Space and the Measurement of Income Segregation

JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007, pp. 255–272 SPACE AND THE MEASUREMENT OF INCOME SEGREGATION Casey J. Dawkins School of Public and International Affairs, Virginia Polytechnic Institute and State University, 301 Architecture Annex, Blacksburg, Virginia 24061. E-mail: dawkins@vt.edu ABSTRACT. This paper proposes a new spatial ordering index that that can be used to quantify the dependence of a given pattern of income segregation on the spatial arrangement of neighborhoods. Unlike other spatial measures of income segregation proposed in the literature, the spatial ordering index is less sensitive to the presence of outliers, satisfies the “principle of transfers,” and is flexible enough to quantify a variety of spatial patterns of segregation. The index can be interpreted in terms of the ratio of two covariances. Properties of the proposed measure are demonstrated using an example from the city of Baltimore, Maryland. 1. INTRODUCTION Recent urban and regional policy debates have emphasized the need to eliminate the spatial concentration of poverty within US metropolitan areas (Massey and Eggers, 1990; Abramson et al., 1995; Goetz, 2003). Despite growing evidence linking residential location to economic inequality (see Ihlanfeldt [1999] for a review), there is still a dearth of information on the severity of poverty concentration and the spatial pattern of household income distributions within US metropolitan areas. Part of this research gap lies in the lack of agreement over how to appropriately quantify spatial aspects of the household income distribution. Existing indices of residential income segregation, derived largely from measures of racial segregation and income inequality, emphasize the unevenness of the income distribution across neighborhoods but tend to ignore the role of spatial proximity as it affects the extent of unevenness. This paper proposes a new spatial ordering index that can be used to quantify the dependence of a given pattern of income segregation on the spatial arrangement of neighborhoods. The measure discussed in this paper extends the Dawkins (2004) standardized spatial Gini index to the case of income segregation. Unlike other measures of income segregation proposed in the literature, the spatial ordering index can be used to quantify the dependence of a given neighborhood income distribution on any specified spatial configuration. All that is required is the prior identification of a spatial ordering that reflects Received: January 2005; revised: January 2006; accepted: July 2006.  C Blackwell Publishing, Inc. 2007. Blackwell Publishing, Inc., 350 Main Street, Malden, MA 02148, USA and 9600 Garsington Road, Oxford, OX4 2DQ, UK. 255 256 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 the spatial relationships among neighborhoods within the region. Unlike other spatial measures of income segregation proposed in the literature, the spatial ordering index is less sensitive to the presence of outliers, satisfies the “principle of transfers,” and is flexible enough to quantify a variety of spatial patterns of segregation. The index can also be interpreted in terms of the ratio of two covariances. The paper proceeds as follows: Section 2 discusses existing measures of neighborhood income segregation, focusing on the sensitivity of existing metrics to the checkerboard phenomenon. Section 3 proposes the concept of a spatial ordering as a way to describe the spatial relationships inherent in a given pattern of neighborhood income segregation. Section 4 proposes a spatial ordering index and discusses its properties, and Section 5 calculates various spatial ordering indices for Baltimore, Maryland. Section 6 offers concluding comments and suggestions for further research. 2. INEQUALITY, SEGREGATION, AND THE CHECKERBOARD PROBLEM When describing spatial dimensions of the income distribution, scholars have generally distinguished between income inequality (variability in individual or household income around a point of central tendency) and neighborhood income segregation (variability in neighborhood per capita income relative to a given pattern of income inequality). The relationship between these two concepts can be illustrated through a simple decomposition of an income inequality metric (I 0 ) into a within-neighborhood component (I W ) which captures the weighted sum of income inequality within each neighborhood and a betweenneighborhood component (I B ) which captures the variability in neighborhood per capita incomes relative to the regionwide mean. While income inequality is concerned with the magnitude of I 0 , income segregation refers to the proportion of total variability in income which can be explained by between-neighborhood variability. In the extreme case where households are perfectly sorted by income into different neighborhoods, I W will be equal to zero and income inequality can be fully explained by variability between neighborhoods. Several scholars have proposed measures of income segregation which can be expressed in terms of the ratio of I B to I 0 . Jargowsky’s (1996) neighborhood sorting index (NSI) is equal to the square root of