A multi-level modeling approach to understanding residential segregation in the United States

Abstract

A well-known limitation of commonly used segregation measures is their inability to describe patterns at multiple scales. Multi-level modeling approaches can describe how different levels of geography contribute to segregation, but may be difficult to interpret for non-technical audiences and have rarely been applied in the US context. This paper provides a readily interpretable description of multi-scale Black–non-Black segregation in the United States using a multi-level modeling approach and the most recent Census data available. We fit a three-level random intercept multi-level logistic regression model predicting the proportion of the population that is Black (Hispanic and non-Hispanic) at the block group level, with block groups nested in tracts and tracts nested in Metropolitan Statistical Areas (MSAs). For the 102 largest MSAs in the United States, we then estimated the extent to which micro- versus meso-level variability drives overall racial residential patterning within the MSA. Finally, we created a typology of racial residential patterning within MSAs based on the total proportion of the MSA population that is Black and the relative contribution of block groups (micro) versus tracts (meso) in driving variation. We find that nearly 80% of the national variation in the geographic concentration of Black residents is driven by within-MSA, tract-level processes. However, the relative contribution of small versus larger scales to within-MSA segregation varies substantially across metropolitan areas. We detect five meaningfully different types of metropolitan segregation across the largest MSAs. Multi-level descriptions of segregation may help planners and policymakers understand how and why segregated residential patterns are evolving in different places and could provide important insights into interventions that could improve integration at multiple scales.

Keywords

Multi-level modeling segregation United States urban planning metropolitan regions cluster analysis

Understanding patterns of segregation in the United States has been of immense scholarly interest over the past 30 years, beginning with the introduction of easy to compute and interpret indices of dissimilarity, isolation, and exposure, among others. Using these classic measures, researchers have described how the extent of residential segregation has evolved over time, varies over space, and differs between different racial/ethnic group pairs (Iceland et al., 2014; Massey and Denton, 1987). In turn, describing the residential patterns of different racial/ethnic groups has enabled researchers to study the causes and consequences of segregation (Bayer et al., 2014; Cutler et al., 1999; Ellen et al., 2016; Reardon et al., 2008). Descriptive work also helps inform policy intended to increase diversity and opportunity in US neighborhoods (de Briggs, 2006; Smedley and Tegeler, 2016).

Despite these critical contributions, the most commonly used measures of segregation suffer well-known limitations, including an inability to describe the multi-scalar nature of how different populations are concentrated within geographic units (Jones et al., 2015; Kaplan and Holloway, 2001; Logan and Parman, 2017). Reardon et al. argue, for example, that segregation in some US metropolitan areas, including Atlanta and Chicago, primarily takes place at large geographic scales and is characterized by the presence of physically large, racially homogenous areas within regions. In contrast, other metropolitan areas, including Phoenix and Pittsburgh, are thought to be segregated predominantly through racial isolation at micro scales, with small racially isolated areas distributed more evenly throughout regions (Reardon et al., 2008).

Policymakers, planners, and other stakeholders often care about differences in these patterns at small geographic scales such as Census tracts and block groups. Moreover, conclusions about the impact of planning decisions on racial residential segregation may hinge on the scale at which the assessment takes place. As one example, Massachusetts General Laws Chapter 40B allows, under many circumstances, housing developers to receive comprehensive permits to construct affordable units in municipalities where less than 10% of the housing stock is affordable, enabling some development projects to bypass local zoning bylaws and other rules and regulations that could otherwise slow or halt construction (Regnante and Haverty, 2003). Public sentiment about the statute itself, and about specific projects permitted under the law, is thought to be heavily influenced by the prospect of both economic and racial/ethnic integration in largely White, affluent areas, despite the fact that the class and race of prospective 40B residents are rarely discussed openly in public debate (Girouard, 2016). A frequently contentious debate about Chapter 40B projects concerns the distribution of affordable units within municipalities, with residents of market-rate housing often advocating that small numbers of units be distributed across block groups, arguing, for example “… it’s better if the housing is dispersed – three units here, 10 units here, five over here – just not in a big cluster, where there’s a stigma attached. They get labeled as ‘the projects’” (Rattigan, 2016). Developers, on the other hand, may propose larger housing developments, claiming that municipalities prefer affordable housing be concentrated in one location (Buote, 2016), likely a single block group. In practice, singular large developments are sometimes described as politically convenient because they can be pushed to the edge of municipalities’ borders where neighboring towns will share traffic, environmental, and social impacts (Girouard, 2016).

