Journal of Quantitative Criminology

, Volume 28, Issue 4, pp 559–586

Race, Space, and Violence: Exploring Spatial Dependence in Structural Covariates of White and Black Violent Crime in US Counties

Authors

    • Department of Sociology and Crime, Law and JusticeThe Pennsylvania State University
  • Casey T. Harris
    • Department of Sociology and Criminal JusticeUniversity of Arkansas
Original Paper

DOI: 10.1007/s10940-011-9162-6

Cite this article as:
Light, M.T. & Harris, C.T. J Quant Criminol (2012) 28: 559. doi:10.1007/s10940-011-9162-6

Abstract

Objectives

To join the literature on spatial analysis with research testing the racial invariance hypothesis by examining the extent to which claims of racial invariance are sensitive to the spatial dynamics of community structure and crime.

Methods

Using 1999–2001 county-level arrest data, we employ seemingly unrelated regression models, spatial lag models, and geographically weighted regression analyses to (1) compare the extent of racial similarity/difference across these different modeling procedures, (2) evaluate the impact of spatial dependence on violent crime across racial groups, and (3) explore spatial heterogeneity in associations between macro-structural characteristics and violent crime.

Results

Results indicate that spatial processes matter, that they are more strongly associated with white than black violent crime, and that accounting for space does not significantly attenuate race-group differences in the relationship between structural characteristics (e.g., structural disadvantage) and violent crime. Additionally, we find evidence of significant variation across space in the relationships between county characteristics and white and black violent crime, suggesting that conclusions of racial invariance/variation are sensitive to where one is looking. These results are robust to different specifications of the dependent variable as well as different units of analysis.

Conclusions

Our study suggests the racial invariance debate is not yet settled. More importantly, our study has revealed an additional level of complexity—race specific patterns of spatially heterogeneous effects—that future research on social structure and racial differences in violence should incorporate both empirically and theoretically.

Keywords

Racial invarianceViolent crimeSpatial econometricsSpatial heterogeneity

Introduction

Within the past two decades, there has been a resurgent interest in the different community circumstances of whites and blacks and their impact on a number of important social outcomes (Sampson et al. 2002; Krivo et al. 2009). On one hand, perhaps the most notable growth in research in recent years has been the upsurge in macro-social studies exploring the racial invariance hypothesis: that structural conditions and particularly structural disadvantage indicators predict crime in the same ways for all racial groups (Sampson and Wilson 1995). A sizable literature now exists comparing the effects of structural disadvantage (e.g., poverty, female headship) and other traits on black and white rates of violent crime across discrete geographic units (for reviews see Ousey 2000; Hannon et al. 2005; Parker 2008; Steffensmeier et al. 2010).

On the other hand, recent years have also witnessed a growing interest in the social ecology of crime that considers how broader spatial processes—functional relationships between proximate geographic units related to location and distance—affect crime. Stemming from the early work of “Chicago School” sociologists (Park et al. 1925; Shaw and McKay 1942), this line of inquiry is concerned, in particular, with whether structural conditions and crime in one geographic unit are affected by the spatial contexts in which the unit is encompassed (Baller et al. 2001). Utilizing new analytic techniques, this research confirms that the crime rate in a given location is strongly related to the crime rate and structural conditions in surrounding areas (Baller et al. 2001; Morenoff et al. 2001) and moreover, that the effects of key macro-social measures (e.g., population diversity, immigration, etc.) on crime often vary significantly over space, as well (Cahill and Mulligan 2007; Graif and Sampson 2009).

Surprisingly, the above two research agendas have yet to be joined so that several gaps exist in the criminological literature. To our knowledge, no published research has explored the racial invariance thesis while assessing the effects of spatial processes on race-specific rates of violence, including (1) whether the dynamics of space matter above and beyond traditional measures of structural disadvantage and other key macro-structural traits, (2) whether spatial processes affect crime differently for whites and blacks, (3) whether models accounting for spatial processes yield comparable findings to models which account only for characteristics of discrete geographic units, or (4) whether and how structural predictors of crime vary across space for particular racial groups.

Accounting for the broader spatial context in which violent crime occurs captures one of the key components of the racial invariance thesis—that black communities in the US tend to be unique ecological environments characterized by concentrated structural disadvantage unprecedented in white communities (Sampson and Wilson 1995). And recent research confirms that this “racial-spatial” divide persists beyond the borders of discrete geographic units and into surrounding communities as well, suggesting that broader ecological processes may impact crime uniquely across racial groups (Peterson and Krivo 2010).

However, it remains to be seen whether spatial processes explain differential rates of criminal offending and (to the authors’ knowledge) no research has explored whether these processes have invariant effects across racial groups or whether the effects are the same across geographic locations. To the extent that spatial dependence matters, explicitly modeling it and assessing spatial variability in the effects of structural predictors of crime can make important contributions by demonstrating whether or not spatial dependence itself has unique relationships with crime for particular racial groups, how accounting for space changes the overall picture of racial invariance in structural predictors of crime, and whether claims of racial variation/invariance are sensitive to where one is looking.

Our research contributes to this important strand of criminological inquiry in three ways. First, we assess whether crime is spatially patterned in a sample of contiguous counties and, if so, whether spatial dependence affects violent crime uniquely for whites and blacks. Second, we explore whether accounting for spatial processes significantly alters estimates of racial invariance in the structural predictors of crime. We focus specifically on the relationship between structural disadvantage and white and black violent crime, comparing traditional racial invariance models to models accounting for spatial processes. Third, we explore whether the relationships between these structural characteristics and crime vary across space. The focus here is on whether relationships between community structure and crime are stable across space (i.e., whether structural disadvantage has the same relationship with violence in all counties), which, as we discuss in more detail below, has important implications for our understanding of differences in the community circumstances of whites and blacks. Moreover, aside from evaluating the spatial stability of these relationships, our examination provides important insights as to whether and how the relationships between community structure and crime are spatially patterned (i.e., the spatial variation is clustered as opposed to random), and perhaps more importantly, where these relationships have heterogeneous effects.

Overall, our goal is to incorporate the methodological insights of spatial processes into the macro-level study of racial differences in criminal offending. In doing so, we distinguish macro-structural models that account for contextual measures within a geographic unit (“models of place”) from models accounting for the geographic interdependence between units (“models of space”) and explain the methodological and theoretical import of incorporating spatial dependence into tests of racial invariance. We begin by briefly reviewing the literatures on the racial invariance thesis and spatial analysis of crime, focusing in particular on the distinction between “place” and “space.” Second, we utilize county-level Uniform Crime Report (UCR) arrest data from 1999 to 2001 to model white and black county-level violent crime. We employ seemingly unrelated regression (SUR), exploratory spatial data analysis (ESDA), and spatial econometrics, assessing (a) the consistency of findings across models and (b) the extent to which “space” predicts white and black violent crime in unique ways. Third, we use geographically weighted regression (GWR) to explore spatial heterogeneity in structural predictors of crime (i.e., variation in associations across space). Fourth, we conduct a series of robustness checks on our results and conclude by discussing the importance of our findings for macro-level studies of crime in general and racial invariance research in particular, as well as directions for future research.

Prior Literature

The Racial Invariance Thesis

Originating with the “Chicago School,” the racial invariance thesis—that structural predictors of crime operate in the same manner for all racial groups—can be traced to early research seeking to explain aggregate-level variations in crime and delinquency (Shaw and McKay 1942). Findings from this early research indicated that crime rates remained high in certain structurally disadvantaged areas in Chicago regardless of which racial or ethnic group occupied those areas. Characterized by what the Chicago sociologists termed a “social disorganization perspective,” this finding is consistent with a number of distinct theories that fall within the structural tradition of sociology and criminology (e.g., anomie/strain perspectives). As Ousey (1999: 407) argues, these theories all share two central propositions: (1) that community socioeconomic deprivation is positively related to community crime rates and (2) that the positive association between socioeconomic deprivation and crime is invariant across racial categories (see review in Steffensmeier et al. 2010).

The second proposition has received considerable attention in recent years and bears directly on the racial invariance thesis by utilizing race-specific measures of crime and structural characteristics. Including a wide range of measures (i.e., residential mobility, low educational attainment, racial segregation, etc.) and outcomes (i.e., homicide, robbery, violent crime, etc.), there is now a sizable body of literature comparing the effects of structural disadvantage on crime across racial groups. However, results remain inconclusive and there is considerable debate with some reviewers arguing that the preponderance of evidence supports the racial invariance thesis (e.g., Krivo and Peterson 2000; Peterson and Krivo 2005), while other reviewers see the results as much more mixed and thus view the racial invariance issue as largely unsettled (Ousey 1999; Parker 2008; Phillips 2002).

