## Segregation Indices

## Segregation Indices

# SEGREGATION INDICES

Residential segregation has been a prominent topic in sociology since Burgess (1928) published his landmark study more than seventy years ago, and for almost as long, sociologists have argued about how to measure it. The debate has ebbed and flowed, and for a time the issue seemed to have been settled. In 1955, Duncan and Duncan published a landmark article (Duncan and Duncan 1955) demonstrating that there was little information in any of the prevailing indices that had not been captured already by the index of dissimilarity. For twenty years afterward, that measure was employed as the standard index of residential segregation.

This *Pax Duncanae* came to an abrupt end in 1976 with the publication of a critique of the dissimilarity index by Cortese and colleagues, ushering in a period of debate that has not ended (Cortese et al. 1976). Over the ensuing decade, a variety of old indices were reintroduced and new ones were invented, yielding a multiplicity of candidates. In an effort to bring some order to the field, Massey and Denton (1988) undertook a systematic analysis of twenty segregation indices they had identified from a review of the literature. They argued that segregation is not a unidimensional construct but encompasses five distinct dimensions of spatial variation. No single dimension is intrinsically more "correct" than any other; each reflects a different facet of the spatial distribution of social groups.

The five dimensions they identified are evenness, exposure, clustering, concentration, and centralization. To verify that conceptualization, Massey and Denton (1988) carried out a factor analysis of indices computed from 1980 census data for U.S. metropolitan areas. Their results showed that each index correlated with one of five factors corresponding to the dimensions they postulated. On theoretical, empirical, and practical grounds, they selected a single "best" indicator for each dimension of segregation. The dimensional structure of segregation and Massey and Denton's (1988) selection of indices have been reaffirmed using 1990 census data (Massey et al. 1996).

The first dimension of segregation is evenness, which refers to the unequal distribution of social groups across areal units of an urban area. A minority group is segregated if it is unevenly spread across neighborhoods. Evenness is not measured in an absolute sense but is scaled relative to another group. It is maximized when all areal units have the same relative number of minority and majority members as the city as a whole and is minimized when minority and majority members have no areas in common.

The index of dissimilarity quantifies the degree of departure from an even residential distribution. It computes the number of minority group members who would have to change neighborhoods to achieve an even distribution and expresses that quantity as a proportion of the number that would have to change areas under conditions of maximum unevenness. The index varies between zero and one, and for any two groups X and Y it is computed as:

where *x*i and *y*i are the number of group X and group Y members in areal unit *i* and *X* and *Y* are the number of group X and group Y members in the city as a whole, which is subdivided into *n* areal units.

Among its properties, the index is inflated by random factors when the number of minority group members is small relative to the number of areal units (Cortese et al. 1976). It is also insensitive to the redistribution of minority group members among areal units with minority proportions above or below the city's minority proportion (James and Taeuber 1985; White 1986). Only transfers of minority members from areas where they are overrepresented (above the city's minority proportion) to areas where they are underrepresented (below the minority proportion) affect the value of the index.

The property means that the dissimilarity index fails the "transfers principle," which requires that segregation be lowered whenever minority members move to areas where they constitute a smaller proportion of the population. This and other problems led James and Taeuber (1985) to recommend using another measure of evenness, the Atkinson index (Atkinson 1970). Massey and Denton (1988), however, pointed out that the Atkinson index and dissimilarity indices are highly correlated and generally yield the same substantive conclusions. Moreover, the Atkinson index is actually a family of indices, each of which gives a slightly different result, creating problems of comparability. Given that *D* has been the standard index for more than thirty years, that its use has led to a large body of findings, and that that index is easy to compute and interpret, Massey and Denton (1988) recommended using it to measure evenness in most cases.

White (1986) points out, however, that another index may be preferred in measuring segregation between multiple groups, since the dissimilarity index is cumbersome to compute and interpret when the number of groups exceeds two. Thus, if one wants to generate an overall measure of segregation between ten ethnic groups, separate dissimilarity indices will have to be computed between all possible pairs of groups and averaged to get a single measure. An alternative index is Theil's (1972) entropy index, which yields a single comprehensive measure of ethnic segregation. The entropy index also can be expanded to measure segregation across two or more variables simultaneously (e.g. ethnicity and income) and can be decomposed into portions attributable to each of the variables and their interaction (see White 1986).