the ratio of betweenneighborhood income variance to total income variance. Jahn, Schmid, and Schrag (1947) proposes a similar measure based on the Gini index (Gini, 1912) that has been utilized extensively to measure racial segregation and more recently by Kim and Jargowsky (2005) to measure economic segregation. Other inequality metrics such as Theil’s (1967) entropy index which are decomposable into between- and within-group components can also be used to measure the magnitude of I B relative to I 0 . Although measures constructed as (I B /I 0 ) are often described as “spatial” metrics, such indices do not generally consider the spatial arrangement of  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 257 FIGURE 1: Cities A and B. neighborhoods. To illustrate the nature of this omission, consider the following two hypothetical cities in Figure 1, each with neighborhoods indexed a through p. The color of each square neighborhood represents the per capita income for that neighborhood, with darker squares indicating that residents earn higher per capita incomes. Assume, for simplicity, that each neighborhood has the same number of residents. Also assume that income inequality is the same for City A and B. By construction, both of the cities in Figure 1 would generate identical values of the most common measures of neighborhood income segregation, because the overall distribution of neighborhood per capita incomes is the same for both cities. The spatial arrangement of neighborhood per capita incomes clearly differs between the two cities, however. City A exhibits a pattern of income segregation that we would likely describe as “clustered” with respect to neighborhood a, while City B exhibits a pattern of income segregation that we would more likely describe as spatially random. In the segregation literature, the failure to distinguish between these two spatial patterns using common segregation metrics is known as the “checkerboard problem” (White, 1983; Morrill, 1991; Dawkins, 2004). Those seeking to quantify the extent of the checkerboard phenomenon generally have two options. First, measures of spatial autocorrelation such as Moran’s I (Moran, 1950) and Geary’s C (Geary, 1954) can be used to compare variability in per capita incomes among nearby neighborhoods with total variability in per capita incomes regionwide. Another approach is to rely on a spatially weighted measure of segregation such as those proposed by Chakravorty (1996), Morgan (1983), Morrill (1991), and Wong (1993). Both types of measures are generally constructed as the ratio of a spatially weighted measure of variability in neighborhood per capita income to total between-neighborhood variability in neighborhood per capita income. While such measures can be used to examine the contribution of spatial proximity towards a given pattern of neighborhood income segregation, relationships with the underlying household income distribution are often ignored. With a few exceptions, all such measures rely on a spatial weights matrix to quantify spatial relationships. Spatial weights matrices are usually constructed so that each row, column element is equal to a measure of distance between a  C Blackwell Publishing, Inc. 2007. 258 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 given pair of neighborhoods. Contiguity matrices assign a value of 1 if a given pair of neighborhoods is adjacent and zero otherwise, while inverse distance matrices assign a value equal to 1/d ij where d ij is the distance between neighborhoods i and j. Reardon and Firebaugh (2002) discuss a generalized social distance metric which incorporates both between-neighborhood segregation and spatial proximity among neighborhoods. The authors limit their discussion to the case of segregation among discrete unordered groups, however. Since income is a continuous interval-level variable, their approach may not be appropriate for measuring economic segregation. In developing an agenda for the field of segregation measurement, the authors suggest that the “field needs segregation indices—both aspatial and social distance-based—that measure segregation among ordered groups, such as groups defined by income categories or educational attainment, for example” (Reardon and Firebaugh, 2002, p. 99). The spatial weights matrix approach is generally thought to suffer from two limitations. First, since such matrices are of dimension J×J, where J is the number of neighborhoods, it is often computationally burdensome to calculate spatial weights matrices for regions with a large number of neighborhoods. As Massey and Denton (1988) observe, the New York SMSA would generate a contiguity matrix of some 6.5 million cells if neighborhoods are based on census tract boundaries. If neighborhoods are defined using smaller geographic units such as census blocks, this problem is further exacerbated. Although large matrix manipulations are now feasible with recent advances in computing power, there are still many advantages to relying on more efficient algorithms to represent spatial relationships, particularly if one wishes to quantify spatial relationships among small geographic units such as census blocks or perform calculations for a large