While Steffensmeier et al. (2010) concluded that much of this ambiguity is the product of scope and conceptual uncertainty, including a lack of clarity regarding the dependent variables, structural predictors, unit(s) of analysis, and race/ethnic groups to which the invariance thesis applies, one aspect of the racial invariance debate that has received scant attention is the role of spatial context (but see Peterson and Krivo 2010 for a recent detailed exploration of this issue). As Heitgerd and Bursik (1987: 776) highlighted over 20 years ago, the key critical shortcoming of research on macro-structural studies of crime in general and social disorganization theory in particular is “an overriding emphasis on the internal dynamics of local communities that wholly ignores the external contingencies that may be important in shaping the nature of these dynamics.” As we note in the following section, it could be that failure to account for different spatial context between races accounts for divergent findings between racial invariance tests because within-unit structural measures only capture part of the different ecological circumstances of white and black communities.

“Place” Versus “Space”

Racial invariance scholars have made significant progress in exploring race group differences in the effects of important structural predictors of crime across discrete geographic units; however, models from these studies presuppose that crime and its ecological correlates in one unit have no effect on any other unit. Such “purely structural interpretations” implicitly assume that once key measures (structural disadvantage, racial/ethnic heterogeneity, and residential mobility) are taken into account, the structural conditions in surrounding areas have no impact on conditions in a specific geographic unit (Baller et al. 2001: 565). Even analytic techniques designed to account for the dependence of samples (e.g., seemingly unrelated regression) and which are now standard in racial invariance research capture only characteristics of a discrete “place.”

In contrast, the space and crime literature suggests that traditional racial invariance models may miss the important effects of conditions in geographically proximate units. Though focusing almost exclusively on total (i.e., not race-specific) crime, geographers and criminologists exploring the space-crime relationship have produced a sizable literature indicating that broader spatial processes—functional relationships between proximate geographic units—affect crime. In comparison to racial invariance models capturing only characteristics of “place” (or “structural similarity”—see Baller et al. 2001), spatial models treat both the structural conditions and aggregate criminal processes among geographic units as interdependent (Mears and Bhati 2006).

Analytically, this suggests an important distinction between traditional macro-social models which account only for measures associated with discrete geographic units, and models which capture the spatial dynamics of crime and its correlates. Relying on the insights of prior literature (Baller et al. 2001; Johnston et al. 1981), we refer to this distinction as the difference between “models of place” and “models of space.” Figure 1 illustrates:
https://static-content.springer.com/image/art%3A10.1007%2Fs10940-011-9162-6/MediaObjects/10940_2011_9162_Fig1_HTML.gif
Fig. 1

Comparison of “Place” and “Space” models. Note: figures are recreated from Baller et al. (2001: 564); while there are alternative “space” models (e.g., spatial error), we show the spatial lag because, as we discuss in later sections, this model is more theoretically informative and is used in our analysis

Panel A depicts a “model of place” as is typically found in racial invariance research and more broadly in macro-social research on the structural predictors of crime. Briefly, this model suggests that the level of crime (Yi) in geographic unit i is affected only by structural traits within that same geographic unit (Xi). In contrast, panel B displays a “model of space,” which captures the functional relationship between proximate geographic units. Here, crime (Yi) in geographic unit i is not only affected by its own structural traits (Xi), but is also influenced by the structural characteristics (Xj) and crime (Yj) of the adjacent geographic unit j, and vice versa.

Stated simply, racial invariance scholars have overwhelmingly utilized models of “place” rather than models of “space.” However, recent years have witnessed a growing interest in the social ecology of crime that bears directly on the “place” versus “space” distinction. We turn now to a brief review of this literature.

Space and Crime

Sociologists and criminologists have known for nearly a century that aggregate levels of crime and socioeconomic disadvantage tend to be geographically clustered (Park et al. 1925; Shaw and McKay 1942). That is, neither crime nor measures of disadvantage (poverty, inequality, etc.) are randomly distributed across space and that geographic units are not independent of each other. As Sampson and Morenoff (2004: 146) state, “neighborhoods are interdependent and characterized by a functional relationship between what happens at one point in space and what happens elsewhere” (see also Sampson and Bean 2006).

Though rarely concerned with race-specific patterns of crime, scholars using spatial econometric methods have consistently demonstrated the importance of accounting for the interdependence of geographic units. For example, Messner et al. (1999) utilized exploratory spatial data analysis (ESDA) and found evidence of significant spatial dependence of homicide independent of traditional structural measures (see also Mears and Bhati 2006). Anselin et al. (2000) reviewed studies with similar findings for prostitution and drug dealing, while Baller et al. (2001) showed that county-level homicide victimization is strongly clustered in space and that this clustering cannot be completely explained by common measures of structural similarity. Finally, Deane et al. (2008) demonstrate that robbery rates among a sample of US cities are spatially dependent.

Most recently, Peterson and Krivo (2010) have shown that various measures of the spatial concentration of disadvantage help explain differential rates of criminal offending in white and minority neighborhoods in major metropolitan cities. However, while their research is the most thorough treatment of understanding how spatial processes impact crime, important questions remain because they do not evaluate the correlates of white and minority crime separately or test for whether the relationships between structural measures and crime are invariant across geographic space. Taken together, this body of research suggests that ignoring spatial processes may miss important mechanisms driving patterns of crime (e.g., exposure, diffusion) and inferences will be inaccurate if spatial processes operate and are not accounted for (Baller et al. 2001).

More recently, geographers and criminologists have also begun exploring differences in the effects of structural predictors of crime across space, or what is known as spatial heterogeneity. As opposed to traditional regression models (i.e., OLS, logistic, negative binomial, etc.) or even spatial econometric models, which report one “global” effect and assume stationarity—that the effects of a given measure are invariant across space—scholars have begun using advanced methods to explore the “local” effects of community structure on crime. For example, Cahill and Mulligan (2007) used geographically weighted regression (GWR) to show that many structural covariates of violent crime (family structure, residential stability, population heterogeneity) vary across block groups in Portland, Oregon. More recently, Graif and Sampson (2009) demonstrated significant spatial heterogeneity in the effects of population diversity and immigration on homicide in Chicago neighborhoods. Overall, these studies suggest that not only is it crucial to explicitly model the spatial processes affecting crime and its correlates, but that global models may mask a criminal landscape that is “highly diversified and spatially stratified” (Graif and Sampson 2009: 19).

Race and Space

To date, the methodological insights of modeling spatial dependence have not been fully incorporated into the study of racial invariance. The distinction between “models of place” and “models of space” implies that traditional racial invariance models which account only for structural characteristics within geographic units may overlook important effects between geographic units. But while previous research has shown that “space” matters in predicting crime (Sampson and Morenoff 2004), little is known about whether spatial processes affect white and black crime differently.

There are several reasons to suspect that space uniquely impacts crime for particular racial groups and that patterns of spatial heterogeneity (i.e., “local effects”) differ across groups, as well. For one, white and black communities vary tremendously along a host of social indicators, particularly socioeconomic disadvantage (Sampson and Wilson 1995), and some scholars have suggested that black neighborhoods experience levels of deprivation at or above a critical threshold where the impact of disadvantage on crime is considerably diminished (Krivo and Peterson 2000; McNulty 2001). Drawing on these insights, black communities characterized by extreme disadvantage may be less affected by the levels of disadvantage within surrounding communities, thus predicting that geographic proximity to highly disadvantaged and high-crime areas will be less consequential for black communities than white ones.

On the other hand, more advantaged communities may be better able to insulate themselves from processes of criminal diffusion (or spillover) and exposure from nearby areas. As the qualitative research by Pattillo-McCoy (1999) demonstrates, even black middle class communities are placed at a higher risk of violent crime due to their proximity to inner-city neighborhoods, whereas white middle class communities tend to locate at greater distances and thus shield themselves from the social deterioration of minority communities. In other words, some community borders may be more porous than others and these differences likely fall along racial lines.

Regarding spatial heterogeneity, to our knowledge, there is no empirical research to date exploring how key macro-structural measures may vary across space in predicting white and black rates of violence. As it pertains to the racial invariance question, failing to assess whether structural predictors of crime have “local effects” ignores the spatial sensitivity of racial invariance claims and may, in part, explain the inconsistent findings in the literature. This is because spatial heterogeneity in structural predictors of crime can yield conclusions of both racial invariance and variance, depending on where one is looking. For example, if the relationship between structural disadvantage and crime (1) varies significantly over space for one racial group but not another, or (2) if the pattern of variation across space is not the same for each racial group (e.g., the effect is positive for blacks in the same places where it is negative for whites), then evidence of racial invariance from models that yield “global” (or average) effects is undermined.