As a result, the question of how Chapter 40B changes racial/ethnic or economic segregation cannot be sufficiently answered without considering scale. The construction of affordable housing in affluent, predominantly White suburbs likely decreases racial/ethnic and economic segregation at the metropolitan area-level, but increases segregation within smaller geographies, either at the Census tract- or block group-level, depending on how the development is cited. How integration happens at these various scales is of intense local interest, but it cannot be described using typical segregation measures.

In particular, classic segregation measures require the a priori selection of specific geographic levels at which to compute indices. For example, the Dissimilarity Index is among the most commonly employed measures of the evenness with which two mutually exclusive groups are distributed across a geographic area (Duncan and Duncan, 1955; Leckie et al., 2012). It describes the relative segregation of two racial/ethnic groups across one area, based on the distribution of these two groups within smaller component areas that comprise the larger geography. For example, to describe the distribution of Non-Hispanic Black and Non-Hispanic White residents using the Dissimilarity Index, higher and lower level geographies must be specified a priori. One could calculate Black–White segregation across census tracts of a metropolitan area, for example, as follows

(1 / 2) \sum | b_{i} / B - w_{i} / W |

where

b_i =

the non-Hispanic Black population of the ith area unit (in this case, a given Census tract),

B =

the total non-Hispanic Black population of the city for which the Dissimilarity Index is calculated,

w_i =

the non-Hispanic White population of the ith area unit (the given Census tract), and

W =

the total non-Hispanic White population of the city for which the Dissimilarity Index is calculated.

The minimum value of the Dissimilarity Index is zero and the maximum value is 1. Higher values indicate greater segregation or a more uneven distribution of residents by race. If an area’s Black–White Dissimilarity Index were .25, this would indicate that 25% of Black or White residents would have to change Census tracts within that area to achieve complete integration. While this statistic is powerful, it cannot describe the extent to which various geographic levels contribute to overall variation in the concentration of Black or White residents across the metropolitan area.

Newer and less commonly used methods exist to examine the multi-scalar aspects of segregation, with the strengths and limitations of many of these innovations described by Reardon et al. (2008). Methods include measuring clustering as an aspect of segregation (Massey and Denton, 1988) and comparing the contribution of various geographic levels to segregation using Theil’s H, an index of the evenness of population distributions. Theil’s H has been applied to both residential (Fischer et al., 2004; Fowler et al., 2016) and educational (e.g. Reardon et al., 2000) contexts. As an added contribution, Theil’s H allows for multi-group comparisons that measures such as the Dissimilarity Index cannot accommodate.

In addition, researchers have begun using hierarchical modeling approaches to describe segregation patterns, where a proportion of interest, such as the proportion of an area that is Black, is estimated according to a multi-level binomial response model that includes random effects for relevant levels of geography (Goldstein and Noden, 2003; Leckie et al., 2012). The group-level random effects are described by variance parameters whose estimates increase as the degree of dissimilarity between units nested within the group goes up, essentially providing a measure of segregation at that level. Formally, Harris (2017) shows that the Dissimilarity Index can actually be calculated from the multi-level residuals of a regression model whose outcome is the proportion of residents in one group when the model contains no intercept and an offset is included for the proportion of the population that belongs to the comparison group.

Multi-level modeling has been used to describe social segregation across London schools (Leckie and Goldstein, 2015; Leckie et al., 2012) and residential segregation among school children across England (Harris, 2017). The modeling approach has also been generalized to capture multi-group, multi-scale processes, for example examining residential segregation across multiple ethnic groups and geographic levels in London (Jones et al., 2015). In the United States, multi-level binomial response models have been used to examine the multi-scalar aspects of political polarization between 1992 and 2012 (Johnston et al., 2016), but we are not aware of other US examples.

