligible for FSM and nonfsmi is
number not eligible
FSM is the total number of pupils
eligible in LEA; NONFSM is number
not eligible
D-= 0, schools are evenly mixed;
0.3 = 30% of pupils move to get
evenness
NB based on OBSERVED proportions and ‘little or nothing is know
about the sampling properties of segregation measures’ (Reardon
and Firebaugh, 2002, 100)
The need to go beyond an Index
Consider a pair of schools where we measure proportion eligible for FSM
and define segregation as the absolute difference between the pair :
Diff Index = p1 – p2
What values can we get for Index when there is no real change, just
stochastic fluctuations?
Simulate data and calculate Index when no real change:
- 3000 pairs of schools, representing two time points
- true underlying proportion is 0.15 for both time points
- no of pupils in entry cohort in each school is 20 (n)
900
800
Distribution of Diff
700
Index
Mean of distribution is 0.079
Apparent substantial change!
F requenc y
600
500
400
300
200
100
0
0.0
0.1
0.2
Index
0.3
0.4
Expected value of the Difference Index
(if just stochastic fluctuations)
E | p1  p2 |~1.12  (1   ) / N
where
 is underlying proportion, N is number of pupils in each school.
Expected value of Diff Index for 3
different true proportions; by n
0.11
0.15
0.20
0.10
0.25
E xp e cte d va lu e
0.09
0.08
0.07
Diff of 3% even when n =
200
0.06
0.05
0.04
0.03
0
100
200
n: pupils in school
The same thing applies to other Indices……….
E (D) (Duncanand Duncan)

n
100
150
200
0.15
0.079
0.064
0.055
0.20
0.071
0.058
0.050
0.25
0.064
0.053
0.046
E: the expected value for D if there was NO segregation;
Structured: higher D when small schools and more extreme proportion
Model-based approaches
•Traditional index construction uses definitions based upon observed
proportions.
• By contrast, a statistical model-based approach allows us to make
inferences about underlying processes by allowing random
fluctuations that are unconnected with the difference of interest
• Extract parameters (‘signal’) from the stochastic ‘noise’
• Either use parameters as natural measure of segregation OR
simulate from parameter and use indices after taking account of
random fluctuations
•Moreover Multilevel model ….
Benefits of multilevel approach
• Explicit and separate modelling of trends and segregation; fixed part of
model gives general trend; variance between schools gives
segregation
• Simultaneous modelling of segregation at any level: eg decreasing at
LA (local economy?), but increasing School (admission policies?)
• Segregation for different types of areas: not just variances, but
variances as a function of variables
• Explicit modelling of binomial fluctuations
• Confidence intervals
• BUT: “the approach is retrograde, and of no clear practical value”
(Gorard, 2004)
 jk
Anatomy of a simple model
Dependent variable: observed
FSM or not, in 2001 for pupil i
in school j
Model Log-odds of
propensity
  
loge

1  
School differences
assumed to come
from a Normal
distribution
Between pupil variance:
allows for stochastic
fluctuations determined
by n and

Distributed as a Binomial variable with a
denominator equal to no of pupils in each school,
with an underlying propensity of having a FSM,
As an underlying average