While we note that our examination of spatial heterogeneity is largely exploratory, there are reasons to suspect that key macro-structural measures may vary across space in predicting white and black rates of violence and that these effects may be spatially patterned (clustering of “local effects”) uniquely for particular racial groups. Though we do not test a specific theoretical mechanism, we are guided by two primary insights. First, a small yet growing body of research has begun examining distinct patterns of structural covariation at various levels of analysis, highlighting how neighborhoods, counties, and states interact with the larger spatial geography in which they are embedded to produce varying effects across geographic space (Baller et al. 2001; Graif and Sampson 2009; Deane et al. 2008).

Second, a long line of ecological theory and research has demonstrated how levels of disadvantage and inequality fall along racial lines, and that residential environment, both urban and rural, differ markedly for white and minority residents (Shaw and McKay 1942; Sampson and Wilson 1995; Tickamyer and Duncan 1990). Combined, these two insights might suggest that certain environments may interact with the larger spatial geography in unique ways depending on the racial makeup. For example, regional differences in historical legacies of hardship and sub-cultural adaptations to it may lead to racially diverging responses to contemporary disadvantage and produce varying effects across geographic space (Lane 1986). As such, the spatial patterning of the relationships between structural traits and crime themselves may exhibit racial variation.

Overall then, there is a pressing need to extend prior research on the racial invariance thesis to incorporate a spatial framework, including whether and how space matters for particular racial groups and whether patterns of spatial variation (clustering of “local effects”) are conditioned by race.

The Current Study

Data

The data for our main analysis are taken from several sources. First, we utilize the Uniform Crime Reporting Program’s (UCR) detailed county-level arrest data from 1999 to 2001 for information on race-specific arrest rates. Second, Summary Files 1 and 3 from the 2000 Census provide information on demographic and social characteristics of counties. Finally, we use US state and county geographical shape files to provide spatial boundaries and derive distances between counties.

Because our analysis is concerned with race-specific measures, our sample is limited to the maximum number amount of counties that are reliably available and satisfy the needs of a spatial analysis. While the UCR is a nation-wide database, not all counties provide race-specific data and many counties in the US have too few African American residents to provide reliable information.1 We therefore limit our analysis to counties that reported at least one violent crime (i.e., murder, rape, robbery, assault) between 1999 and 2001 and that had at least 200 whites and 200 blacks residents as of the 2000 census. Moreover, since missing data is especially problematic for spatial analysis (i.e., the primary insight gained by spatial econometrics is the relationship between units that are spatially proximate), we use only those counties for which arrest information are available for a significant number of neighboring counties, as well. This contiguous sample of counties is composed largely of Western and South-Western counties (from Washington to New Mexico), Southern counties (from Texas east through Georgia) and East Coast counties (from South Carolina north through New Hampshire). The final sample size is 1,315 counties that had UCR violent arrest information for the 1999–2001 period and met our population requirements (map of data available upon request).2 While we recognize that this sample of counties covers only 42% of all counties in the United States, it does cover nearly 70% of the US population. Moreover, we demonstrate below that this sample overlaps significantly with the nation as a whole in key dependent and independent measures and, therefore, does not appear to be unduly influenced by our data availability or selection criteria (see Table 1).
Table 1

Descriptive statistics for sample of counties (N = 1,315)

 

Sample of counties

United Sates

 

Means

SD

Moran’s Ic

Means

White

Black

White

Black

White

Black

White

Black

Violent index ratea

70.95

306.33

57.06

248.39

.53***

.28***

101.78

386.58

Violent index rate (ln)a

3.96

5.42

.85

.86

.34***

.23***

4.62

5.96

Structural disadvantagea,b

.00

.00

1.00

1.00

.52***

.41***

  

Povertya

10.93

27.90

4.29

10.73

8.12

24.84

Unemployment

4.81

11.92

2.09

6.90

2.79

6.86

Female headshipa

4.76

16.37

1.12

6.79

5.23

23.50

Low educationa

21.36

33.25

7.31

12.37

14.55

27.57

Residential instabilitya

20.32

23.84

7.65

15.04

.34***

.35***

19.05

15.77

Entropy

.55

.55

.28

.28

.69***

.69***

.48

.48

% Hispanic

6.55

6.55

11.25

11.25

.82***

.82***

12.52

12.52

Male pop. 15–24a

6.70

9.70

2.10

5.73

.06**

.15***

6.36

7.86

Police (per 1,000)

18.67

18.67

11.32

11.32

.30***

.30***

17.16

17.16

Population structureb

.00

.00

1.00

1.00

.51***

.51***

Total population (ln)

10.92

10.92

1.26

1.26

Population density (ln)

4.64

4.64

1.39

1.39

4.38

4.38

% Urban

50.28

50.28

29.34

29.34

68.35

68.35

** p < .01; *** p < .001

aRace-specific measure

bCombines multiple structural traits using principal component analysis

cMoran’s I calculated using the Queen’s 1st order weights matrix

We rely on counties as our units of analysis because (a) they provide a large enough sample size to include an adequate number of covariates in our models and still retain statistical power to detect effects, (b) they allow us to explore spatial patterning across a large proportion of the United States, and (c) extant literature has used counties in prior research on crime and spatial processes (Baller et al. 2001; DeFronzo and Hannon 1998; Mencken and Barnett 1999; Messner and Anselin 2004). However, we want to make several important points. First, spatial analyses and the treatment of spatial effects may be sensitive to the unit of analysis (Openshaw 1983) and counties may be too large if violent crime varies primarily among smaller units or, conversely, too small if it is a regional phenomenon (Baller et al. 2001). Second, much of the previous research has relied on smaller geographic units such as cities or census tracts. While our county-level analysis may miss these within-county spatial dynamics and could be seen as a shortcoming, the inverse is that previous research may have overemphasized the ecological context within cities while missing the broader county, state or regional dynamics that may be occurring as well. Thus, rather than viewing our county-level analysis as a weakness of testing the racial invariance thesis, we view our research as a complement to existing studies by examining whether larger level spatial dependence helps understand racial differences in criminal offending. If our analysis does reveal regional differences in the spatial dynamics of white and black crime, this may suggest that larger level ecological contexts condition lower level criminogenic processes.

However, we think understanding whether and to what extent our results are dependent on the unit of analysis is an important consideration. As such, we supplement our main analysis by re-estimating our substantive models using race-specific tract-level homicide data from the Homicides in Chicago Study (see Block and Block 1992; see also Mears and Bhati 2006; Velez 2006).3 As we review below, the relationships between disadvantage, spatial proximity, and crime from these models are remarkably similar to our findings when using counties, leading us to conclude that our results are robust across levels of analysis.

Dependent Variables

The dependent variables in this study are county-level white and black violent index (sum of arrests for homicide, aggravated assault, forcible rape, robbery) offending rates per 100,000. We use three-year averages to reduce the potential bias of a single high- or low-crime year and, given the skew of our dependent variables, we log transform our rates.

Though the use of arrest data is subject to well-known criticisms, it is generally believed that UCR violent index arrest statistics are reasonable proxies for criminal offending (Krivo and Peterson 1996). For one, violent crimes are viewed as serious criminal offenses and are more likely to be reported to the police and result in arrest relative to other types of crime (i.e., less serious offenses, property crime, white collar crime, etc.). Additionally, UCR arrest data are better suited for the research questions studied here, providing an adequate number of geographic units for both racial groups under study. Self-report data, which are often based on samples of households, are less well suited to this type of aggregation than are arrest data. Nevertheless, we test the robustness of our results using several different dependent variable specifications (described below) that are less prone to reporting bias.

Key Independent Variables

Our primary focus is on race-specific measures of structural disadvantage, including poverty (measured as the percentage of county residents below the poverty line), unemployment (measured as the percentage of the county civilian labor force between the ages of 16 and 59 that is unemployed), female headship (measured as the percentage of county families with children under 18 years old that are headed by a female), and low education (measured as the percentage of county residents over 25 years old with less than a high school education or equivalent). Because poverty, unemployment, low education, and female headship tend to be highly correlated in aggregate data, estimating their unique effects, and race-ethnic differences in those effects, can be problematic due to multicollinearity. Based on standard principal components methods (see Land et al. 1990), we extracted race-specific structural disadvantage factors based on the four race-specific disadvantage measures (i.e., one structural disadvantage factor represents the combined influence of poverty, unemployment, education, and female headship for each group).