number of metropolitan areas. A second problem is that the spatial weights matrix approach is limited to the representation of pair-wise relationships between neighborhoods. Other spatial patterns such as distance from a single location within the region are generally not amenable to a spatial weights matrix representation. In the next section, I discuss an approach that allows one to define spatial relationship using a single vector of “spatial orderings.” The approach can also be shown to be consistent with a spatial weighting approach under certain conditions. Apart from common reliance on a spatial weight matrix, existing metrics differ substantially in the manner in which spatial segregation is quantified. Measures, such as Moran’s I, which are designed to measure spatial autocorrelation and not necessarily spatial segregation, rely on the variance as a basis for measuring between-neighborhood segregation (I B ). It is well known that the variance suffers from two important limitations as a measure of variability. First, the variance is sensitive to mean income for the region. As a result, one neighborhood per capita income distribution may exhibit higher relative variability than another but still have a smaller variance if the mean of the distribution is smaller (Sen, 1997). A second property of the variance is its  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 259 sensitivity to outliers and extreme values. Given that the distribution of per capita incomes is more likely than not to come from skewed non-normal distributions, this property may render variance-based measures of spatial segregation undesirable. An alternative approach is to apply a spatial weights matrix to a traditional segregation index such as Duncan and Duncan’s (1955) dissimilarity index. The indices proposed by Chakravorty (1996), Morrill (1991), and Wong (1993) rely on some version of this approach. Morgan (1983b) modifies the exposure index using a distance-based measure of spatial proximity. To date, with the exception of Chakravorty (1996), spatially weighted segregation indices such as the ones proposed by these authors have only been applied toward the measurement of discrete variables such as racial composition. It is generally not known how such indices behave when measuring spatial variability in a continuous intervallevel variable such as income. As a basis for spatial segregation measurement, the dissimilarity index may not necessarily be the most desirable choice. It is well known that the dissimilarity index does not always satisfy the Pigou (1912) – Dalton (1920) “principles of transfers.” In the case of neighborhood income segregation, this principle implies that a transfer of income from a neighborhood with a higher per capita income to one with a lower per capita income should always result in a decline in the chosen measure of neighborhood income segregation (Sen, 1997). The dissimilarity index satisfies the principle of transfers only when income is transferred from neighborhoods with per capita incomes that are greater (or less than) the average for the entire metropolitan area to one that is less than (or greater than) the metrowide average (James and Tauber, 1985). If the variability in neighborhood per capita income is a function of the distance between neighborhoods, this transfer bias takes on a spatial dimension, because neighborhoods at either extreme of the neighborhood per capita income distribution are more likely to be spatially clustered. This implies that the dissimilarity index will tend to underestimate the impact of transfers among adjacent neighborhoods relative to more distant neighborhoods (Dawkins, 2004). Unlike the dissimilarity index, the Gini (1912) index does not violate the principle of transfers. While the use of the Gini index in segregation measurement dates at least to Jahn, Schmid, and Schrag (1947), it has more commonly been applied toward the measurement of racial segregation. Kim and Jargowsky (2005) propose a measure of income segregation in the spirit of Jahn, Schmid, and Schrag (1947) that is constructed as the ratio of a betweenneighborhood Gini index to a Gini index of overall household income inequality. Their examination is limited to the case of between-neighborhood income segregation, however, and does not consider the role of spatial proximity among neighborhoods. As a basis for the measurement of spatial segregation, the Gini index has two desirable properties. First, Gini-based measures provide an intuitive link between income inequality (I 0 above), neighborhood income segregation (I B  C Blackwell Publishing, Inc. 2007. 260 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 above), and patterns of spatial income clustering among adjacent neighborhoods. Second, the between-neighborhood Gini index has desirable statistical properties which make it less sensitive to deviations from normality, compared to measures based on the variance (Yitzhaki, 2003). The next two sections examine a new measure of income segregation that is based on a spatial version of the Gini index. 