0
& allowed to vary school
difference u0 j
With a variance of
KEY measure of
segregation; betweenschool variance on logit
scale; if assumption met,
complete summary, not
arbitrary index
Results from simple model
Logit: -1.84 when transformed median of 0.137
(95% CI’s 0,133 and 0.142); and mean of 0.182
(0.177 and 0.187)
“Significant” between school segregation;
Equivalent to a D of 0.374 (see next slide)
Distributional assumptions
for school differences
Linking models to indexes
• Using model parameters we can derive expected values
of any function of underlying school probabilities
• Consequently, derive index by simulation from model
parameters.
Segregation curves for a range of values for the Variance and the D-Index
Cumulative % of FSM
100
Var
0.0
0.1
0.2
0.3
0.5
0.7
1.0
1.5
2.0
2.5
3.0
4.0
5.0
6.0
7.0
80
60
40
20
0
0
20
40
60
80
Cumulative % of Non-FSM
100
D-Index
0.000 Evenness
0.124
0.173
0.209
0.262
0.303
0.350
0.408
0.451
0.484
0.512
0.556
0.590
0.616
0.638
EG: Converting logit
Variance to D
(simulate 500k Logits with a
given underlying mean and
variance; convert to
proportions, and calculate
Index)
Variance of 0.7
equals DIndex of 0.30
Behaviour of the indexes
Using simulation
Relating five segregation indices to the variance (median proportion:0.15)
0.6
D-Index
G-Index
0.5
Gini
H-Index
I-Index
In d e x
0.4
0.3
0.2
0.1
0.0
0
1
2
Variance on the Logit scale
3
Gorard G index
Relating G-index to the variance for different proportions
0.5
0.05
0.10
0.15
0.4
0.20
G-In d e x
0.25
0.3
0.2
0.1
0
1
2
3
Variance on the Logit scale
Note how a change can be either due to changing dispersion or mean
Back to Results from simple model
Logit: -1.84 when transformed median of 0.137
(95% CI’s 0,133 and 0.142); and mean of 0.182
(0.177 and 0.187)
“Significant” between school segregation;
Equivalent to a D of 0.374
Distributional assumptions
for school differences
Results for simple model repeated for each
entry cohort 2001-2006
Median: small
improvement
Segregation:
changes smaller than uncertainty
Three-level model: partitioning between LA,
and between school variance
3 Changes
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Pupils (i) in schools (j) In LA’s (3)
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Average + LA difference + School
difference
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Between LA difference
Within LA, between school
Modelling at two scales simultaneously
Results for 3 level model
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3 level model applied to each cohort separately
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compared with Goldstein and Noden (earlier and
overall school and not entry cohort)
Between LA and within LA,
between school segregation 1994-2006
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Greater segregation
between schools than
between LA’s
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LA’s: trendless fluctuations
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Continued increasing
between-school
segregation
LACohort
0.7
LAG&N
S egregatio n
SchoolCohort
SchoolG&N
0.6
0.5
1995
2000
Years
2005
Area characteristics 1
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Are LA’s that are selective (Grammar/Secondary) more
segregated than totally Comprehensive systems?
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3 level model, with a different variance for schools within
different LA characteristics
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Average FSM
- for English pupils living in a nonselecting LA
- for English pupils living in a
selecting LA
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Between LA variance
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Within LA
- between school variance for schools
located in a non-selecting LA
- between school variance for schools
located in a selecting LA
Results for Non and Selecting LA’s
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Pupils going to school in Selecting
LA’s are less likely to be in poverty
Slight decline in poverty in both
types of area
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Schools in Selecting areas
are more segregated
Slight evidence of an increase
Area characteristics 2
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Is there more segregation in areas that are selective and where less schools
are under LA control in terms of admission policies?
Variance function for Selective/Non-selective, structured by the proportion of
pupils in an LA who go to Community or Voluntary Controlled schools (contra
Voluntary Aided,Foundation, CTC’s, Academies)
FSM over the period 2001-6
• Average FSM in selecting and nonselecting LA’s and how this changes with
degree of LA control
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Between LA variance
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Within LA between schools
- variance function for non-selecting LA
- variance function for selecting LA
Results for Non and Selecting LA’s
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Pupils going to school in NonSelecting LA’s with low LA
control are more likely to be in
poverty
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Schools in Selecting areas
are more segregated
Segregation decreases with
greater LA control for both
types of LA
Area characteristics 3