Control Variables

We control for race-specific discrete measures of residential instability (measured as the percentage of the county population who lived in a different county within the past 5 years) and the percentage of the county population that is male and between the ages 1524, as well as the percent Hispanic (a global measure of the percentage of the total county population this is Hispanic). We also include a global (not race-specific) factor for population structure, which includes a county’s population size, population density, and the percentage of the population that is living in an urban area.4 Finally, we include controls for racial diversity as measured by the entropy index5 and police per capita6 as a control for variations across counties in law enforcement activity (see Schwartz 2006).7

Analytic Strategy

We begin our analysis by, first, replicating prior racial invariance research regressing violent crime on our key discrete structural measures (characteristics of “place”) with a particular focus on the association of our race-specific structural disadvantage components with white and black violent crime. These models are now standard in racial invariance research and represent the baseline to which we can compare subsequent spatial models. As prior research has shown, when subgroup (e.g., black and white) equations are estimated from the same geographic units, ordinary least squares regression (OLS) violates assumptions of independence and homoscedasticity (Ousey 1999). In contrast, seemingly unrelated regression (SUR) simultaneously estimates models predicting both white and black violent crime while accounting for the correlated errors associated with white and black crime sharing interdependent unmeasured causes (see Schwartz 2006 for a review). To directly assess the extent of racial invariance, we conduct z-tests of the equality of coefficients to explore whether our key structural predictors vary in their effects on white and black violence (Clogg et al. 1995; Paternoster et al. 1998).

Second, we investigate the extent to which our baseline SUR models fail to capture the spatial relationships between violent crime and key structural characteristics by mapping the residuals from our baseline models (maps available upon request) and testing for spatial autocorrelation by calculating Moran’s I statistics.8 If crime rates are determined solely by the structural factors included in the model there should be no spatial patterning of crime beyond that created by socio-demographic similarities of geographically proximate counties. We build upon our baseline analysis by utilizing spatial econometric models that take into account the effects of space on violent crime for whites and blacks in contiguous counties. This entails first creating spatial weights matrices9 and then estimating separate white and black models that explicitly account for the spatial processes impacting violent crime.

We estimate spatial dependence by constructing spatial lag models of white and black violent crime.10 Formally, the spatial lag model can be expressed as follows:
$$ y = \rho Wy + X\beta + \varepsilon $$
(1)
where y is a N by 1 vector of observations on the dependent variable (in our case, county-level violent crime), Wy is the corresponding spatially lagged county-level violent crime for weights matrix W, X is an N by K matrix of observations on the explanatory variables, ε is a N by 1 vector of normally distributed error terms, ρ is a spatially autoregressive parameter, and β is a K by 1 vector of regression coefficients (Anselin 1988: 35). Because estimating both the focal and neighboring levels of violent crime creates problems of endogeneity in standard OLS estimation, maximum likelihood estimation is utilized in our spatial lag models.

As with our baseline models, we conduct z-tests of the equality of coefficients to directly assess the extent to which our key structural predictors are racially invariant. Results from these “models of space” can be compared to those of our baseline SUR “models of place” to determine (1) whether space matters, (2) whether space matters differently for predicting white and black crime, and (3) whether other key structural traits (i.e., structural disadvantage) differ in their association with violent crime across models.

Third, we utilize geographically weighted regression (GWR) models to assess whether our race-specific structural effects vary across space and, if so, whether the patterns of variation differ for whites and blacks (Fotheringham et al. 2002). Unlike our baseline SUR and spatial econometric models, GWR recognizes that just as the levels of disadvantage and other structural traits vary from one location to the next, the relationships between structure and crime may vary across space as well. Conventional regression models and spatial econometric models estimate one global parameter for the relationship between each independent variable and the dependent variable, and these relationships are assumed to be constant across a study area (i.e., stationary). The GWR approach goes beyond the standard regression framework to estimate local rather than global parameters and to indicate where effects are observed. That is, as opposed to calculating a single regression equation, GWR generates a separate equation for each observation in the analysis and each equation is calibrated using a different weighting of the observations contained in the data. This model can be expressed as follows:
$$ y_{i} = \beta_{0} (u_{i} , v_{i} ) + \sum\limits_{k} {\beta_{k} (u_{i} , v_{i} )x_{ik} + \varepsilon_{i} } $$
(2)
where (uivi) represents the coordinate location of the county centroid, and βk (uivi) is the realization of the continuous function βk (uv) at point i. We estimate β using the following equation:
$$ \hat{\beta }(u_{i} , v_{i} ) = \left( {X^{T} W(u_{i} , v_{i} )X} \right)^{ - 1} X^{T} W(u_{i} , v_{i} ) y $$
(3)
where \( \hat{\beta } \) is an estimate of β, and W(ui, vi) is an n by n matrix whose off-diagonal elements are zero and whose diagonal elements denote the geographical weighting of the n observed data for regression point i (Fotheringham et al. 2002). Because each observation is weighted according to its proximity to i, as the model moves across space, the weight of an observation is no longer constant in the calibration, as in OLS or weighted least squares (WLS), but varies with i. We use a spatially adaptive weighting function based on the Akaike Information Criterion (AIC) minimization procedure, which provides a trade-off between goodness-of-fit and degrees of freedom. Similar to the logic of kernel weighting in propensity score matching, data observations close to i are weighted more than data from observations further away.
In order to determine the number of nearest neighbors from the focal county i, we utilize a spatially adaptive bandwidth in the following bi-square function:
$$ \begin{aligned} W_{ij} & = \left[ {1 - \left( {\frac{{d_{ij} }}{{g_{i} }}} \right)^{2} } \right]^{2} \quad {\text{if}}\;d_{ij} < g_{i} \\ & = 0\;{\text{otherwise}} \\ \end{aligned} $$
(4)
where gi denotes the distance (bandwidth) of the Nth nearest neighbor from i, and dij represents the distance between counties i and j. This function provides a continuous, near-Gaussian weighting function up to distance g and then weights any county beyond g zero (Fotheringham et al. 2002).

Because some of the local variability in the effects across space may result from sampling variation we utilize the Monte Carlo approach to test if the observed variation in a parameter is sufficient to reject the null hypothesis of a globally fixed parameter (see Graif and Sampson 2009 for more details on the GWR framework and its application to criminology). Overall then, GWR tests the assumption of stationarity in structural covariates of crime by estimating local effects of each of our key structural traits on violent crime and assessing whether and to what extent these relationships vary across spatially contiguous counties.

Testing for spatial variability in the effects of structural predictors of crime marks an important advance in research on structural covariates of crime in general and the racial invariance thesis specifically. Extant statements of the racial invariance thesis do not discuss whether the effects of the structural predictors of violence need to be invariant across space. However, we argue that spatial variability is an important dimension of the larger racial invariance hypothesis. If the effects of structural predictors are (a) invariant across space or (b) vary in similar ways for each group, then global models common in racial invariance research may be appropriate for capturing invariance/variance in the structural predictors of crime. In contrast, without spatial stability in effects within groups or similar patterns of variability across groups, the ability to say that a particular structural effect is invariant/variant depends greatly on where one is looking. Thus, an analysis that can shed light on this understudied area of the racial invariance thesis makes important contributions to the race and crime literature.

Results

Table 1 provides descriptive statistics for our sample of contiguous counties and, to illustrate the generalizability of the sample, the levels of each key variable for the nation as a whole. Several notable patterns emerge.

First, the violent crime rate is substantially greater for blacks than whites. The mean untransformed black rate of roughly 306 violent offenses per 100,000 is over four times greater than the mean white rate of about 71 violent offenses per 100,000. Second, Table 1 reveals striking disparities in socioeconomic status where blacks fare worse than whites on virtually every measure. For example, mean black poverty (27.90) is far greater than that of whites (10.93), as is unemployment (11.92–4.81), low education (33.25–21.36), and female headship (16.37–4.76, respectively). These racial disparities in violence and disadvantage are very similar to those observed for the entire United States (see the final two columns of Table 1), suggesting that our sample of counties is representative of the nation as a whole. Overall then, the picture painted by Table 1 is one of high black involvement in violent crime and exposure to structural disadvantage relative to whites in our sample of counties—a pattern consistent with that of the broader United States.11

Additionally, as an initial indication of spatial dependence in both the structural characteristics and crime rates among counties, we tested for spatial autocorrelation using Moran’s I statistics for all measures in our analysis. These results show that for whites and blacks, there is significant spatial clustering on all measures. While suggestive of the need for models that incorporate the dynamics of space, more formal tests are needed to show that this spatial clustering is not due to structural similarity in proximate counties.