3. SPATIAL ORDERING To quantify the similarity between a given neighborhood income distribution and a predefined spatial arrangement of neighborhoods, I propose the concept of a “spatial ordering.” Spatial orderings have been suggested by Kelejian and Robinson (1992) as a way to quantify spatial autocorrelation, but the literature offers no formal treatment of such orderings in the context of income segregation. Given an initial “parade” (Penn, 1971) of neighborhood per capita incomes (y j ) ranked such that y 1 ≤ y 2 ≤ . . . ≤ y j ≤ . . . y J , a spatial ordering constitutes a reranking of the original parade in a way that reflects the j th neighborhood’s spatial position. (For the remainder of the paper, “neighborhood per capita income” is assumed to refer to “average neighborhood income per household.”) The spatial ordering index discussed in the next section is based on a comparison of the original neighborhood income parade with the spatially ordered income parade. Although many such spatial ranking schemes are possible, two are emphasized in this paper. Nearest Neighbor Spatial Ordering: Holding neighborhood per capita incomes constant, assign each neighborhood the income parade ranking of the most spatially proximate nearby neighborhood. Monocentric Spatial Ordering: Define a new variable, X, which is equal to the distance from a given point within the region to the centroid of each neighborhood. Holding neighborhood per capita incomes constant, rank the neighborhood income parade by ascending or descending values of X. While many other spatial orderings are also possible, these two are likely to be most useful to researchers, given the interest in identifying patterns of clustering of the neighborhood income distribution (Hardman and Ioannides, 2004) and given the importance of distance from the central business district in shaping regional patterns of income segregation (Muth, 1969). These cases also illustrate two different approaches to ordering two-dimensional observations along a one-dimensional array. The nearest neighbor spatial ordering defined above can be implemented in one of two ways. One approach is to hold each neighborhood’s rank within the overall neighborhood income distribution constant while switching incomes among spatially adjacent neighborhoods. An equivalent approach is to  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 261 hold neighborhood per capita incomes constant while reranking per capita incomes according to the rank of each neighborhood’s nearest neighbor. For example, assume the original neighborhood per capita income parade can be written as [$10,000(1,4); $15,000(2,6); $25,000(3,5); $30,000(4,1); $40,000(5,3); $45,000(6,2)]. The numbers in parentheses refer respectively to the neighborhood’s original per capita income rank and the income rank of the neighborhood’s nearest neighbor, assuming a nearest neighbor pattern such that neighborhoods ranked 1 and 4 are nearest neighbors, the neighborhoods ranked 2 and 6 are nearest neighbors, and the neighborhoods ranked 3 and 5 are nearest neighbors. With this pattern of nearest neighbors, the spatially ordered income parade then becomes [$30,000(4,1), $45,000(6,2), $40,000(5,3), $10,000(1,4), $25,000(3,5), $15,000(2,6)]. Notice that the reordered parade can be implemented either by switching incomes or by switching ranks. The monocentric ordering is different in that its ranking is defined by a separate variable that is equal to the distance from a given point within the region. Thus, the only way to implement this spatial ordering is to assign a new rank to each element of the income parade that reflects the neighborhood’s distance from a given point. It is possible, however, to define this ranking in terms of ascending or descending values of the distance variable X. As demonstrated in the following sections, while different spatial orderings may have different interpretations, the spatial ordering approach is very general and can be used to compare an observed ordering to other, sometimes artificial cases, such as random permutations. 4. THE SPATIAL ORDERING INDEX (S r ) This section proposes an index constructed from the spatial orderings described above and explores several properties of the spatial ordering index. The calculation of the index using matrix algebra is examined in more detail by Dawkins (2004) in the context of racial segregation. This section extends the matrix algebra interpretation of the index to the measurement of economic segregation and explores alternative covariance-based approaches to calculating the index. Expressing the spatial ordering index as a ratio of covariances provides a straightforward interpretation of the index and provides an alternative computational formula that is easily implemented by any standard linear regression software package. To measure the importance of spatial ordering of the neighborhood income distribution (distribution of neighborhood per capita incomes), I propose the following modification of the standardized spatial Gini index proposed by Dawkins (2004): (1) Sr = Gr /G B where S r is a spatial ordering index calculated from the rth spatial ordering, G B is a Gini index of between-neighborhood income segregation, and G r is a spatial Gini index calculated from either a nearest neighbor or a monocentric spatial  C Blackwell Publishing, Inc. 2007. 