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The Great Divide: fixed vs. random effects in an education context Claire Crawford with Paul Clarke, Fiona Steele & Anna Vignoles and funding from ESRC ALSPAC Large Grant

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Introduction I Our strand concerned with determinants of educational achievement Number of substantive research questions: – Impact of SEN – Impact of school size – Joint determination of cognitive and non-cognitive skills (ECM agenda)

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Introduction II Thinking about appropriate models – Pupils clustered within schools hierarchical models Two popular choices: fixed and random effects Which approach is best in which context? – Idea is always to move closer to a causal interpretation Choice of model: – Often driven by discipline tradition – May depend on whether primary interest is pupil or school characteristics

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Outline of talk Why SEN? Fixed and random effects models in the context of our empirical question Data and results Tentative conclusions

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Introduction to SEN One in four Y6 pupils in England identified as SEN – With statement (more severe): 3.7% – Without statement (less severe): 22.3% SEN label means different things in different schools and for different pupils – Maximum means special school or full time teaching assistant (i.e. additional resources) – Minimum means close monitoring or annual review – Recognition that SEN is not a treatment

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Why adjust for school effects? Want to estimate causal effect of SEN on pupil attainment no matter what school they attend Need to adjust for school differences in SEN labelling – e.g. children with moderate difficulties more likely to be labelled SEN in a high achieving school than in a low achieving school (Keslair et al, 2008; Ofsted, 2004) – May also be differences due to unobserved factors Hierarchical models can account for such differences – Fixed or random school effects?

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Fixed effects vs. random effects Long debate: – Economists tend to use FE models – Educationalists tend to use RE/multi-level models But choice must be context and data specific

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Basic model FE: u s is school dummy variable coefficient RE: u s is school level residual – Additional assumption required: E [u s |X is ] = 0 That is, no correlation between unobserved school characteristics and observed pupil characteristics Both: both models assume: E [e is |X is ] = 0 – That is, no correlation between unobserved pupil characteristics and observed pupil characteristics

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Relationship between FE, RE and OLS FE: RE: Where:

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How to choose between FE and RE Very important to consider sources of bias: – Is RE assumption (i.e. E [u s |X is ] = 0 ) likely to hold? Other issues: – Number of clusters – Sample size within clusters – Rich vs. sparse covariates – Whether variation is within or between clusters What is the real world consequence of choosing the wrong model?

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Sources of selection Probability of being SEN may depend on: – Observed school characteristics e.g. ability distribution, FSM distribution – Unobserved school characteristics e.g. values/motivation of SEN coordinator – Observed pupil characteristics e.g. prior ability, FSM status – Unobserved pupil characteristics e.g. education values and/or motivation of parents

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Intuition I If probability of being labelled SEN depends ONLY on observed school characteristics: – e.g. schools with high FSM/low achieving intake are more or less likely to label a child SEN Random effects appropriate as RE assumption holds (i.e. unobserved school effects are not correlated with probability of being SEN)

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Intuition 2 If probability of being labelled SEN also depends on unobserved school characteristics: – e.g. SEN coordinate tries to label as many kids SEN as possible, because they attract additional resources; Random effects inappropriate as RE assumption fails (i.e. unobserved school effects are correlated with probability of being SEN) FE accounts for these unobserved school characteristics, so is more appropriate – Identifies impact of SEN on attainment within schools rather than between schools

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Intuition 3 If probability of being labelled SEN depends on unobserved pupil/parent characteristics: – e.g. some parents may push harder for the label and accompanying additional resources; – alternatively, some parents may not countenance the idea of their kid being labelled SEN Neither FE nor RE will address the endogeneity problem: – Need to resort to other methods, e.g. IV

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Other considerations RE model may be favoured over FE where: – Number of clusters is large e.g. ALSPAC vs. NPD – Most variation is between clusters e.g. UK (between) vs. Sweden (within) – Have rich covariates