Replicating Prior Racial Invariance “Models of Place:” Seemingly Unrelated Regression

Following prior research on the racial invariance hypothesis (Ousey 1999; Steffensmeier et al. 2010), we next utilize SUR to simultaneously estimate white and black models of violent crime. Again, we focus in particular on the relationship between structural disadvantage and crime given the importance of this association in prior racial invariance literature. Panel A of Table 2 provides these results.
Table 2

Comparison of models regressing white and black violent index rates on county-level structural traits (N = 1,315)

 

(A) “Place” model (SUR)

(B) “Spafe” models (spatial lag)

Whites

Blacks

Z test for diff.

Whites

Blacks

Z test for diff.

b

B

b

B

b

B

b

B

Constant

4.243***

(.080)

 

6.204***

(.082)

  

2.981***

(.134)

 

4.944***

(.174)

  

Structural disadvantagea

.242***

(.021)

.285

.110***

(.020)

.128

4.552***

.223***

(.027)

.262

.121***

(.025)

.141

2.772**

Residential instabilitya

009***

(.003)

.081

−.005**

(.001)

−.087

4.427***

.013***

(.003)

.117

−.001

(.004)

−.017

2.800**

Entropy

−.255**

(.075)

−.084

−1.090***

(.086)

−.355

7.318***

−.242***

(.071)

−.080

−1.001***

(.089)

−.326

6.667***

% Hispanic

.010***

(.002)

.132

−.002

(.002)

−.026

4.243***

.007***

(.002)

.093

−.001

(.002)

−.013

2.828**

Male pop. 15–24a

−.065***

(.010)

−.161

−.009**

(.003)

−.060

−5.364***

−.068***

(.012)

−.168

−.013**

(.004)

−.087

−4.348***

Police (per 1,000)

.002

(.002)

.027

.001

(.002)

.013

.354

.002

(.002)

.027

.002

(.002)

.026

.001

Population structure

−.191***

(.023)

−.225

−.200***

(.023)

−.233

.277

−.162***

(.023)

−.191

−184***

(.023)

−.214

.676

Spatial lagb

.309***

(.027)

.309

.216***

(.026)

.216

2.481*

R2

.212

 

.164

  

.307

 

.218

  

Breusch–Pagan

χ2 = 614.886, p < .001

    

χ2 = 77.804, p < .001

 

χ2 = 55.365, p < .001

  

AIC

5,285.396

    

2,873.700

 

3,045.720

  

p < .05; ** p < .01; *** p < .001 (two-tailed)

Standard errors in parentheses

aRace-specific measure

bStandardized and unstandardized coefficients are identical because measure is already in the metric of the dependent variable

Important findings are as follows. First, structural disadvantage has a statistically significant, positive association with white violent crime, net of other important structural characteristics, and is the strongest predictor in the model (B = .285). In contrast, structural disadvantage does have a statistically significant effect on black violent crime, net of other important structural traits, but the effect is relatively modest in comparison to other measures in the models (e.g., entropy and population structure). That is, higher levels of structural disadvantage are associated with higher violent crime rates, particularly for whites. Among the other predictors, only population structure (negative), the size of the male crime-prone population (negative), and racial heterogeneity (negative) have statistically significant relationships for both whites and blacks in the same direction.

Second, turning now to the central focus of our analysis, we find a statistically significant difference in the association between disadvantage and violent crime for whites and blacks. The z-tests in the final columns of panel A in Table 2 reveal that structural disadvantage (as well as several other factors) differs in its relationship with violent crime across groups (z = 4.552, p < .001). Specifically, the association of structural disadvantage with white violent crime is significantly stronger (roughly twice the size) than with black violent crime. Among other measures, we find differences in relationships with violent crime across groups for all of our measures except police per capita and population structure.

Overall then, there appear to be important differences between whites and blacks in the associations between key structural traits (especially structural disadvantage) and violent crime in our sample of counties. In fact, we find more differences than similarities in our county-level models, with five out of seven cross-group comparisons statistically significant at the p < .05 level. However, these SUR models provide a baseline for our assessment of racial invariance because they measure characteristics within geographic units but do not capture the relationships between geographically proximate areas. This is supported by exploratory spatial data analysis, including formal Moran’s I tests and maps of the residuals from these baseline SUR models (available upon request), which indicate that both white and black residuals are spatially auto-correlated. Overall, the ability of our baseline SUR models to predict white and black violent crime in a given county depends greatly on surrounding counties and suggests that there is a meaningful spatial relationship that our baseline models are failing to capture.

In order to (a) determine whether space matters above and beyond “place”, (b) explore whether space impacts violent crime uniquely for whites and blacks, and (c) assess the consistency of findings between “models of place” and “models of space,” we turn now to explicitly modeling spatial processes.

Building a “Model of Space”: Spatial Econometric Models12

As mentioned previously, our model of space is based on a spatial lag approach. The underlying spatial relationship is captured by the spatial lag term for the models reported in panel B of Table 2. This “spatial lag” can be interpreted as the weighted effect of violent crime in adjacent counties on the level of violent crime within a particular county. Fit statistics revealed that these spatial regression models significantly improved upon our baseline SUR models.13

Key findings are as follows. First, we find that “space” matters above and beyond characteristics of “place” for both whites and blacks. The spatial lag term is statistically significant and positively associated with violent crime for both race groups after controlling for other important macro-structural traits. That is, proximity to counties with high rates of white and black violent crime corresponds to higher white and black rates of violence, respectively. Second, the spatial lag term is significantly stronger for whites than it is for blacks (roughly 1.4 times larger), indicating that spatial processes impact white violent crime to a greater extent than black violent crime. A z-test (see the final column of Table 2) confirms that the association of spatial dependence with violent crime is significantly different across race groups (z = 2.481, p < .05).

Third, as in our baseline SUR models, we find that structural disadvantage is positively associated with both white and black violent crime, net of other important structural traits. However, the effect varies across groups—it remains stronger for whites than for blacks—and this difference is statistically significant (z = 2.772, p < .01). That is, accounting for “space” does not attenuate the difference in association between structural disadvantage and violent crime across race groups observed in our traditional SUR models of “place.” Overall, where we find differences across race-groups in our baseline models for structural disadvantage and other structural traits, these differences remain. Moreover, “space” itself differs in its association across groups, as well. Thus, accounting for “space” yields similar conclusions as models accounting only for characteristics of “place.”

Spatial Variability in Relationships: Geographically Weighted Regression

All of the models discussed thus far yield “global” effects—the coefficients reported in each model are means of relationships that are assumed to be constant across geographic units. The final step in our analysis uses geographically weighted regression (GWR) to test the assumption of stationarity (spatial homogeneity) by examining whether the associations of our county characteristics with violent crime differ across our study area. Table 3 reports the quartile distribution of the estimates for each structural trait for whites (columns 1–5) and blacks (columns 7–11), as well as the significance tests for overall spatial variability and the direction (in parentheses) in which statistically significant effects are observed (column 6 for whites, column 12 for blacks). In essence, Table 3 displays the distribution of “local effects” of each structural trait across the sample of contiguous counties for both whites and blacks, indicating whether those effects vary significantly across space, whether they have either negative or positive effects on violent crime, and whether these effects are statistically significant.14
Table 3

Quartiles and significance tests of spatial variability in the within-race effects of key county-level structural traits on white and black violent crime (N = 1,315)

 

White

Black

Min

Q1

Median

Q3

Max

Sig. (dir.)

Min

Q1

Median

Q3

Max

Sig. (dir.)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Structural disadvantagea,b

−.022

.200

.271

.375

.544

* (+)

−.052

.103

.167

.226

.645

*** (+)

Residential instabilitya

−.028

−.006

.007

.017

.061

*** (+/−)

−.028

−.012

−.003

.005

.029

*** (+/−)

Entropy

−.994

−.407

−.177

.176

1.159

* (+/−)

−2.216

−1.376

−1.196

−.977

−.646

** (+/−)

% Hispanic

−.098

−.001

.012

.031

.131

*** (+/−)

−.067

−.007

.005

.035

.094

*** (+/−)

Male pop. 15–24a

−.220

−.095

−.054

−.030

.141

* (+/−)

−.059

−.023

−.015

.004

.028

* (−)

Police (per 1,000)

−.035

−.006

.003

.011

.035

*** (+/−)

−.029

−.003

.002

.009

.030

*** (+/−)

Population structureb

−.573

−.293

−.175

−.094

.129

*** (−)

−.364

−.257

−.208

−.109

.095

*** (−)

Constant

3.222

3.892

4.168

4.519

5.606

*** (+)

5.082

6.005

6.197

6.524

6.973

* (+)

p < .05; ** p < .01; *** p < .001 (two-tailed)

Significance tests indicate statistically significant within-race spatial variation in effects and are based on the Monte Carlo test; Q1 and Q3 represent lower and upper quartile boundaries, respectively

aRace-specific measure

bCombines multiple structural traits using principal component analysis

The key finding from Table 3 is that, for both whites and blacks, all of our key structural traits vary significantly across the study area in their associations with violent crime. Of the 14 county-level traits examined (7 for whites, 7 for blacks), all display statistically significant variation across our sample of counties and 9 of these have both positive and negative statistically significant effects in various geographic units. Turning to the focus of our study, structural disadvantage varies significantly across space in its impact on both white and black violent crime—that is, the positive association between disadvantage and crime varies significantly across counties. Overall then, Table 3 suggests that not only do key structural factors affect violent crime differently for whites and blacks (as shown in panel B of Table 2), but that these structural traits also vary across space for each race group.