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Which of England’s LA’s have the most segregated school system?
Model with 144 averages and 144 variances, one for each LA!
LA analysis FSM 2001-6
4
Reading
Non
Select
3
Va ria nc e
Hammersmith and Fulham
Buckinghamshire
2
1
Southend-on-Sea
Slough
Traf f ord
Oldham
Calderdale
Sutton
Telf
ord
and
Wrekin
Solihull
Barnet
Wirral
Know sley
Milton Keynes
Croydon
Stockton-on-Tees
Birmingham
Liverpool
Kent
BexleyPlymouth
Salf ord
DudleyWolverhampton
Gloucestershire
Lincolnshire
Bromley
Walsall
Bolton
Havering
Kingston-upon-Thames
Torbay
Enf
ield City
Bradf
ord of
Bristol,
City
North
of Peterborough
East
Leeds
Kensington
Lincolnshire
and Chelsea
Essex
Sef ton
St
Helens
Bracknell Forest
Lancashire
MedwCumbria
ayRedbridge
Kirklees
Warw ickshire
Bury
Thurrock
Shef
f
ield
Wokingham
Northamptonshire
Hillingdon
Blackburn
w ith Darw en Camden
Cheshire
Lambeth Manchester
Derby
NorthHertf
Yorkshire
Warrington
Bournemouth
ordshire
Gateshead
Stoke-on-Trent
Coventry
Redcar and ClevelandNew castle-upon-Tyne Hackney
Hampshire
Sw indon
Portsmouth
Haringey
Derbyshire
South
Tyneside
Wigan
Waltham
Forest
Westminster
Barnsley
City
of
Kingston-Upon-Hull
Stockport
Hartlepool
Ealing
Staf
East
fYork
Sussex
Darlington
Rochdale
Brent
Wiltshire
North
Surrey
Somerset
Brighton
&
Hove
Oxf
ordshire
Suf
fordshire
olk
Wakef
ield
City
of Nottingham
Somerset
Doncaster
Luton
Bath
and
North
NE
Somerset
Lincolnshire
Durham
Hounslow
Middlesbrough
East
Riding
of
Yorkshire
West
Sussex
Rotherham
Sunderland
Leicester
Halton Wandsw
City
orth
North
Tameside
Tyneside
Worcestershire
Greenw
ich
Merton
Cambridgeshire
Norf
Nottinghamshire
olk
Blackpool
Lew
isham
Leicestershire
Heref
ordshire
Barking
Southw ark
Sandwand
ell Dagenham
Dorset
DevonRichmond-upon-Thames
Shropshire
Southampton
New ham Islington
Windsor
and
Maidenhead
South
Gloucestershire
Rutland
West
Berkshire
Cornw all
Harrow
Tow er Hamlets
0
0.0
0.1
0.2
0.3
0.4
0.5
Median Proportion FSM 2001-6
0.6
LA’s with highest segregation
(not including estimates lees than 2* SE)
LA
Variance
D equiv
Index
Median prop
FSM
2001-6
Select
Prop LA control
Buckinghamshire
2.12
0.46
0.03
Select
0.77
Southend-on-Sea
1.92
0.45
0.09
Select
0.21
Slough
1.76
0.43
0.11
Select
0.37
Trafford
1.75
0.43
0.08
Select
0.40
Oldham
1.72
0.43
0.18
Non
0.75
Calderdale
1.59
0.42
0.12
Select
0.32
Sutton
1.50
0.41
0.05
Select
0.39
Telford &Wrekin
1.46
0.40
0.15
Select
0.53
Solihull
1.42
0.40
0.08
Non
0.85
Barnet
1.42
0.40
0.16
Select
0.41
Knowsley
1.38
0.40
0.34
Non
0.67
Wirral
1.38
0.40
0.18
Select
0.74
Milton Keynes
1.36
0.39
0.12
Non
0.43
Croydon
1.30
0.39
0.16
Non
0.31
Stockton-on-Tees
1.29
0.39
0.16
Non
0.69
Extensions of the model-base approach
• multi- categorical responses: eg ethnic group
segregation.
• Multiple and crossed (non-nested levels) eg schools
and neighbourhoods simultaneously
• Multiple responses in a multivariate model eg. model
jointly the variation in the proportion FSM & proportion
entering with high levels of achievement
• Modelling spatial segregation: with MM models
High
8
High
8
HiMed
7
HiMed
Low
7
Low
LowMed
LowMed
6
5
N o rth
N o rth
6
4
5
4
3
3
2
2
1
1
1
2
3
4
5
East
6
7
8
1
2
3
4
5
East
6
7
8
The Composition of Schools in England
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What they did
Calculate D for LA’s in 1999 and 2007 (ignoring sampling variability)
Regress D for LA’s on variables EG prop of LA in Grammar schools; prop of faith schools, prop with
FSM; compare R2’s
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What they found
The level of FSM segregation increased for most LAs, but the average increase was relatively small.
Levels of FSM primary segregation more associated with the prop of FSM than any other LA
characteristics.
Levels of FSM secondary segregation more associated with the proportion in grammar schools than
any other LA characteristics.
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Some difficulties
Sampling variability and n
– ignores the nature of the Index that a more extreme proportion will produce higher D (eg Poole:
highest increase in segregation but also highest drop in FSM 1999-2007); scale artefact
- school size differs by type, and D index related to size of school
Levels: no recognition of within and between
- eg does not address: is there more segregation among schools within LAs for faith schools
Regression models:
-Focus on R2’s, but variation in D that cannot be explained, again not taken account of size
References
• Allen, R. and Vignoles, A. (2006). What should an index of school
segregation measure? London, Institute of Education.
• Duncan, O. and B. Duncan (1955). A methodological analysis of
segregation indexes American Sociological Review 20: 210-217.
• Hutchens, R. (2004). One measure of segregation. International
Economic Review 45: 555-578.
• Goldstein, H. and Noden, P. (2003). Modelling social segregation.
Oxford Review of Education 29: 225-237
• Gorard, S. (2000). Education and Social Justice. Cardiff, University
of Wales Press.
• Gorard (2004) Comments on 'Modelling social segregation' by
Goldstein and Noden, Oxford review of Education, 30(3), 435-440
• Reardon, S and Firebaugh, G (2002) Response: segregation and
social distance- a generalised approach to segregation
measurement Sociological Methodology, 32, 85-101.