The second dimension of segregation is exposure, which refers to the degree of potential contact between groups within the neighborhoods of a city. Exposure indices measure the extent to which groups must physically confront one another because they share a residential area. For any city, the degree of minority exposure to the majority is defined as the likelihood of having a neighborhood in common. Rather than measuring segregation as a departure from an abstract ideal of "evenness," however, exposure indices get at the *experience* of segregation from the viewpoint of the average person.

Although indices of exposure and evenness are correlated empirically, they are conceptually distinct because the former depend on the relative size of the groups that are being compared, while the latter do not. Minority group members can be evenly distributed among the residential areas of a city but at the same time experience little exposure to majority group members if they constitute a relatively large share of the population of the city. Conversely, if they constitute a small proportion of the city's population, minority group members tend to experience high levels of exposure to the majority regardless of the level of evenness. Exposure indices take explicit account of such compositional effects in determining the degree of segregation between groups.

The importance of exposure was noted early by Bell (1954), who introduced several indices. However, with the establishment of the *Pax Duncanae* in 1955, sentiment coalesced around the dissimilarity index and exposure was largely forgotten until Lieberson reintroduced the *P** index in the early 1980s (Lieberson 1980, 1981). This index has two basic variants. The interaction index (x*P**y) measures the probability that members of group X share a neighborhood with members of group Y, and the isolation index (x*P**x) measures the probability that group X members share an area with each other.

The interaction index is computed as the minority-weighted average of each neighborhood's majority proportion:

where *x*i, *y*i, and *t*i are the numbers of group X members, group Y members, and the total population of unit *i*, respectively, and *X* represents the number of group X members citywide. The isolation index is computed as the minority-weighted average of each neighborhood's minority proportion:

Both indices vary between zero and one and give the probability that a randomly drawn group X member shares a neighborhood with a member of group Y (in the case of x*P**y) or with another group X member (in the case of x*P**x). Values of y*P**x and y*P**y can be computed analogously from equations (2) and (3) by switching the *x* and *y* subscripts. When there are only two groups, the isolation and interaction indices sum to one, so that x*P**y + x*P**x = 1.0 and y*P**x + y*P**y = 1.0. The interaction indices are also asymmetrical; only when group X and group Y constitute the same proportion of the population does x*P**y equal y*P**x.

*P** indices can be standardized to control for population composition and eliminate the asymmetry (Bell 1954; White 1986). Standardizing the isolation index yields the well-known correlation ratio, or eta2 (White 1986). Stearns and Logan (1986) argue that eta2 constitutes an independent dimension of segregation, but Massey and Denton (1988) hold that it straddles two dimensions. Since it is derived from *P**, eta2 displays some properties associated with an exposure measure, but standardization also gives it the qualities of an evenness index. Massey and Denton (1988) demonstrate this duality empirically and argue that it is better to use *D* and *P** as separate measures of evenness and exposure. Nonetheless, Jargowsky (1996) has shown that one version of eta2 yields a better and more concise measure of segregation when one wishes to measure segregation between multiple groups simultaneously (e.g., between income categories).

The third dimension of segregation is clustering, or the extent to which areas inhabited by minority group members adjoin one another in space. A high degree of clustering implies a residential structure in which minority areas are arranged contiguously, creating one large enclave, whereas a low level of clustering means that minority areas are widely scattered around the urban environment, like a checkerboard.

The index of clustering recommended by Massey and Denton (1988) is White's (1983) index of spatial proximity, SP. It is constructed by calculating the average distance between members of the same group and the average distance between members of different groups and then computing a weighted average of those quantities. The average distance, or proximity, between group X members is

and the average proximity between members of group X and group Y is

where *Y* is the number of group Y members citywide, *x*i and *y*j are the numbers of group X and group Y members in units *i* and *j*, and *c*ij is a distance function between these two areas, defined here as a negative exponential: *c*ij = exp (−*d*ij). The term *d*ij represents the linear distance between the centroids of units *i* and *j*, and *d*ij is estimated as (.6*a*i)·5, where *a*i is the area of the spatial unit. Use of the negative exponential implicitly assumes that the likelihood of interaction declines rapidly as the distance between people increases.

Average proximities also may be calculated among group Y members (*P*yy) and among all members of the population (*P*u) by analogy with equation (4). White's SP index (1983) represents the average of intragroup proximities, *P*xx,/*P*tt and *P*yy/*P*tt, weighted by the fraction of each group in the population:

SP equals one when there is no differential clustering between group X and group Y and is greater than one when group X members live nearer to each other than they do to group Y members. In practice, SP can be converted to a zero-to-one scale by taking the quantity SP−1 (Massey and Denton 1988). White (1984) also has proposed a more complex standardization by taking *f*(*d*ij)=*d*ij2, which yields a statistic equivalent to the proportion of spatial variance explained.

