Session 2: Modelling Social SegregationMonday 30th June 2008

Modelling Segregation Using Multilevel Models: FSM in England 2001-6

Outline• 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)

Following 1988 Education Reform Act with emphasis on choice, league tables, competition expectation of INCREASED segregation

Un-attractive to High status

parents

Apparent worseningperformance

Apparent poor

performance

Virtuous and Vicious circles

Choice increased polarization in terms of ability

Attractive to High status

parents

Apparent improved

performance

Apparent high

performance

Choiceincreased polarization in terms of socio-economic background; poverty; ethnicity etc

Motivation: are we become a segregated society? EG in relation to schools

Research QuestionsFSM 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

The current eligibility criteria are that parents do not have to pay for school lunches if they receive any of the following:

• 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

FSM: Only statutory available information on economic disadvantage

Cumulative % of Non-FSM

Cum

ulat

ive

% o

f FS

M

100806040200

100

80

60

40

20

0

D-Index of 0.3

Evenness

Variable

Characteristic segregation curve

fsmi is number of pupils in school i eligible for FSM and nonfsmi is number not eligibleFSM 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)

Measuring segregation: traditional Index-based 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

The need to go beyond an IndexConsider 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)

Mean of distribution is 0.079Apparent substantial change!

0.40.30.20.10.0

900

800

700

600

500

400

300

200

100

0

Index

Fre

quen

cy

Distribution of DiffIndex

Expected value of the Difference Index(if just stochastic fluctuations)

1 2| |~ 1.12 (1 ) /E p p N

where is underlying proportion, N is number of pupils in each school.

The same thing applies to other Indices……….

0.15

0.20

0.25

2001000

0.11

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

n: pupils in school

Exp

ect

ed v

alu

e

different true proportions; by nExpected value of Diff Index for 3

Diff of 3% even when n = 200

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

Duncan) and(Duncan )(DE

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)

Anatomy of a simple model

jk

Dependent variable: observedFSM or not, in 2001 for pupil iin school j

1loge

Model Log-odds ofpropensity

School differencesassumed to come from a Normal distribution

With a variance of

KEY measure of segregation; between-school variance on logit scale; if assumption met, complete summary, not arbitrary index

Between pupil variance:allows for stochasticfluctuations 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

& allowed to vary school difference ju0

0

Results from simple model

Distributional assumptionsfor school differences

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)

Cumulative % of Non-FSM

Cu

mu

lati

ve %

of

FS

M

100806040200

100

80

60

40

20

0

2.5 0.4843.0 0.512

4.0 0.5565.0 0.5906.0 0.616

7.0 0.638

0.0 0.000 Evenness

0.1 0.1240.2 0.173

0.3 0.2090.5 0.262

0.7 0.3031.0 0.350

1.5 0.4082.0 0.451

Var D-Index

Segregation curves for a range of values for the Variance and the D-Index

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 D-Index of 0.30

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.

Behaviour of the indexesUsing simulation

D-Index

G-Index

Gini

H-Index

I-Index

0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Variance on the Logit scale

Index

Relating five segregation indices to the variance (median proportion:0.15)

Gorard G index

0.05

0.10

0.15

0.20

0.25

0 1 2 3

0.1

0.2

0.3

0.4

0.5

Variance on the Logit scale

G-Index

Relating G-index to the variance for different proportions

Note how a change can be either due to changing dispersion or mean

Back to Results from simple model

Distributional assumptionsfor school differences

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

Results for simple model repeated for each entry cohort 2001-2006

Segregation: changes smaller than uncertainty

Median: small improvement

Three-level model: partitioning between LA,

and between school variance 3 Changes

• Pupils (i) in schools (j) In LA’s (3)

• Average + LA difference + School difference

• Between LA difference • Within LA, between school

Modelling at two scales simultaneously

Results for 3 level model• 3 level model applied to each cohort separately

• compared with Goldstein and Noden (earlier and overall school and not entry cohort)

• Greater segregation between schools than between LA’s

• LA’s: trendless fluctuations

• Continued increasing between-school segregation

LACohort

LAG&N

SchoolCohort

SchoolG&N

1995 2000 2005

0.5

0.6

0.7

Years

Seg

rega

tion

between school segregation 1994-2006Between LA and within LA,

Area characteristics 1• Are LA’s that are selective (Grammar/Secondary) more

segregated than totally Comprehensive systems?