This paper addresses two gaps in the literature examining multi-scalar residential segregation. First, we offer the first description of multi-scale Black–non-Black segregation nationally in the United States using the most recent Census data available. Second, we address one important limitation of existing approaches to summarizing multi-scale segregation, which is interpretability and interest for non-technical audiences. Despite the well-known limitations of simple segregation measures such as the Dissimilarity Index or Isolation Index, these indices continue to dominate policy conversations because they are easy to understand and allow for comparisons between specific metropolitan areas. For example, metropolitan area-specific segregation rankings, rather than national summary measures of segregation at multiple scales, garner population media attention (24/7 Wall St, 2016: 7). We demonstrate one approach to summarizing multi-level segregation patterns as single summary measures that reflect multidimensional information, acknowledging that Reardon et al. (2008) have previously created visual presentations of segregation profiles that characterize metropolitan areas according to the spatial scale at which segregation occurs. Our approach differs from Reardon’s segregation profiles in many ways, including by eliminating the need for readers to interpret a set of unique, continuous scores for metropolitan areas, and instead replaces scores with a typology.

Analytically, we addressed these aims in three parts. First, we estimated the contribution of each geographic level to the overall variation in how the Black population is geographically distributed in the United States. To do this, we fit a three-level random intercept multi-level logistic regression model that predicted the proportion of the population that is Black (Hispanic and non-Hispanic) at the block group level, where block groups were nested in tracts and tracts were nested in MSAs. Second, for the largest MSAs in the United States, we estimated the extent to which micro- versus meso-level variability drives overall racial residential patterning within the MSA. We fit a series of MSA-stratified, two-level random intercept multi-level logistic regression models that predicted the proportion of the population that is Black (Hispanic and non-Hispanic) at the block group level, with block groups nested in tracts. Finally, we created a typology of racial residential patterning within MSAs based on the total proportion of the MSA population that is Black and the relative contribution of block groups (micro) versus tracts (meso) in driving variation.

Methods

Data

We used block group-level population counts by race from the 2010 US Census, accessed through the National Historic Geographic Information System’s (NHGIS) online database (Minnesota Population Center, 2016). Because this paper aims to model multi-level patterns of segregation across US metropolitan areas, we identified block groups located within an MSA using crosswalk files provided by the Bureau of Economic Analysis (US Department of Commerce, n.d.) and excluded all that fell outside MSAs. Approximately 85% of the US population was captured in this approach. Our analytic sample included 178,078 populated Census block groups (level 1) nested within 60,519 Census tracts (level 2) and 382 MSAs (level 3), as shown in Figure 1.

Figure 1.

Multi-level sample size description.

Analytic approach

Random intercept, multi-level logistic regression models predicted the proportion of the population that is Black (Hispanic and non-Hispanic) at the block group level, adjusting for total block group population. This specification effectively modeled the log odds of a resident being Black in each block group j. We also included fixed effects for Census divisions. Our motivation for including Census divisions in the fixed portion of the model was to ensure that striking regional differences in racial/ethnic composition and patterning were not attributed entirely to the MSA-level alone. Rather, we present the relative contributions of the MSA-, tract-, and block group-levels to segregation within comparable groups of MSAs, as determined by Census division. The random portion of the model specified “cells” as an omitted level 0 in order to estimate the proportion of total variance in the concentration of Black residents that is attributable to geographic levels and not to random variability. Level 0 functions as a “pseudo-level” that allowed us to model total population by race at the block group-level as count data on individuals nested within block groups. Variance estimated at level 0 captures stochastic Poisson variability and is therefore not interpreted (Browne et al., 2005; Jones et al., 2015). That is, under a model-based approach, observed population counts are viewed as arising from both an underlying probability distribution and additional natural variability, where pseudo-level 0 accounts for the latter. Traditional measures of segregation, in contrast, provide no mechanism for accommodating random variability and are therefore well known to produce overestimates of segregation in the presence of small counts (Allen et al., 2015; Jones et al., 2015).

First, we combined data across all 382 MSAs and fit a pooled, three-level random intercept model with cells indexed by i functioning as an omitted pseudo-level (level 0) and whose variance cannot be interpreted, block groups indexed by j at level 1, tracts indexed by k at level 2, and MSAs indexed by l at level 3. The basic specification of the null model has been described in detail by Leckie et al. (2012) and can be written as follows:

logit ({blackrace}_{ijkl}) = β_{0} X_{0 ijkl} + (f_{0 l} + v_{0 kl} + u_{0 jkl})

All three random terms f, v, and u are assumed to be independently distributed with a mean of 0 and an unknown variance, represented by σ². The random term f_0l represents the precision-weighted difference between the average log odds that a resident is Black in a given Census division, and the average log odds that a resident is Black in MSA l, which is distributed normally with a mean of 0 and variance

σ_{l}^{2}

. Likewise, v_0kl is the precision-weighted difference between the average log odds that a resident is Black in MSA l and the average log odds that a resident is Black in a tract k, while u_0jkl represents the precision-weighted differential between the log odds that a resident is Black in tract k and the log odds a resident is Black in block group j.