Perhaps the most valuable facet of this analysis, geographically weighted regression also allows us to examine where spatial variability in these relationships is observed. Given our focus on the relationship between structural disadvantage and violent crime, Fig. 2 displays the t values showing statistically significant variation across space in the association between white structural disadvantage and white violent crime (panel a) and black structural disadvantage and black violent crime (panel b), net of all other measures (parameter maps available from authors). Our focus here is on the patterns of spatial variation (i.e., where the relationships between disadvantage and violence cluster in space) unique to the race-specific relationships between white and black structural disadvantage and white and black violent crime, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs10940-011-9162-6/MediaObjects/10940_2011_9162_Fig2_HTML.gif
Fig. 2

Spatial variability in the effect of a white structural disadvantage on white violent crime and b black structural disadvantage on black violent crime (T values)

Panel a of Fig. 2 shows the t values for the white structural disadvantage-white violent crime relationship, net of other important county-level traits. The obvious pattern, as indicated by Table 3, is that there are large, distinct clusters of counties where white structural disadvantage has a statistically significant, positive association with white violent crime, with most other counties having a non-significant association (net of other traits). We find that white structural disadvantage is positively associated with white violent crime in areas all along the western coast of the United States, eastern Texas through southwestern Louisiana, Mississippi, Alabama, Georgia, the Carolinas, Virginia, and the mid-Atlantic.

As a means of comparison, panel b of Fig. 2 displays the t values for the relationship between black structural disadvantage and black violent crime, net of other county characteristics. The pattern is one of concentrated, statistically significant positive effects of black structural disadvantage on black violent crime. Though many of the counties in our sample show no statistically significant black structural disadvantage-crime relationship (net of other measured structural traits), we find evidence of statistically significant positive “local effects” along the Pacific coast, the southwestern US, eastern Texas, southeastern Louisiana and southern Mississippi and Alabama, and from North Carolina north through the New England states.

Perhaps more importantly, when taken together, panels a and b of Fig. 2 suggest that drawing conclusions as to whether a macro-structural predictor of crime (e.g., structural disadvantage) is invariant/variant across racial groups is sensitive to where one makes comparisons. For example, looking only at counties in New England—where the disadvantage effect on violence is significant and strong for blacks but non-significant for whites—one would likely find evidence of racial variation (i.e., disadvantage affecting black violence but not white violence). In contrast, if an analytic sample included only the Pacific coast or Texas, the likelihood of observing positive disadvantage-violence relationships would be greater for both whites and blacks and, by extension, there would be less of a difference between groups for this relationship. Overall then, concluding that structural characteristics exert racially invariant/variant effects ignores substantial differences across space—in some places it may be invariant, while in others it may be significantly different.

Examining the Robustness of Our Findings: Alternative Outcomes and Data Sources

While we rely on previous research to guide our study, our main analysis has two shortcomings that warrant consideration. First, arrest data are subject to well-known criticisms, including bias from under-reporting and differences across jurisdictions in the use of police discretion in charging (Mosher et al. 2010). As such, our dependent variable may not accurately reflect violent offending and, as a result, our result may not hold for (a) more reliably reported crimes (e.g., homicide) or (b) race-specific patterns of victimization. Second, because much of the racial invariance research has focused on neighborhoods and cities, counties may be too expansive to examine the spatial processes (and spatial heterogeneity) that may operate on crime among smaller units (Baller et al. 2001). In other words, we may have a mismatch between our unit of analysis at the theoretical level and our unit of analysis at the analytical level, and it is possible that the racial differences we observe among key structural predictors may not be observed among smaller units of analysis.

In order to demonstrate the robustness of our findings, we replicated our analyses in several ways. First, rather than using an overall measure of violence, we constructed models and assessed patterns of spatial heterogeneity using white and black homicidearrest rates (from the UCR) and homicide death rates (constructed from the National Vital Statistics at Center for Disease Control and Prevention 2000).15 Second, we constructed models and explored spatial heterogeneity using tract-level homicide victimization data from Chicago using the Homicides in Chicago dataset (Block and Block 1992; see also Mears and Bhati 2006; Velez 2006). We use Chicago because it is the site of many previous examinations of the racial invariance hypothesis and because it represents one of the few sources of race-specific crime statistics available at the tract level. Moreover, it serves as a means of comparison for our primary models at the larger county-level and provides a powerful test of whether our results hold at a smaller unit of analysis.

Table 4 presents the results from our supplemental models using homicide as an outcome. Here, we have substituted (1) white and black homicide arrest rates and (2) white and black homicide death rates for our county-level white and black violent arrest rates. Given our focus on structural disadvantage as the focal predictor in prior racial invariance research, we present only the coefficients for disadvantage and our spatial lag measures (though all models include a full set of controls).
Table 4

Comparisons of key coefficients and tests for differences between whites and blacks for models regressing (1) homicide arrest rates and (2) CDC vital statistics homicide death rates on county-level structural traits

 

(A) “Place” model (SUR)

(B) “Space” model (spatial lag)

Whites

Blacks

Z test for diff.

Whites

Blacks

Z test for diff.

b

B

b

B

(1) Homicide arrest ratea

 Structural disadvantage

.133***

(.018)

−.006

(.019)

5.311***

.156***

(.019)

.016

(.020)

5.075***

 Spatial lag

 

.170***

(.025)

.087***

(.021)

2.542*

(2) Homicide death ratea

 Structural disadvantage

.124***

(.016)

.006

(.020)

4.607***

.153***

(.017)

.025

(.022)

4.604***

 Spatial lag

 

.201***

(.027)

.087***

(.025)

3.098**

p < .05; ** p < .01; *** p < .001

Each model is estimated with a full set of controls

aThough currently unable to be estimated using geo-spatial software, alternative “place” models were run using counts of homicide arrests and deaths (from CDC Vital Statistics). Given the count nature of these dependent variables and significant evidence of over-dispersion, we employed negative binomial regression to estimate these models. Substantive results are identical

Key findings are as follows. First, for both of the models of “place,” structural disadvantage is significantly and positively associated with white but not black homicide rates. Z-tests for differences confirm that this effect for whites is significantly different than the null black effect in both instances (p < .001 for both tests). Second, “space” matters for both whites and black and for both homicide arrest and homicide death rates. However, the effect for whites is stronger than for blacks in both supplemental models and tests for differences confirm this (p < .01 for both tests). Finally, the effect of structural disadvantage on white homicide arrest rates and homicide death rates does not disappear once we account for space. As in our primary models, the impact of disadvantage on white crime remains even after controlling for spatial dependence as does the difference between whites and blacks.

Given space constraints, we do not to display the tables for the geographically weighted regression analysis. However, these results (available from authors) overlap almost identically with our analysis of overall violence shown above. For both whites and blacks, structural disadvantage (and almost all of our other structural characteristics) varies significantly across space (p < .001) in its association with both homicide arrest rates and homicide death rates. Moreover, the clusters of “local effects” for structural disadvantage on both homicide arrests and deaths vary across race groups—that is, there are a number of counties where disadvantage is associated with increased white homicide but not black homicide and vice versa. These patterns are substantively consistent with our previous results of spatial heterogeneity among predictors of overall violent arrests.

We turn now to a robustness check of our findings using race-specific tract-level homicide data from Chicago. Table 5 displays the results from “place” and “space” models regressing structural characteristics on white and black tract-level homicide rates.16
Table 5

Comparison of models regressing white and black homicide rates on Chicago census tract-level structural traits

 

(A) “Place” model (SUR)

(B) “Space” models (spatial lag)

Whites

Blacks

Z test for diff.