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Can tests help? Hausman test: – Commonly used to test the RE assumption i.e. E [u s |X is ] = 0 – But really testing for differences between FE and RE coefficients Over-interpretation, as coefficients could be different due to other forms of model misspecification and sample size considerations (Fielding, 2004) – Test also assumes: E [π is |X is ] = 0

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Data Avon Longitudinal Study of Parents and Children (ALSPAC) – Recruited pregnant women in Avon with due dates between April 1991 and December 1992 – Followed these mothers and their children over time, collecting a wealth of information: Family background (including education, income, etc) Medical and genetic information Clinic testing of cognitive and non-cognitive skills Linked to National Pupil Database

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Looking at SEN in ALSPAC Why is ALSPAC good for looking at this issue? – Availability of many usually unobserved individual and school characteristics: e.g. enjoyment of school, education values of parents, headteacher tenure – In particular: IQ (measured by clinicians) Good measures of non-cognitive skills (including behavioural difficulties) reported by parents/teachers

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Descriptive statistics 18% of sample are SEN at age 10 Individual characteristicsSchool characteristics Standardised KS1 APS-0.104**Independent school-0.102** IQ (age 8)-0.003**% eligible for FSM-0.002** SDQ (age 7)0.012**Hteacher tenure: 1-2 yrs-0.044** Mum high qual vocational-0.028*Hteacher tenure: 3-9 yrs-0.046** Mum high qual O-level-0.021Hteacher tenure: 10+ yrs-0.031 Mum high qual A-level-0.033* Mum high qual degree-0.019Observations5,615 Notes: relationship between selected individual and school characteristics and SEN status. Omitted categories are: mums highest qualification is CSE level; head teacher tenure < 1 year.

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Impact of SEN: full model OLS, RE and FE dont give qualitatively different answers to question of impact of SEN on KS2 APS – Hausman test suggests no difference between FE and RE OLSFERE SEN-0.514** [0.046] -0.494** [0.025] -0.511** [0.025] Std KS1 APS0.399** [0.041] 0.445** [0.012] 0.407** [0.011] Observations5,615 Note: model also controls for vast array of other individual and school characteristics (where appropriate).

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Impact of SEN: NPD only Again OLS, RE and FE dont give qualitatively different coefficients on SEN – But global Hausman test suggests FE and RE are NOT equivalent – May be because there is correlation between SEN and unobserved individual characteristics? SEN coefficients about 0.1 SDs higher than in full model OLSFERE SEN-0.610** [0.060] -0.581** [0.026] -0.597** [0.026] Std KS1 APS0.537** [0.044] 0.580** [0.011] 0.560** [0.011] Observations5,615 Note: model also controls for limited other individual and school characteristics (where appropriate).

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Impact of SEN: girls only Despite halving sample size, OLS, RE and FE again dont give qualitatively different coefficients on SEN – But Hausman test suggests FE and RE are NOT equivalent OLSFERE SEN-0.606** [0.066] -0.624** [0.043] -0.606** [0.042] Std KS1 APS0.377** [0.038] 0.402** [0.017] 0.377** [0.016] Observations2,741 Note: model also controls for vast array of other individual and school characteristics (where appropriate).

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Summary SEN appears to be strongly negatively with progress between KS1 and KS2 – SEN pupils score around 0.5 SDs lower Choice of model does not seem to matter here – OLS, FE and RE all give qualitatively similar results – Suggests correlation between probability of being SEN and unobserved school characteristics is not important – But doesnt mean we dont have to worry about selection on unobserved individual characteristics

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Still to come... More detailed investigation of conditions under which FE and RE are equivalent – Simulation study Do effects of SEN differ across schools?

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Tentative conclusions Approach each problem with agnostic view on model – Should be determined by theory and data, not tradition In reality, choice may not make very much difference – Can our results be generalised? Different questions? Different data? Worth remembering that neither FE nor RE deals with correlation between observed and unobserved individual characteristics

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