We used these estimates to decompose the national variance in the concentration of Black residents into the specific contribution of each geographic level to the total variance. Variance partitioning coefficients (VPCs) represent the proportion of the total variance attributable to each level (Browne et al., 2005; Goldstein et al., 2002) and are calculated by dividing the variance for each level by the total variance across all levels. It is important to note that the relative contribution of each geographic scale to overall segregation can be identified through this multi-level modeling approach because variance at each level is estimated net of variances at the other levels (Harris, 2017; Jones et al., 2015: 20). Estimates were obtained using first-order marginal quasi-likelihood (MQL).

The second series of models aimed to characterize the largest MSAs in the United States according to the extent to which micro- versus meso-level processes drove overall variability in the concentration of Black residents within the MSA. We excluded MSAs with fewer than 500,000 residents (278 MSAs) because this population cutoff has been used in previous policy-relevant efforts to summarize the extent to which major US cities are segregated (Frey, n.d.). We then dropped any large MSAs with fewer than 5000 Black residents (2 MSAs) because even well-integrated population groups can appear highly segregated according to dissimilarity indices when the minority population is small (Allen et al., 2015; Harris, 2017; Jones et al., 2015). In practice, researchers urge caution when interpreting segregation indices for MSAs with fewer than 1000 Black residents (for example, see The Social Science Data Analysis Network, 2001). Excluding MSAs that could be spuriously mischaracterized according to classic measures of segregation allowed us to more directly compare the type of information provided by the Dissimilarity Index and our multidimensional typology. The 102 large MSAs that met our inclusion criteria contain about 66% of the total US population, spread across 137,450 block groups and 16,160 tracts.

We stratified the data by MSA and fit 102 MSA-specific two-level models that included cells, indexed by i and functioning as an omitted level (level 0) whose variance cannot be interpreted, block groups indexed by j at level 1, and tracts indexed by k at level 2:

logit ({blackrace}_{ijk}) = β X_{0 ijkl} + (v_{0 k} + u_{0 jk})

We calculated VPCs for the block group and tract level within each of these 102 large MSAs. MSA-specific estimates were obtained using second-order predictive (or penalized) quasi-likelihood (PQL2; see Goldstein, 2003) to protect MSA-specific estimates from the potential for downward bias sometimes induced by using MQL, although we re-estimated MSA-specific models using MQL in order to check the sensitivity of our results to choice of estimation method. Because results were largely consistent when we used MQL versus PQL2 estimation, and because PQL2 estimates were less biased than were MQL estimates when compared to results obtained through Markov chain Monte Carlo (MCMC) estimation, we report results based on PQL2 methods. To increase comparability of our analysis with previous work, we also transformed variance estimates from the 102 multi-level models into estimates of the Dissimilarity Index, essentially using the model parameters to simulate population counts that are used to compute the Dissimilarity Index (Leckie et al., 2012). Because simulated index values are model-based and therefore less sensitive to stochastic Poisson variation than are descriptive approaches, our simulated index values are not subject to the same upward bias that affects Dissimilatory Index calculations that use observed population counts. All multi-level modeling estimates were calculated using MLwiN 2.34, run via the “runmlwin” command in Stata 14.

Finally, we used cluster analysis, an exploratory data analysis technique, to identify groups of MSAs within the overall population of 102 large MSAs, with group membership based on the percent of the MSA population that was Black, and by the VPCs at the block group and tract levels, calculated as described above. We used a hierarchical agglomerative clustering algorithm, with relationships among MSAs organized by a dendrogram, to visually gauge the number of MSA groupings that could be used to organize the data set of 102 large MSAs. We used an average-linkage clustering algorithm, which computes average Euclidean dissimilarity scores between pairs of MSAs across groups, to identify how closely related groups of MSAs were (Kaufman and Rousseeuw, 2009). We visually examined dendrograms that displayed the hierarchical agglomeration of MSAs, from 102 unique observations to one large group, to identify the k number of groups that efficiently maximized differences between groups. We then implemented a partitional clustering algorithm to assign each MSA to exactly one of the k groups, again based on mean Euclidean dissimilarity scores. We ran all clustering alogrithms in Stata 14 (StataCorp, 2015).