Whites

Blacks

Z test for diff.

b

b

b

b

Constant

−.005

(.149)

1.485***

(.081)

 

.006

(.130)

1.133***

(.113)

 

Structural disadvantagea

.761***

(.084)

.302***

(.030)

5.146***

.580***

(.076)

.242***

(.031)

4.118***

Residential instability

.010**

(.003)

.016***

(.002)

−1.664*

.008**

(.003)

.014***

(.002)

−1.664

Entropy

.542***

(.138)

−.733***

(.093)

7.662***

.269*

(.121)

−.667***

(.091)

6.182***

% Hispanic

−.005*

(.002)

.008***

(.001)

−5.814***

−.005**

(.002)

.007***

(.001)

−5.367***

Male pop. 15–24a

.006

(.008)

−.013**

(.005)

2.014*

.002

(.007)

−.013**

(.005)

1.744

Pop. density

−.038***

(.010)

−.021***

(.005)

−1.521

−.035***

(.008)

−.021***

(.005)

−1.484

Spatial lag

.504***

(.044)

.241***

(.049)

3.994***

R2

.222

.318

 

.403

.359

 

AIC

1,577.639

848.776

 

1,462.340

822.912

 

† p < .10; * p < .05; ** p < .01; *** p < .001 (two-tailed)

Standard errors in parentheses

aRace-specific measure

Key findings are as follows. First, in our models of “place,” structural disadvantage is significantly and positively associated with white and black homicide rates, particularly for whites. A test for difference confirms that this strong effect for whites is significantly different than the weaker black effect (p < .001). Second, “space” matters for both whites and blacks, but the effect for whites is stronger than for blacks as confirmed by the a formal test for difference across groups (p < .001). Finally, the effect of structural disadvantage on both white and black homicide rates does not disappear once we account for space—thus, as with all of our previous analyses, our picture of racial invariance does not change after accounting for spatial processes. This analysis provides a powerful robustness test for our main results, and highlights that our substantive conclusions are not heavily dependent on the unit of analysis employed (or, again, on the dependent variable).

However, we note that while the relationships between disadvantage, spatial proximity, and violent crime are similar between our tract and county analyses, not all effects are identical. For example, the effects for percent young males and percent Hispanic vary (in either direction or significance) depending on the unit of analysis. While a full explication of these differences is beyond the scope of this article, it is possible that the relationships between certain measures and violent crime are not scale invariant in that micro-social processes may exhibit interactive and non-linear affects depending on the larger macro-social context in which they are imbedded (see Krivo et al. 2009; Hipp 2011 for examples). How these types of interactions inform the racial invariance thesis, however, awaits future research.

Geographically weighted regression analyses (available upon request) reveal remarkably similar patterns of spatial variation as those noted for structural characteristics on violence and homicide at the county-level. For both whites and blacks, structural disadvantage (and most of our other structural characteristics) varies significantly across Chicago tracts (p < .001) in its association with homicide rates. While structural disadvantage has statistically significant positive associations with white and black homicide rates, the clustering of local effects varies across race groups. Similar to our main and other supplemental analyses at the county-level, there are a number of tracts where disadvantage is associated with increased white homicide but not black homicide and vice versa. These patterns are substantively consistent with our findings of spatial heterogeneity among predictors of broader violent arrests and with our supplemental models of county-level homicide arrest rates and homicide death rates.

In sum, supplemental analyses utilizing (1) county-level homicide arrest rates, (2) county-level homicide death (victimization) rates, and (3) tract-level homicide victimization data all reveal substantively identical results as those from our analysis of white and black violent index crime at the county-level. Thus, our results are robust to both different dependent variables and different units of analysis. As such, we are confident that our findings that “space” matters (and more so for whites than blacks), that accounting for space does not alter the racial invariance picture, and that there is significant spatial heterogeneity in associations between structural disadvantage and crime, are substantive and meaningful.

Discussion

Broadly, the goals of this study have been to link the upsurge in macro-social studies exploring the racial invariance hypothesis and the growing interest in examining how broader spatial processes affect crime. Critical insights from research on space and crime have yet to be incorporated into racial invariance literature, including that conditions in a particular geographic unit often affect conditions in those surrounding it and that relationships between structural covariates and crime can vary across space.

Moreover, prominent scholars have called for research to move beyond traditional analytic models to explore more than “mean effects” (Maltz 1994) and to incorporate a spatial framework into studies of stratification and inequality (Messner et al. 1999; Sampson and Morenoff 2004; Lobao et al. 2007). Overall, this study builds upon prior research in the macro-structural criminological tradition by applying relatively new analytic techniques to understand how spatial processes affect violent crime above and beyond the typically included measures of “place,” as well as explore the importance of looking at local predictors of crime.

Specifically, this paper examined (1) whether the dynamics of space matter above and beyond typical measures of structural disadvantage and other key macro-structural traits, (2) whether “space” affects crime differently for whites and blacks, (3) whether models accounting for spatial processes yield comparable findings to models which account only for characteristics of discrete geographic units (i.e., characteristics of “place”), and (4) how the effects of structural predictors of crime vary across space for particular racial groups. Our findings indicate, first, that white and black violent crime is indeed spatially patterned (i.e., not randomly distributed) and that traditional models used to assess racial invariance fail to capture this spatial dependence. As our baseline models and subsequent diagnostics reveal, traditional racial invariance models capture characteristics of “place” but fail to leverage the dynamics of “space.”

Second, spatial dependence is an important predictor of white and black violent crime, net of other structural traits, and this association varies across race groups. In our models, proximity to counties with high rates of white and black violent crime corresponds to higher white and black violent crime rates, and this relationship is significantly stronger for whites than for blacks. While more research is needed to determine why spatial proximity to high violence counties and neighborhoods appears to more strongly predict white violent crime than black violent crime, we think previous research on the ecological context of white and black communities may be informative here (Krivo and Peterson 2000; McNulty 2001). As prior work has shown and consistent with our descriptive statistics, black communities are characterized by exceptionally high levels of violent crime and disadvantage compared to white ones. It could be the case that black communities are less sensitive to surrounding concentrations of violence due to the high level of disadvantage and violent crime within the community. In other words, the spatial processes of disadvantage and crime may differ markedly depending on the corresponding levels of these measures within a community. If this is indeed the case, our results suggest that communities with lower levels of crime and disadvantage are most sensitive to higher rates of violence in surrounding areas. While testing this hypothesis is beyond the scope of the current article, we think this is a fruitful avenue for future research.

Third, regarding the racial invariance thesis, our findings suggest that, at both the county and tract-level, modeling “space” yields similar findings as models accounting only for characteristics of “place.” That is, accounting for spatial dependence does not diminish racial variation in the structural disadvantage-violent crime association and we find that differences across race-groups in associations between other structural traits and crime observed in traditional invariance models remained even after accounting for the spatial processes impacting geographically proximate counties.

Fourth, all of our structural predictors vary significantly across space in their associations with violent crime (i.e., are spatially heterogeneous), suggesting that “global” parameters derived from traditional models (OLS, SUR, etc.) overlook important differences in local conditions affecting white and black violent crime. Additionally, many of our structural predictors of violent crime exhibit unique patterns ofvariation across space for each racial group. That is, we observe counties or groups of counties (as well as tracts within Chicago) where structural covariates (a) matter only for one racial group but not the other or (b) impact violent crime in opposite ways for whites and blacks. This suggests that drawing conclusions as to whether a particular structural predictor of violent crime is racially invariant/variant depends on where one looks.

Taken together, our study is more in line with research suggesting that the racial invariance debate remains unresolved (Ousey 1999; Parker 2008; Phillips 2002; Steffensmeier et al. 2010). Across both our county and tract level analyses, we found considerable variation in effects for white and black violence of key macro-structural predictors and “space” itself. Additionally and perhaps the most unique contribution of this study, we found that the pattern of effects showed considerable spatial variation depending on the race group. In merging the literature on spatial analyses and the racial invariance hypothesis, we have revealed another level of complexity to the study of social structure and racial differences in violence that future research should incorporate, both empirically and theoretically.

We note that our study has several important limitations. We rely on official UCR crime statistics for our main analysis, which only include rates of white and black violent crime. One of the major shortcomings of UCR statistics is the inability to disaggregate rates by ethnicity (e.g., non-Hispanic white, non-Hispanic black, etc.). However, with Hispanic rates of violence falling somewhere between those of whites and blacks, if anything, our results here are more likely to be biased in the direction of showing greater similarity between whites and blacks because Hispanic arrests are more likely to be counted as “white,” as opposed to “black” or “other” (Harris et al. 2009). Still, future research should explore spatial processes across additional race/ethnic groups, with particular emphasis on Hispanics.