• 3 level model, with a different variance for schools within different LA characteristics

• Average FSM- for English pupils living in a non- selecting LA- for English pupils living in a selecting LA

• Between LA variance

• 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

• Pupils going to school in SelectingLA’s are less likely to be in poverty

• Slight decline in poverty in both types of area

• Schools in Selecting areas are more segregated

• Slight evidence of an increase

Area characteristics 2• 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 non-

selecting LA’s and how this changes with degree of LA control

• Between LA variance

• Within LA between schools- variance function for non-selecting LA- variance function for selecting LA

Results for Non and Selecting LA’s

• Pupils going to school in Non-Selecting LA’s with low LA control are more likely to be in poverty

• Schools in Selecting areas are more segregated

• Segregation decreases with greater LA control for both types of LA

Area characteristics 3• Which of England’s LA’s have the most segregated school system?• Model with 144 averages and 144 variances, one for each LA!

Non Select

0.60.50.40.30.20.10.0

4

3

2

1

0

Median Proportion FSM 2001-6

Varianc

e

West Sussex

Warwickshire

SurreySuffolkSomersetOxfordshire

Northamptonshire

Norfolk

Lincolnshire

Hertfordshire

Gloucestershire

Cumbria

Cornwall

Telford and Wrekin

Shropshire

City of NottinghamNottinghamshire Blackpool

Blackburn with DarwenLancashireMedway

Kent

WorcestershireHerefordshire

Thurrock

Southend-on-Sea

EssexTorbayPlymouth

Devon

Warrington

Halton

Cheshire

City of Peterborough

Cambridgeshire

Wokingham

Slough

Reading

West BerkshireWindsor and Maidenhead

Bracknell Forest

SwindonWiltshire

Stoke-on-Trent

Staffordshire

Rutland

Leicester CityLeicestershire

Southampton

PortsmouthHampshireBrighton & HoveEast SussexDarlingtonDurham

Bournemouth

Dorset

DerbyDerbyshire

Milton Keynes

Buckinghamshire

LutonYork

North Yorkshire

North Lincolnshire

North East Lincolnshire

East Riding of YorkshireCity of Kingston-Upon-Hull

Stockton-on-Tees

Redcar and Cleveland

MiddlesbroughHartlepool

South Gloucestershire

North Somerset

Bristol, City of

Bath and NE Somerset SunderlandSouth Tyneside

North Tyneside

Newcastle-upon-TyneGateshead

Wakefield

LeedsKirklees

Calderdale

Bradford

Sheffield

RotherhamDoncasterBarnsleyWigan

Trafford

TamesideStockport

Salford

Rochdale

Oldham

ManchesterBury

Bolton

Wirral

SeftonSt Helens

LiverpoolKnowsley

WolverhamptonWalsall

Solihull

Sandwell

Dudley

Coventry

Birmingham

Waltham Forest

Sutton

Richmond-upon-Thames

Redbridge

NewhamMerton

Kingston-upon-Thames

Hounslow

Hillingdon

Havering

Harrow

Haringey

Enfield

Ealing

Croydon

Bromley

Brent

Bexley

Barnet

Barking and Dagenham

WestminsterWandsworth Tower Hamlets

SouthwarkLewisham

Lambeth

Kensington and Chelsea

Islington

Hammersmith and Fulham

Hackney

Greenwich

Camden

LA analysis FSM 2001-6

LA’s with highest segregation (not including estimates lees than 2* SE)

LA Variance D equivIndex

Median prop FSM2001-6

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

HiMed

Low

LowMed

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

East

North

High

HiMed

Low

LowMed

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

East

North

The Composition of Schools in England • 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

• What they foundThe 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.

• Some difficultiesSampling 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 schoolsRegression 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.