Finally, we mapped the k MSA clusters. In order to provide a comparison to classic methods of measuring racial residential segregation at the MSA level, we also provided a map of Black-non-Black segregation as measured by the Dissimilarity Index. We calculated the Dissimilarity Index at the Census tract level, comparing the geographic distribution of Black and non-Black residents. Index values represent the proportion of Black residents within each MSA that would have to move to a different tract for the concentration of Black residents to be equal in all tracts. We used Stata 14 (StataCorp, 2015) and R 3.2.3 (R Core Team, 2013) for all data management and descriptive statistics, and ArcGIS software (ESRI, 2011) to produce maps.

Results

VPCs from the pooled model that predicted the log odds that any given resident was Black are shown in Table 1. Examining patterns within Census divisions, macro-scale differences across MSAs account for only 12% of the national variation in the geographic concentration of Black residents while the census tract-level accounts for over 78% of the total variance, suggesting that segregation is a predominantly meso-scale phenomenon nationally.

Table 1.

National VPC estimates.

Random effects	Variance estimate	S.E.	VPC (%)
Level 3: MSA	0.732	0.06	11.7
Level 2: Tract	4.899	0.03	78.4
Level 1: Block group	0.619	0.003	9.9

MQL: marginal quasi-likelihood; MSA: Metropolitan Statistical Area; VPC: variance partitioning coefficient.

Units: 382 MSAs, 60,519 tracts, 178,078 block groups.

Note: Model estimated using MQL, first order. Response variable = proportion Black. Data are 2010 decennial Census population counts provided by National Historic Geographic Information System.

Estimates of the extent to which the micro- versus meso-level differences drive variability in the concentration of Black residents within MSAs are provided in Figure 2. Box plots show the dominant influence of the tract- versus the block group-level in driving total MSA variation in racial composition, with the mean tract-level VPC estimated at nearly three times that of the mean block group-level VPC.

Figure 2.

Distribution of Census tract- and block group-level VPCs across 382 MSAs.

Analyses of the 102 largest MSAs revealed good correspondence between simulated, model-based estimates of the Dissimilarity Index and descriptive index calculations based on observed population counts (Figure 3). Simulated values were, as expected, consistently lower than observed values because of the upward bias stochastic variation introduces into index values that are based on observed population counts.

Figure 3.

Observed versus simulated Dissimilarity Index values for 102 large MSAs.

Finally, a dendrogram (not shown) of how the subset of 102 large MSA agglomerated into progressively larger groups of metropolitan areas that shared similar segregation patterns according to geographic scale visually showed five distinct branches of hierarchically clustered MSAs. Characteristics of the five distinct groups according to the three input clustering variables are displayed in Figure 4.

Figure 4.

Distribution of proportion Black, block group VPC, and tract VPC within MSA clusters.

Of the five clusters detected, Group 1 had the highest share of residents that were Black, and among the highest tract-level VPCs. Group 2 contained MSAs with relatively large shares of Black residents, but micro-scale, block group level processes were more important in racialized residential patterning in Group 2 MSAs than they were in Group 1. Groups 3 and 5 had the lowest shares of residents who were Black, but were differentiated by variation in the geographic concentration of Black residents occurring at the block group- versus tract-level; Group 5 had the highest micro-scale segregation pattern of any cluster we detected, while meso-level processes happening at the scale of Census tracts were more important in Group 3 MSAs. Of all the clusters, Group 4 exhibited the highest average tract-level VPC and was characterized by relatively lower proportions of Black residents.

Table 2 describes how selected MSA-level characteristics that were not included in our clustering algorithm were patterned by cluster. The mean Dissimilarity Index was statistically different across groups, as was the average size of the MSA’s Black population. However, overall population size and median household income were not statistically different across clusters. To ensure that our typology was not simply picking up on multi-level sample size differences that might influence VPC calculations, we also tested whether average Census tract or block group counts within MSAs were different across groups, and found that they were not.

Table 2.

Group characteristics.