262 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 ordering of neighborhood per capita income. S r is a measure of the degree to which a given spatial configuration of neighborhoods results in a reordering of the neighborhood income distribution used to construct G B . As such, it is a direct measure of the importance of the checkerboard phenomena toward an overall pattern of income segregation. S r has several desirable properties. First, as discussed below, the index is a measure of the correlation between neighborhood aggregate or per capita incomes and the spatial rank of neighborhood per capita incomes. As neighborhoods take on spatial rankings that correspond closely to their ranking in the original income parade, S r approaches one. Similarly, as neighborhoods take on rankings that differ from their rank in the original income parade, S r approaches negative one. Finally, in the case where neighborhoods’ spatial ranking is unrelated to their ranking in the original income parade, S r approaches zero. Since any reranking of the elements used to generate G r results in an index that is bounded below by −G B and above by G B , S r is, in general, unit free and ranges between −1 and 1 (see Dawkins [2004] for further properties of this index in the context of racial segregation). The literature suggests at least two possible approaches for calculating S r , both offering unique insights into the interpretation of the index. We begin with an examination of the covariance method and later compare this with a matrix algebra approach. Assume that a given region is divided into J neighborhoods. Also assume that H households reside within the region. G r and G B can then be calculated using the following covariance-based formulas:    yj 2 Jj=1 R̄ j(n) − H+1 2 (2) Gr = HY 2 (3) GB = J j=1  R̄ j − HY H+1 2  yj where: y j = Aggregate household income earned by residents of the j th neighborhood Y = Aggregate household income earned by the residents of the region R̄ j = Average rank of per capita income earned by neighborhood j within the overall neighborhood per capita income distribution R̄ j(n) = Average spatial rank of per capita income earned by neighborhood j within the overall neighborhood per capita income distribution Calculation of R̄ j and R̄ j(n) requires somewhat more effort when neighborhoods have populations that are unequal. To calculate R̄ j , first rank neighborhoods in order of per capita income so that the neighborhood earning the lowest per capita income is ranked 1 and the neighborhood earning the  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 263 highest per capita income is ranked J, or y 1 ≤ y 2 ≤ . . . ≤ y j ≤ . . . y J . Then, calculate the cumulative number of households within each successive position within the neighborhood income parade and define this cumulative total as N j . The cumulative total of households’ one rank lower within the income parade can similarly be defined as N j−1 . The j th neighborhood’s average rank is then equal to: (N j + N j−1 + 1)/2. To calculate R̄ j(n) for a nearest neighbor spatial ordering, pair each neighborhood with its most spatially proximate neighbor ( j (n)), and assign the j th neighborhood a rank equal to the rank of j (n). The cumulative number of households within each successive position within the spatially ordered income parade can now be defined as N j (n) , and the cumulative total of households one rank lower within the spatially ordered income parade can similarly be defined as N j−1(n) . The j th neighborhood’s average rank is then equal to: (N j (n) + N j−1(n) + 1)/2. A similar procedure can be utilized to implement a monocentric ranking if one ranks neighborhoods, not by the rank of their most spatially proximate neighbor, but by distance from a given point within the region. Observing that (H + 1)/2 in Equations 2 and 3 is equal to the mean per J H+1 j=1 ( R̄ j − 2 )y j is equal capita income rank for the entire region establishes that H to Cov(y j , R̄ j ), or the covariance between neighborhood aggregate income and the average rank of neighborhood j. This result allows one to express G B as (2/Y)∗ Cov(y j , R̄ j ) and G r as (2/Y)∗ Cov(y j , R̄ j(n) ). Stuart (1954) was the first to establish the relationship between the Gini index of inequality and the covariance, and others (see, e.g., Yitzhaki and Lerman 1991) have extended this toward the case of between-group inequality, or segregation. This section illustrates that a similar relationship holds for G r . Dividing G r by G B yields the following computational formula for S r :  J  H+1 yj j=1 R̄ j(n) − 2   Sr =  (4) J H+1 yj j=1 R̄ j − 2 Since (2/HY) cancels out as a result of the division, S r , unlike G r and G B , is not influenced by aggregate income or the number of households within the region. Written in this manner, it is clear that S r is a measure of association that is similar to the Gini correlation measure (G xy ) proposed by Schechtman and Yitzhaki (1987). Their measure is defined as follows: Cov[F(x), y] (5) Gxy = Cov[F(y), y] where y = household income, F(y) = cumulative distribution of y, x is a second variable, and F(x) = cumulative distribution of the second variable x. Schechtman and Yitzhaki (1987) present the Gini correlation as a measure of association that is a combination of the nonparametric Spearman’s coefficient and the parametric Pearson product moment correlation coefficient. As such, it can be interpreted as a normalized measure of association between a given variable and the cumulative distribution of a second variable. The spatial ordering  C Blackwell Publishing, Inc. 2007. 