Jakubs (1981) and Morgan (1983a, 1983b) have proposed that *D* and *P** be adjusted to incorporate the effects of clustering. Massey and Denton (1988) argue against this procedure because it confounds two different dimensions of segregation. They maintain that it is better to measure clustering directly as a separate dimension than to try to adjust other measures to reflect it.

The fourth dimension of segregation is centralization, or the degree to which a group is located near the center of an urban area. In the postwar period, African-Americans became increasingly isolated in older central cities as whites gravitated to the suburbs. Centralization is measured by an index that reflects the degree to which a group is spatially distributed close to or far away from the central business district (CBD). It compares a group's distribution around the CBD to the distribution of land area around the CBD by using a formula adapted from Duncan (1957):

where the *n* areal units are ordered by increasing distance from the CBD and *X*i and *A*i are the respective cumulative proportions of group X members and land area in unit *i*.

In most circumstances, the centralization index varies between plus one and minus one, with positive values indicating a tendency for group X members to reside close to the city center and negative values indicating a tendency for them to live in outlying areas. A score of zero means that the group has a uniform distribution throughout the metropolitan area. The index states the proportion of group X members who would have to change their area of residence to achieve a uniform distribution around the CBD.

The last dimension of segregation is concentration, or the relative amount of physical space occupied by a minority group in the urban environment. Concentration is a relevant dimension of segregation because discrimination restricts minorities to a small set of neighborhoods that together account for a small share of the urban environment. The index of concentration takes the average amount of physical space occupied by group X relative to group Y and compares that quantity to the ratio that would obtain if group X were maximally concentrated and group Y were maximally dispersed:

where areal units are ordered by geographic size from smallest to largest, *a*i is the land area of unit *i*, and the two numbers *n*1 and *n*2 refer to different points in the rank ordering of areal units from smallest to largest: *n*1 is the rank of the unit where the cumulative total population of units equals the total minority population of the city, summing from the smallest unit up, and *n*2 is the rank of the areal unit where the cumulative total population of units equals the majority population, totaling from the largest unit down. *T*1 equals the total population of areal units from 1 to *n*1, and *T*2 equals the total population of areal units from *n*2 to *n*. As before, *t*i refers to the total population of unit *i* and *X* is the number of group X members in the city.

In most circumstances, the resulting index varies from minus one to plus one; a score of zero means that the two groups are equally concentrated in urban space, and a score of minus one means that group Y's concentration exceeds group X's to the maximum extent possible; a score of positive one means the converse. In certain circumstances, however, Egan et al. (1998) demonstrate that whenever the number of group X members is very small and the areas in which they live are very large, the index becomes unbounded in the negative direction. Thus, caution should be used in measuring the concentration of groups with very few members.

Which of these five indices of segregation is chosen for a particular application depends on the purpose of the study. All are valid measures, and arguments about which one is "correct" or "best" are meaningless, since they measure different facets of segregation. *D* provides an overall measure of evenness that is highly comparable with prior work, widely understood, readily interpretable, and independent of population composition. *P** captures the degree of inter- and intragroup contact likely to be experienced by members of different groups and directly incorporates the effect of population composition. Most recent work has relied most heavily on these two segregation measures (Massey and Denton 1993; Frey and Farley 1994, 1996; Massey and Hajnal 1995; Peach 1998).

Neither *D* nor *P** is inherently spatial, however, and each may be applied to study nongeographic forms of segregation, such as segregation between men and women across occupations (see Jacobs 1989). The remaining three dimensions are relevant whenever it is important to know about the physical location of a group in space. If the extent to which group members cluster is important, SP should be computed; if it is important to know how close to the city center a group has settled, CE may be calculated; and if the amount of physical space occupied by a group is relevant, CO is the appropriate index.

The most comprehensive understanding of residential segregation is achieved, however, when all five indices are examined simultaneously. That multidimensional approach yields a fuller picture of segregation than can be achieved by using any single index alone. Thus, Massey and Denton (1989) found that blacks in certain U.S. cities were highly segregated on all five dimensions simultaneously, a pattern they called "hypersegregation." Denton (1994) has shown that this pattern not only persisted to 1990 but extended to other metropolitan areas. By relying primarily on the index of dissimilarity, prior work overlooked this unique aspect of black urban life and understated the severity of black segregation in U.S. cities.

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Douglas S. Massey