Group	Population, Mean (SD)	Black population, Mean (SD)	Median household income, Mean (SD)	Index of dissimilarity, Mean (SD)	Number of MSAs
1	1,580,535 (1,364,208)	518,930 (432,553)	51,495 (7,077)	57.6 (6.3)	11
2	1,745,035 (1,817,901)	342,093 (299,271)	51,199 (5,201)	48.1(6.8)	8
3	1,543,353 (1,361,837)	116,315 (174,991)	53,321 (6,829)	51.0 (8.0)	32
4	2,860,646 (3,820,615)	432,825 (648,986)	54,930 (1,0155)	65.2 (6.5)	38
5	1,166,738 (1,025,525)	53,085 (71,468)	56,165 (1,2443)	39.6 (7.3)	13

Finally, we created maps to visualize the geographic distribution of MSA cluster types, and to allow for comparison between the typology and the typical approach to mapping metropolitan area segregation using the Dissimilarity Index (Figure 5). Our typology was geographically patterned, with Group 1 MSAs, which were relatively small metropolitan areas with high proportions of Black residents and strong meso-level segregation patterns, were largely clustered in the Southeast. Group 2 MSAs were physically proximate to Group 1 MSAs and exhibited similar population sizes and overall levels of segregation, but showed stronger micro-scale patterning in the concentration of Black residents relative to Group 1. Group 3 MSAs accounted for a disproportionate share of Midwestern region and Pacific division metropolitan areas. These MSAs had low proportions of Black residents, and most of the variance in how Black residents were geographically concentrated occurred at the tract-level. Group 4 MSAs contained most of the country’s largest cities, including New York, Los Angeles, and Chicago. These MSAs were the most segregated according to the Dissimilarity Index, on average, and showed the highest relative contribution of meso- versus micro-level patterning in how Black residents were concentrated. Group 5 MSAs were clustered in Mountain division states. Mean population size and the proportion of Black residents were lowest in this group. Of the five clusters, micro-scale variation in the concentration of Black residents was most important in Group 5 MSAs, on average. The geographic patterning of our typology was similar to that created by mapping Dissimilarity Index levels, but the multidimensional typology differentiated between relatively low segregation areas on the east versus west coasts, and created more distinct bands of MSAs running southwest to northeast across the East North Central division and Northeast region (Group 4), and across the East South Central and South Atlantic divisions (Group 1).

Figure 5.

A total of 102 large MSAs according to Cluster (top) and Dissimilarity Index (bottom).

Discussion

Our analyses show that, after accounting for Census divisions, nearly 80% of the national variation in the geographic concentration of Black residents is driven by within the MSA, meso-level processes. By comparison, differences between MSAs and micro-scale differences within Census tracts each account for approximately 10% in the total variation. However, we also show that the relative contribution of small (block groups) versus larger (tracts) scales to within-MSA segregation varies substantially across metropolitan areas. For example, in some MSAs, the block group-level is more important than is the tract-level in driving overall variation in how Black residents are concentrated within the MSA.

Finally, we summarize these important variations in the relative contribution of the meso and micro geographic scales for 102 large MSAs, a population of MSAs frequently ranked according to the Dissimilarity Index. We detected five meaningfully different types of large MSAs based on the overall Black composition of the MSA and the relative importance of the scales at which the Black composition varied across the MSA. While there are many reasons the spatial profile approach Reardon et al. (2008) used to create multi-scalar segregation profiles for the 40 largest MSAs in the United States cannot be expected to align closely with our typology – including differences in which racial groups were compared, the use of different decennial census years, calculations performed across different boundaries, and our incorporation of overall racial composition in the typology – the two sets of results were in substantial agreement. For example, Reardon et al. (2008) characterized Detroit, Cleveland, St. Louis, and Newark as places that had the highest relative contribution of large versus small geographic scales in driving Black–White segregation. In our analysis, these metropolitan areas all fell into Group 4, the cluster with the highest relative contribution of the meso- versus micro-scale variation in the concentration of Black residents. Similarly, San Jose and Phoenix, which had among the lowest relative contributions of large versus small geographic scales in driving Black–White segregation according to Reardon’s spatial profiles, fell into Group 5, the cluster with a substantially lower relative contribution of the meso- versus micro-scale variation in the concentration of Black residents. For other metropolitan areas, results were poorly aligned. For example, our typology partitioned Pittsburgh into Group 4 with Detroit, Chicago, and other places whose segregation patterns were dominated by the meso-scale, while Reardon et al. described Pittsburgh as having among the strongest micro-scale segregation patterns.