264 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 index, S r , is equivalent to G xy if we consider F(x) to be the spatial reranking of y. One qualification to this interpretation is that nearest neighbor spatial orderings will produce a correlation-based index with slightly different properties than one created from a monocentric ordering. With a monocentric ordering, x in Equation 5 is simply the rank of distance from a given point within the region. Thus, like the Gini correlation index G xy , it is possible to define two S r indices under a monocentric ordering, one which measures the correlation between income and the rank of distance with respect to a given spatial location, and another which measures the correlation between distance from a given location and rank of income. In general these two indices will not be identical. The properties of asymmetric Gini correlation measures such as this are discussed in further detail by Schechtman and Yitzhaki (1987). In the case of nearest neighbor spatial ordering, however, the correlation between income and neighbor’s rankings is identical to the correlation between neighbor’s income and the original income ranking, which implies that G xy = G yx . Further research is needed to evaluate the implications of this result. The covariance interpretation of G r and G B allows one to calculate each using the slope coefficient (␤ 1 ) from a bivariate regression with the following form: (6) yij = ␤0 + ␤1 Rij + eij where y ij is equal to household income earned by the ith household living in neighborhood j, assuming that each household within j is assigned the per capita income of their neighborhood of residence, and R ij is equal to the rank of income earned by household i, j. Applying a result from Shalit (1985) toward the measurement of between-neighborhood income segregation, G B can be calculated as follows:    2 H −1 ␤1 (7) GB = ȳ 6H where ȳ is equal to average household income for the entire region, and H is the total number of households. In other words, G B is equal to ␤ 1 times a constant. A similar approach can be used to calculate G r if one replaces R ij with R ij(n) , where R ij (n) is the spatial ranking of household i, j. This implies that S r can be interpreted as a ratio of two slope coefficients. This representation of the G B and G r is useful, because it allows for an intuitive interpretation of the G B as a measure of the average increase in per capita income associated with moving up one rank in the neighborhood income parade (Lerman and Yitzhaki, 1984). The interpretation of G r depends on the approach taken to implement the spatial ranking. In the case of a nearest neighbor spatial ordering, G r can be interpreted as a measure of the average increase in neighbor’s per capita income associated with moving up one rank in the original neighborhood income parade. In the case of monocentric ordering, G r can be interpreted as a measure of the average increase in per capita income  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 265 FIGURE 2: Spatial Ordering Example associated with moving one neighborhood away (or one neighborhood closer) to a given point within the region. An example is displayed in Figure 2. For simplicity, the example assumes that each neighborhood has a population equal to one. The dark gray bars in Figure 2 refer to the original income parade (y j ) for the example displayed in section 3: [$10,000(1,4); $15,000(2,6); $25,000(3,5); $30,000(4,1); $40,000(5,3); $45,000(6,2)]. Assuming the same pattern of nearest neighbors, a nearest neighbor spatial ordering can be represented with the lighter gray bars (y j(n) ). The slope of the trend line for each distribution is equal to ␤ 1 in Equation 6. Notice that the slope of y j(n) with respect to the original income parade ranking is negative. This implies that for a given position within the original neighborhood per capita income parade, nearby neighborhoods earn per capita incomes that are dissimilar. Another approach for calculating S r is the matrix algebra formulation proposed by Dawkins (2004): (8) Sr = [hJ(n) /H, hJ−1(n) /H, . . . , h j(n) /H]G [yJ(n) /Y, y J−1(n) /Y, . . . , y j(n) /Y]′ / [hJ/H, hJ−1 /H, . . . , h j /H]G [yJ/Y, y J−1 /Y, . . . , y j /Y]′ where: G = Silber’s (1989) G-matrix, a JxJ matrix with elements equal to zero along the diagonal, −1 above the diagonal and 1 below the diagonal. And all other terms are defined as above. The vectors [h J /H, h J−1 /H, . . ., h j /H] and [y J /Y, y J−1 /Y, . . ., y j /Y] in Equation 8 are each ranked in descending order  C Blackwell Publishing, Inc. 2007. 