Modeling Social Segregation Using Multilevel Models

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

2. 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)

3. Motivation: are we become a segregated society? EG in relation to schools Virtuousand Vicious circles Following 1988 Education Reform Act with emphasis on choice, league tables, competition expectation of INCREASED segregation Choice increased polarization in terms of ability increased polarization in terms of socio-economic background; poverty; ethnicity etc Choice

4. 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?

5. 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,

6. Greater than 25% departure from 6 year median

7. 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: • 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

8. 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 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)

9. 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) Mean of distribution is 0.079 Apparent substantial change!

10. Expected value of the Difference Index(if just stochastic fluctuations) where is underlying proportion, N is number of pupils in each school. Diff of 3% even when n = 200 The same thing applies to other Indices……….

11. E: the expected value for D if there was NO segregation; Structured: higher D when small schools and more extreme proportion

12. 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 Multilevelmodel ….

13. 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)

14. Anatomy of a simple model Distributed as a Binomial variable with a denominator equal to no of pupils in each school, with an underlying propensity of having a FSM, Dependent variable: observed FSM or not, in 2001 for pupil i in school j Model Log-odds of propensity As an underlying average & allowed to vary school difference With a variance of School differences assumed to come from a Normal distribution KEY measure of segregation; between-school variance on logit scale; if assumption met, complete summary, not arbitrary index Between pupil variance: allows for stochastic fluctuations determined by n and

15. Results from simplemodel 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

16. Using model parameters we can derive expected values of any function of underlying school probabilities Consequently, derive index by simulation from model parameters. Linking models to indexes 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

17. Behaviour of the indexesUsing simulation

18. Gorard G index Note how a change can be either due to changing dispersion or mean

19. 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

20. Results for simple model repeated for each entry cohort 2001-2006 Segregation: changes smaller than uncertainty Median: small improvement

21. 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

22. 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

23. 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

24. Results for Non and Selecting LA’s • Schools in Selecting areas are more segregated • Slight evidence of an increase • Pupils going to school in Selecting • LA’s are less likely to be in poverty • Slight decline in poverty in both types of area

25. 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

26. Results for Non and Selecting LA’s • Schools in Selecting areas are more segregated • Segregation decreases with greater LA control for both types of LA • Pupils going to school in Non-Selecting LA’s with low LA control are more likely to be in poverty

27. 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!

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

30. 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

31. 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 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. • 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

32. 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 Review20: 210-217. Hutchens, R. (2004). One measure of segregation. International Economic Review45: 555-578. Goldstein, H. and Noden, P. (2003). Modelling social segregation. Oxford Review of Education29: 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.