The scope of our analysis is limited by several important factors. First, there are multiple meaningful geographies below the MSA-level that were excluded from the analysis. For example, counties and Census designated places could have served as intermediate levels at which to further parse meso-level processes. There are very few counties within MSAs typically, which prevented us from including this level. However, places could be included as an intermediate level in future work. Also, this analysis ignores multi-group segregation, despite the substantive importance of understanding multi-group residential patterning and the fact that the relative concentration of small and larger geographic scales will change for different group comparisons. Finally, we did not provide data on trends in the relative importance of block groups versus tracts over time, although future work should examine temporal changes in segregation patterns.

Within the scope of our analysis, we made key methodological decisions that, while useful in some ways, introduced their own weaknesses. For example, our inclusion of fixed effects for Census divisions in the pooled model allowed us to control for baseline differences in racial composition regionally but depressed our MSA-level VPC estimate. Also, because of the computational burden associated with running the national multi-level binomial response model on nearly 200,000 block groups within nearly 61,000 tracts, we did not use MCMC estimation methods to obtain national variance estimates. Our inclusion of overall proportion Black as an input into the MSA typology also means that our clusters do not purely summarize the scalar patterning of segregation within MSAs. We included information about the proportion of each MSA’s population that was Black because areas with low shares of Black residents are more likely to exhibit micro-scale patterning simply because there are not enough residents to homogenously populate large areas. Including the racial composition indicator brought into greater relief differences between places that had similar Black composition but distinct spatial patterning regardless.

Finally, we note several potential barriers that may hinder adoption of this approach to describe segregation in the US. First, although the analysis described in this paper produces a readily interpretable typology, practitioners and researchers unfamiliar with multi-level modeling and/or cluster analysis may prefer indices that are based on more transparent calculations. Plain language descriptions of the advantages of a multi-scale approach, and of statistical methods where possible, should allow readers to decide for themselves which segregation measures are most appropriate for their needs. Second, we note the need for a thorough investigation of the VPC’s properties as a segregation measure, particularly with respect to the principal of transfers, compositional invariance, and symmetry in groups and types (Allen and Vignoles, 2007; Hutchens, 2004). Although there are many papers describing the properties of traditional segregation indices (e.g. Allen and Vignoles, 2007; Massey and Denton, 1988; Mora and Ruiz-Castillo, 2005; Reardon and Firebaugh, 2002), we are not aware of any comparable resources for model-based approaches.

Despite these limitations and tradeoffs, this paper demonstrates the potential for multi-level modeling approaches to provide rich data on how segregation occurs at different geographic levels across the United States. The approach can be customized or extended to look at relationships between different pairs, or even multiple groups, of populations, or to track trends over time. This type of descriptive work may help advance policy conversations beyond simply asking which metropolitan areas are most segregated in the United States and towards examining the level(s) at which that segregation occurs. This multi-level perspective may help housing developers, planners, and policymakers understand how and why segregated residential patterns are evolving in different places and provide important insights into interventions that could improve integration at multiple scales. The creation of a simple typology that groups MSAs together based on the scale at which segregation occurs could further help regions share challenges and lessons with peers facing similar difficulties.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Mariana Arcaya is an assistant professor of Urban Planning and Public Health in the Department of Urban Studies and Planning at the Massachusetts Institute of Technology. She is a social epidemiologist and urban planner whose work explores dynamic relationships between geographic contexts, particularly neighborhoods, and health. She holds a Doctorate of Science from the Harvard School of Public Health, and a Master of City Planning from MIT's Department of Urban Studies & Planning.

Gabriel Schwartz is a doctoral student in the Department of Social and Behavioral Sciences at the Harvard T.H. Chan School of Public Health. His research focuses on the ways neighborhoods, social policy, and health inequalities intersect (particularly as pertains to race). His professional experience includes rigorous policy evaluation of nutrition and education programs, as well as health advocacy in low-income clinics, connecting families to the resources they need to keep their families healthy.

SV Subramanian is a professor of Population Health and Geography at Harvard University. He has published over 550 articles, book chapters, and books on topics linking health, disease, and society. His current research interests are: understanding the causes and consequences of child development (with a focus on India), understanding and modeling individual and population heterogeneities in health, and understanding the reciprocal association between place and health. He was named among Thomson Reuters Highly Cited Researchers (top 1% of citations in Web of Sciences) in 2015, 2016, and 2017. Subramanian is the Co-Editor-in-Chief and Co-Senior Editor (Social Epidemiology) for the international journal Social Science and Medicine (SSM), and the founding Co-Editor-in-Chief of an Open Access Online journal, SSM Population Health.

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