266 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 (in contrast to the ascending order ranking above) by the ratio [(y j /Y)/(h j /H)], while the vectors [h J(n) /H, h J−1(n) /H, . . ., h j(n) /H] and [y J(n) /Y, y J−1(n) /Y, . . ., y j(n) /Y] are ranked in descending order by the ratio [(y j(n) /Y)/(h j(n) /H)]. An advantage of this formula is its computational simplicity. All that is required to calculate G r is a sorting vector which gives the ranking of either the nearest neighborhoods’ positions in the income parade or distances from a given point within the region. The matrix algebra interpretation of S r can be used to illustrate the relationship between S r and Moran’s I statistic (Moran 1950), a common measure of spatial autocorrelation. Moran’s I statistic for a given variable, X, is calculated as follows (Anselin, 1988): I = (N/wi j )(e′ We/e′ e) (9) where, x i = value of X at location i, ␮ = mean of X, “e is a vector of residuals with elements equal to (x i − ␮),” N = total sample size, and W = spatial weight matrix with elements w ij . The matrix W is typically defined using a contiguity matrix, where the elements of W are equal to 1 if neighborhood i and j are nearest neighbors and 0 otherwise. Values of I become larger as spatial autocorrelation increases, and the spatially weighted residuals (e′ We) become large relative to the total variability in X as captured by the term e’e (Dawkins, 2004). The spatial ordering index can be given a similar interpretation as a ratio of a spatially weighted covariance to an overall measure of variability. To see this, consider the following representation of the nearest neighbor S r suggested by Dawkins (2004): (10) Sr = [hJ/H, hJ−1 /H, . . . , h j /H]PGP′ [y J/Y, y J−1 /Y, . . . , y j /Y]′ / [hJ/H, hJ−1 /H, . . . , h j /H]G[yJ/Y, y J−1 /Y, . . . , y j /Y]′ The matrix P is a JxJ permutation matrix which serves to reorder the elements of the vectors [h J /H, h J−1 /H, . . ., h j /H] and [y J /Y, y J−1 /Y, . . ., y j /Y] according to a nearest neighbor spatial ordering. In this case, P serves a function that is similar to W in Equation 9. The primary difference between W and P is that P is a biostochastic permutation matrix where each row and each column has J−1 zeros and one “1”. In the restrictive case where all neighborhoods have only one neighbor and neighbor pairs are symmetric such that j is k’s neighbor and k is j’s neighbor for all j and k, P is equal to W. While this may seem to imply that W is more general than P, W can be used to quantify pair-wise relationships only. As discussed above, P can incorporate pair-wise orderings as well as other types of spatial orderings, provided such orderings can be represented by a single vector of ranks. In cases where P and W are equal, Equations 9 and 10 differ only in one important way: while Moran’s I is a measure of the correlation between per capita income and the per capita income of nearby neighborhoods, the nearest neighbor S r is a measure of the correlation between per capita income and the rank of per capita incomes earned by nearby neighborhoods.  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 267 Apart from slight differences in interpretation, this property has two important implications. First, it provides an interpretation of the nearest neighbor G r that is consistent with G B and G 0 , where G 0 is the Gini index of income inequality across households within the region. As shown above, G B can be interpreted as a weighted measure of the covariance between neighborhood aggregate (or per capita) incomes and the average rank of neighborhood per capita incomes. Furthermore, Stuart (1954) and Lerman and Yitzhaki (1984) have shown that G 0 can be interpreted as a weighted measure of the covariance between household incomes and the rank of household incomes. Second, this result suggests that in addition to its usefulness in measuring spatial patterns of segregation, a nearest neighbor S r may also serve as a useful measure of spatial autocorrelation in circumstances when the variable in question is drawn from a non-normal distribution. As demonstrated by Yitzhaki (2003), Gini-based measures place smaller weights on differences among adjacent observations located farther from the median than do measures based on the variance. The implication of this result is that Gini-based measures are less sensitive to the presence of outliers and non-normal distributions than are measures based on the variance. 5. CASE STUDY: THE SPATIAL PATTERN OF INCOME SEGREGATION IN BALTIMORE, MARYLAND This section examines the indices discussed above for the city of Baltimore, Maryland. Neighborhood per capita income is defined as total household earnings divided by the total number of households per census tract. Values are based on those reported in the 2000 US Census of Population and Housing. Baltimore was chosen due to the relative uniformity in census tract sizes within the region. The strong observed pattern of monocentric spatial ordering with respect to the central business district also provides an opportunity to determine if the index can be used to accurately identify a pattern of income segregation that likely characterizes many urban areas within the U.S. Figure 3 displays the observed pattern of earnings per household by census tract for Baltimore. Notice that as one moves away from the central business district (the