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A re-examination of segregation indices in terms of compositional invariance
Chris Taylor is a research associate working on an ESRC-funded study of the social composition of schools at the School of Social Sciences, Cardiff University. Stephen Gorard and John Fitz are both Readers at the School of Social Sciences working on the same project. The team have recently published Questioning the crisis account: a review of evidence for increasing polarisation in schools in Educational Research (2000) 42, 3.
The role and validity of various indices of segregation have been a focus of considerable debate and speculation over the last fifty years in social science research, and have therefore been the subject of several previous issues of Social Research Update (e.g. Blackburn and Jarman 1997, Gorard 1999). Similar debates have occurred in many fields including the analysis of: residential patterns by ethnicity; gendered patterns of occupation; polarised income patterns in family economics, and the social composition of schools in education. During these years it is possible to distinguish at least two index wars. The first of these apparently crowned the Dissimilarity Index as the premier of all measures (Peach 1975). The more recent war seems to have moved the focus of attention away from individual measures of segregation, towards a consideration of composite measures, which identify different elements of segregation (Massey et al. 1996). Despite this shift in the epistemological debate the prevalence of particular indices has remained relatively unchanged since the first index war, 1947-55.
For any area with sub-areas in which segregation may take place, the index of dissimilarity may be defined as:
One basis for the repeated use of D in segregation research, despite criticism (e.g. by Blackburn and Jarman 1997), has been the way that it appears to meet the key criteria as generally agreed for an index of segregation. James & Taeuber (1985), for example, suggested that there were four such criteria for indices to satisfy:
Organisational equivalence -The index should be unaffected by changes in the number of sub-areas, by combination for example of two sub-areas on the same side of the line of no segregation.
Principle of transfers -The index should be capable of being affected by the movement of one individual from sub-area to sub-area.
Composition invariance -The index should be unaffected by scaling of columns or rows, through increases in the raw figures which leave the proportions otherwise unchanged.
Watts (1998) argued that for any analysis of segregation over time both composition invariance and occupation invariance are key to our understanding of a useful measure. These were defined in the following way -Compositional invariance refers to the invariance of the index, following uniform changes in the number of males and females in each occupation reflecting the overall, but typically unequal, percentage changes in male and female employment [...] Occupations invariance requires that the measure of segregation be invariant to changes in the relative size of occupations if the gender composition of these occupations remains constant (1998:490). These two criteria would ensure that the measure of segregation would not be affected by either an increase in the absolute levels of a particular group across all sub-areas, or an increase in the absolute levels of all groups in a particular sub-area (such that the relative composition of each sub-area remained unaltered)
It is criteria like these, which we would support, that have led to the decline of other previously suggested measures of segregation, inequality or polarisation such as the Variance ratio, Information theory index, index of Isolation, and the Atkinson index, and to the pre-eminence of D. The Dissimilarity Index, unlike many of the losers in the war, has long been considered as composition invariant, for even though Duncan & Duncan acknowledge that the proportion of both subgroups is present in the calculation they argue that D is unaffected by changes in either group. For example, Lieberson (1981) claims that D is not affected by population composition, and gives as an example if the number of whites in each subarea was divided by ten, then the index of dissimiliarity would remain unchanged (p.63). One of the primary purposes of this paper is to argue that on a strong interpretation of composition invariance this is not, in fact, so (or that at least D does not meet both of the requirements as described by Watts above).
Table 2 presents a hypothetical example of the number of students in four schools who are eligible for free school meals (FSM is an indicator of families defined as in poverty). D for this set of schools is 0.267. The fourth column shows what proportion of the total number of children in poverty are in each school. The final column shows what proportion of the total number of children in each school are in poverty
Obviously, in the trivial case where all of the numbers in Table 2 are scaled (so that School A takes 20 FSM students from a total of 200 for example), D remains the same. Additionally, as Lieberson and others have pointed out, if the number of students eligible for free school meals is doubled in each school, perhaps reflecting a period of economic recession, then D remains the same (Table 3). This is so despite changes in the proportion of students in poverty in each schools (column 5) since the proportion in each school of the total in poverty remains the same as in Table 2 (column 4). However, it should be noted that this invariance only applies if the number of students not eligible for free school meals is held constant (and this proviso is seldom acknowledged in verbal descriptions of the index properties). This is what we term here weak composition invariance.
If, instead, the number of students in poverty rises as a proportion of an existing school population but in such a way that the relative distribution of students in poverty remains unchanged between schools, then D varies. In Table 4, D increases to 0.4, which suggests that segregation has increased even though the proportion of the total students eligible for free school meals is the same for each school as it was in Tables 2 and 3. Put simply, a doubling of the figures for column 5 leads to an increase in D, yet it is far from clear that the schools in Table 4 are any more segregated (i.e. with FSM more unevenly distributed between schools) than those above. What D is picking up here is simply an increase in poverty across all schools.
These three hypothetical examples illustrate one potential misinterpretation of figures of segregation whether in school intakes, as represented here, or in ethnicity of cities or the gendered division of labour, in situations with differing composition. To be strongly composition invariant an index must be unaffected by changes in the relative frequency of the groups being measured. As an example, an occupation containing 20% of the total workforce but only 10% of the women in the workforce cannot be said to be more or less segregated simply because the overall number of women in the workforce changes, but only if the 10% and 20% figures change. The point is similar in many respects to that made about achievement gaps in Gorard (1999). Simple scaling of the numerator should not lead to changes in either achievement gaps or measures of segregation. Yet this apparently simple rule leads to paradox whereby either the figures in Table 3 or the figures in Table 3 are seen as differently segregated to those of Table 2.
The key difference is in the base figure used to compare the distribution of any particular group. Hence, while D compares the proportion of two groups with each other by sub-area, S compares the proportion of one group with the total for that sub-area. This means that even if the proportion of students eligible for free school meals is altered, S remains unchanged as long as they are distributed to each of the schools in the same proportions as the original figures. This is illustrated in Figure 1 which shows the effects on both indices of artificially changing the overall proportion of students eligible for free school meals across the whole of one local education authority (Camden in 1994), while retaining the initial proportion of students eligible for free school meals in each school. As can be seen, S remains constant irrespective of changes to the absolute levels of students eligible for free school meals. However, the effects of such changes on D are clearly evident and curvilinear.
The relationship between the two indices can be expressed as:
The segregation index proposed here was devised in just such an empirical manner. The original form in which it was published betrays its derivation from a verbal definition of what segregation between sub-areas actually is (see Gorard and Fitz 1998). The original proposal also included another technique, described as the segregation ratio, which combined well with the index in measuring aspects of the process of segregation which the overall index is less sensitive to (for example identifying the sub-areas in which segregation is worst). The chief recommendation for the segregation index is that it is strongly composition invariant, making it particularly appropriate for a study of changes in FSM over time since while poverty has increased dramatically over the last ten years the school population has not. The segregation index is the only index we have encountered which is thus able to separate the overall relative growth of FSM from changes in the distribution of FSM between schools. It is suitably ironic that some commentators in educational research have turned this situation on its head and argued that our index is sensitive to changes in composition, while the decomposed index of isolation (Noden 2000) or even unscaled percentage point differences (Gibson and Asthana 2000) are composition invariant. That is how wars start!
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