Local indices of segregation: a case study of London secondary schools

Local indices of isolation and of concentration with application to social segregation between London secondary schools

What is a local index of segregation?
Characteristics of such an index?
How to form one.
Case study: social segregation in London secondary schools.
Conclusions

Introduction

Segregation indices are used to measure how various social or ethnic groups of people are distributed across a study region and whether there is evidence they are separated or separating from each other or not.
Broadly, two types
Differential
Probabilistic

e.g. p obs is the proportion of the Free School Meal (FSM) eligible pupils that are within the school, p exp is the contrast group, the summation is across all n schools within the local education authority (LEA) and k is a scaling constant
definition of p EXP can be, inter alia , the proportion of all pupils within the LEA that are FSM eligible, the proportion of pupils within the school that are not FSM eligible or the proportion of all pupils in the LEA that attend the school.
These formulations produce a measure of within LEA variation, an index of dissimilarity and the Gorard index, respectively

e.g. the index of isolation: p obs is the proportion of all FSM eligible pupils within the LEA that are in each school, and p exp is the number of FSM pupils as a proportion of all pupils in the school.
It is the average probability of having selected a particular school then finding a FSM eligible pupil within it.

The indices do not consider where a school is located other than in terms of the area unit of analysis (the LEA)
Any sub-LEA patterns of association, dependence or interaction are lost.
Essentially an averaging problem.
Implies that geographies of school choice and local schools markers are coterminous with LEA boundaries.

“ in trying to establish whether or not the marketization of education has had a polarizing effect, the unit of analysis must […] be the local market within which schools (and parents faced with placement decisions) actually operate.”
Gibson & Asthana (2000, p.139)

… will
Be a function of how one place (school) compares with other places around it (competing schools)
Be calculated separately for each place
Therefore permit the distribution of the index values across the study region to be examined.

Case Study: social segregation between secondary schools

student Primary school Secondary school Attributes 1 A I … 2 A II … 3 B I … 4 C I … 5 C II … 6 B III … … … … …

primary primary primary primary secondary secondary

Because i and j are “competing”
w ij > 0
Specifically
w ij = p i (i ∩ j) × p j (i ∩ j)
the weight is equal to the proportion of secondary school i’s intake that is drawn from the primary schools shared with school j, multiplied by the proportion of secondary school j’s intake that is drawn from the same, with the weights then scaled (row-standardised) so that the sum of the weights for any school is equal to one.

Local index of dissimilarity (difference)
Local index of concentration
Local index of isolation

The local index of isolation is not composition invariant.
Will also use a modified index that scales for the local context
Year FSM, i FSM, j 1-FSM, j LID LII 1 0.4 0.3 0.7 0.10 0.28 2 0.6 0.5 0.5 0.10 0.30

p l(i) is the proportion of FSM eligible pupils for the sub-region, the LEA, within which the school i is located, and p ALL is the proportion of FSM pupils across the entire study region and/or study period

Application

Note: only |LID| is unidirectional (it determines a difference but not which school has the greater proportion of FSM-eligible pupils. All the others rise with FSM-eligibility per school. LII MLII LID |LID| LII 1 0.843 0.766 0.313 MLII 0.843 1 0.819 0.264 LID 0.766 0.819 1 0.206 |LID| 0.313 0.264 0.206 1

In 2008, the mean percentage of FSM eligible pupils in a London secondary school was 27.
There was, on average, a 7.8 percentage point difference between the percentage of FSM eligible pupils in one school and the weighed average percentage of its competitors.
The typical range was from a 2 percentage point difference to 11.

The mean index of isolation score was 0.179 with a typical range from 0.111 to 0.243
The mean modified index score was 0.189 with a typical range from 0.125 to 0.240

Note: Due to data swapping, it is not possible to read values for specific schools from their apparent locations on the map

Higher than expected Lower than expected

Directly or indirectly selecting schools tend to have a lower proportion of FSM eligible pupils than their competitors

No evidence that social segregation increased 2003-8. Why?

School 1 had 28 competing secondary schools in 2003 (w ij > 0) but, of these, four provided the first 90% of the sum of weights
ID 2 3 4 5 6 7 8 37 40 … w ij 0.275 0.131 0.015 0.018 0.262 0.029 0.222 0.005 0.002 … ID 2 6 8 3 7 105 5 4 43 … w ij 0.275 0.262 0.222 0.131 0.029 0.021 0.018 0.015 0.007 … ∑ w ij 0.275 0.537 0.759 0.89 0.918 0.939 0.957 0.972 0.979 …

And of the four, all of them were still providing the first 90% of the sum of weights in 2008 (100%)
In other words, not much had changed in terms of the main competition between secondary schools.
School 2 had seven main competitors (at the 90% threshold) in 2003, of which six are still main competitors in 2008 (86%)

Working through all the 361 of 382 secondary schools that can be matched from 2003 to 2008, on average, 69% of the main competitors in 2003 remain so in 2008, with a typical range from 52% to 83%.
Is this indicative of a stable system?

Segregation indices are used to measure how various social or ethnic groups of people are distributed across a study region and whether there is evidence they are separated or separating from each other or not.
A local and spatial index compares one place with others around it, adopting a network model of space.
The index can be calculated for each place permitting the distribution of the index values across the study region to be examined.

Applied to London and to social segregation between secondary schools, differences are observed between the proportion of pupils eligible for FSMs in one school and the proportion for competitors.
The differences are greatest for selecting schools (selection by academic attainment or by the practise of faith).
No evidence that segregation has increased over the period 2003-8.
Because the system is quite stable? The patterns of competition are remaining the same, on average?

Local indices of segregation: a case study of London secondary schools

by Rich Harris

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