census tract adjacent to the tip of the northernmost inner harbor), per capita incomes generally increase. Furthermore, the overall pattern of neighborhood income segregation is one characterized by a high degree of clustering among adjacent census tracts. The extent of clustering becomes even more apparent when we compare Figure 3 with a spatial pattern created by reshuffling and redistributing census per capita incomes to other randomly placed census tracts around the city. This “reshuffled” pattern is shown in Figure 4 and was created by randomly permuting the elements of a census tract index and reassigning per capita incomes based on the randomly permuted index. Comparing Figure 3 and Figure 4, it is clear that Figure 3 exhibits a relatively higher level of clustering and declining per capita incomes with  C Blackwell Publishing, Inc. 2007. 268 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 FIGURE 3: Earnings Per Household, Baltimore, MD. respect to the central business district, compared to the randomly permuted pattern. Several metrics of income segregation are shown in Table 1: a betweenneighborhood segregation index (G B ), a nearest-neighbor spatial Gini index  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 269 FIGURE 4: Spatially Permuted Earnings Per Household, Baltimore, MD. (G N ), a monocentric spatial Gini index (G M ), a spatial ordering index of nearest neighbor ordering (S N ), and a spatial ordering index of monocentric ordering (S M ). Recall that S N is constructed as (G N /G B ), while S M is constructed as (G M /G B ). I also calculate a spatial Gini index and spatial ordering index of  C Blackwell Publishing, Inc. 2007. 270 JOURNAL OF REGIONAL SCIENCE, VOL. 47, NO. 2, 2007 TABLE 1: Measures of Spatial Income Segregation: Baltimore, MD Index Value Between-neighborhood segregation index (G B ) Nearest neighbor spatial Gini index (G N ) Spatial ordering index, nearest neighbor ordering (S N ) Monocentric spatial Gini index (G M ) Spatial ordering index, monocentric ordering (S M ) Permuted nearest neighbor spatial Gini index (GNP) Spatial ordering index, permuted nearest neighbor ordering (SNP) 0.169 0.092 0.545 −0.034 −0.201 0.018 0.106 nearest neighbor ordering for the spatially permuted pattern shown in Figure 4, to illustrate how the indices perform under a pattern characterized by less pronounced nearest neighbor clustering of per capita incomes. The spatial Gini index of nearest neighbor ordering for the permuted spatial pattern is defined as G NP , while the spatial ordering index for the spatially permuted pattern is defined as S NP . Each of these indices is calculated using the matrix algebra approach outlined in Equation 8 above. The S-Plus programs used to calculate these indices are available from the author upon request. The index values largely correspond with the visual displays in Figures 3 and 4. The relatively high value of S N (0.545) points to high nearest neighbor clustering in per capita incomes. The magnitude of S N becomes more apparent when one compares this to the value of S NP created from the randomly permuted spatial pattern. In addition to providing a basis for visual comparison in the current case, random permutations of the original vector of per capita incomes suggest a possible way to test hypotheses regarding the value of S N . If one calculated S NP from a larger number of such random permutations, the rank of the observed value of S N in relation to the empirical distribution of S NP can be used to establish p-values for hypothesized values of S N . Such permutation tests have been proposed as a way to test hypotheses regarding the value of Moran’s I and other spatial autocorrelation statistics. Given that the elements of [h J /H, h J−1 /H, . . ., h j /H] and [y J /Y, y J−1 /Y, . . ., y j /Y] are ranked in descending order by the value of [(y j /Y)/(h j /H)] when calculating G B and ranked in ascending order of distance from the region’s central business district when calculating G M , the negative value of S M implies that census tract per capita incomes are an increasing function of the rank of distance from the region’s central business district. Given recent evidence which points to high spatial concentrations of metropolitan wealth within suburban areas (Swanstrom et al., 2004), this pattern is likely ubiquitous across the US urban landscape. 6. CONCLUSION This paper proposed a spatial ordering index as a way to quantify the spatial relationships inherent in a given pattern of income segregation. Given the  C Blackwell Publishing, Inc. 2007. DAWKINS: INCOME SEGREGATION 271 index’s relationship to the Gini index of income inequality, one can utilize the spatial ordering index to examine the importance of spatial proximity as it affects both overall income inequality and between-neighborhood income inequality. The relationship between the spatial ordering index and the covariance provides an intuitive interpretation of the spatial ordering index as a correlation measure which quantifies the association between the neighborhood per capita income distribution and a spatial reordering of the neighborhood income distribution. 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