Future research should also examine the nature of how structural characteristics interact with the spatial geography of counties and cities to produce varying effects across geographic space. Above all else, such investigations require significant theoretical accounts to explain why certain structural features of distinct places produce varied effects depending on their surrounding community and geographic location. In line with this view, more research is needed to explicate the mechanisms through which geographic contiguity impacts patterns of violent crime. While our study highlights the importance of geographic contiguity in explaining both white and black rates of violence, we are unable to determine, with the available data, the mechanisms through which these patterns operate.

With these caveats in mind, we argue that understanding the spatial dynamics of criminogenic processes is important to advancing our knowledge of the structural predictors of crime. In line with this perspective, we think that GWR can be a useful diagnostic tool to help understand community-level variation in the effects of structural predictors. Rather than identifying regions or communities a priori, our approach estimates spatially moving clusters of structural variation (Graif and Sampson 2009). Using this method we have identified certain clusters of counties in the US and neighborhoods within Chicago where disadvantage has varied effects on violence. Thus, our study helps answer one of the more “curious anomalies” in race-specific studies discussed by Messner and Rosenfeld (1999: 37) who noted that “poverty emerges as a major determinant of [violence] for Whites but not Blacks.” Our results suggest that depending on where one looks, this statement can be true or false. The critical next step is to try and explain why we observe these patterns of results in these particular places. Future research should attempt to explain this observed spatial variability and ask what contextual factors might explain why certain structural characteristics would matter more or less in some places and not others.

Footnotes
1

For example, of the 3,141 counties in the United States, only 1,906 have at least 200 blacks living in them.

 
2

We utilize the maximum number of counties while taking into account (1) the under-reporting (or the absence of reporting) of race-specific crimes in the UCR’s county-level database, (2) the reliability of UCR crime counts that were provided, (3) the necessity of including counties with a reasonably reliable number of both whites and blacks, and (4) our spatial requirement of having relatively contiguous counties. As such, we arrived at our sample of 1,315 counties after imposing what we see as reasonable selection criteria. It should be noted, however, that we estimated all of our models on a smaller sample of counties which used a more stringent constraint on our spatial requirements and limited our sample to only 1,114 counties, which largely excluded the West and South-West counties. Our substantive results were identical using this sample.

 
3

We thank an anonymous reviewer for this helpful suggestion.

 
4

At the county-level, these three measures are highly correlated and load well on one factor using principle component analysis (all factor loadings above .88). In supplemental analysis we ran our SUR models using each measure independently and found substantively similar results.

 
5

Our entropy measure is calculated for each county (Reardon and Firebaugh 2002). The entropy index (E) is a measure of the diversity of a geographic area, calculated as:

\( \mathop E\limits_{m = 1}^{M} = \sum {\pi_{m} \ln (1/\pi_{m} )} \, \)

where, πm is the proportion of people in race m (e.g., proportion black) and M is the total number of racial groups (here, white and black). E has a minimum value of 0 when a census place has no diversity and is composed entirely of one racial group and a maximum value of 1 when groups are equally represented. Racial heterogeneity scores were divided by their maximum values (1.099) to impose a range of 0 to 1 for E.

 
6

We recognize that police per capita is an imperfect measure of police activity and may capture the effect of police force size (i.e., larger police forces) rather than the likelihood of arrest. To verify the robustness of our findings we estimated all of our models using an alternative measure, clearance ratios (arrests made divided by offenses known). Our substantive results were identical to those reported in the tables (available from authors). We thank an anonymous reviewer for this suggestion.

 
7

Given its importance in research on the spatial analysis of crime (Baller et al. 2001), we (a) ran supplemental baseline SUR models with a “South” dummy variable and (b) conducted Lagrange Multiplier tests to determine if the spatial relationship (spatial error vs. spatial lag) differed in Southern counties than in non-Southern counties for our exploratory spatial analyses. Results suggest that there is little meaningful difference between South and non-South counties—a “South” dummy variable has trivial or non-significant effects in our baseline SUR models and the functional form of the spatial relationship (i.e., lag vs. error) is the same in Southern counties as in non-Southern counties. Therefore, we omit the South/non-South distinction in our final models.

 
8

Using a spatial weights matrix, Moran’s I statistic assesses the extent to which the pattern of values are spatially random. Rejection of this null hypothesis indicates significant spatial clustering.

 
9

The spatial weights matrix used in all spatial analyses is the queen’s 1st order, where each county is weighted by the values of all of its direct contiguous neighbors. Alternative weights matrices, including the rook’s 1st order and 10 nearest neighbors showed similar results with the latter displaying slightly less spatial autocorrelation. For example, the Moran’s I statistic for the white violence index rate was .34, .34, and .30 using the queen’s 1st order, the rook 1st order, and the 10 nearest neighbors weights matrices, respectively. The corresponding statistics for the black violence index were .23, .23, and .18, respectively. We utilize the queen’s 1st order matrix for two reasons. First, there are several areas where counties have few contiguous neighboring counties that have complete data (specifically in western Texas and Northern Kentucky). As a result, the nearest neighbor criterion weights the effects of counties that, while closest to that county, are in fact a significant distance away and are not theoretically likely to exert a strong influence on that “neighbor.” Moreover, specifying the number of neighbors is an arbitrary choice driven by neither contiguity concerns nor theory. Second, previous research has shown that contiguous counties are likely to have larger influences than non-contiguous counties (Baller et al. 2001).

 
10

While spatial dependence can be treated as a “nuisance” by constructing spatial error models, we chose to construct spatial lag models because (1) supplemental analysis reveals similar substantive results and model fit statistics with spatial error models and (2) choosing the appropriate modeling procedure should be based on theoretical considerations (Ward and Gleditsch 2008), and we conceptualize spatial dependence as substantively meaningful. Because lag models are more theoretically interpretable and appropriate for the hypothesized relationships between crime in one county affecting crime in another, we utilize a spatial lag approach.

 
11

We note that our sample of counties is less urban than the United States as a whole (50.3% compared to 68.4%), which may affect our results. However, our series of robustness tests using different dependent variables and different units of analysis lend confidence that our substantive conclusions regarding disadvantage, spatial proximity, and crime are meaningful.

 
12

Spatial econometrics utilizes two general models, spatial error and spatial lag models. Briefly, the spatial error model evaluates the extent to which the clustering of violence rates not explained by measured independent variables can be accounted for with reference to the clustering of error terms. Thus, it captures the spatial influence of unmeasured independent variables (Baller et al. 2001) and treats spatial dependence as a “nuisance.” In contrast, the spatial lag model incorporates the spatial influence of unmeasured independent variables as well as the effect of neighboring units’ dependent variable (i.e., the lagged dependent variable). The spatial lag model is appropriate when crime in one county is directly influenced by crime in that county’s “neighbors” above and beyond other covariates specific to that county. This model is most compatible with notions of “spillover” processes, implying the influence of neighbors’ violence rates independent of measured or unmeasured independent variables (see in Baller et al. 2001 for visual representation of these different models).

 
13

Since our baseline SUR model produces only a single AIC (Akaike Information Criterion) value and our spatial lag models produce two, we computed AIC values for baseline OLS models for model-fit comparisons. Findings suggest that spatial lag models (white AIC = 2,873.7, black AIC = 3,045.7) are a better fit than baseline models (white AIC = 3,007.5, black AIC = 3,116.4).

 
14

While these models can assess whether the effects of county level characteristics are spatially invariant for white and black models, they cannot test whether this spatial variability is significantly different between models. In other words, while parameter estimates may vary across geographic space in each model, we have no formal way of testing whether these “local parameters” are significantly different between whites and blacks.

 
15

We used vital statistics data on homicide deaths for ages 17 and up because the vital statistics data does not provide an age category with a threshold at age 18 which would allow us to separate out adults (i.e., 18 and up). Additionally, we averaged the number of homicide deaths across the years 1999–2005 in order to leverage a greater number of rare homicides and to avoid suppression of the data for small counties which the CDC employs to protect the anonymity of victims which cannot be assured even without personal identifiers.

 
16

The dependent variable for this analysis is the race-specific rate of homicide victimization in each tract for the years 1985–1995. These years were used because the dataset only covers homicides from 1965 to 1995 and to limit the number of tracts that had no homicides in any 1 year. All independent variables were taken from the 1990 census. While we tried to utilize the same measures as our main analysis, certain information was unavailable. For example, the census did not collect race-specific mobility information at the tract level in 1990. Because of this, our measure of residential instability in the Chicago supplement is a total (i.e. not race-specific) measure of instability. Also, because all of Chicago is urban, we used a measure of population density as opposed to our population structure variable. Finally, clearance ratios and police per capita information were not available at the tract level. All other measures are identical to those used in our main analysis. The sample is limited to tracts that had had at least 100 whites for the white models (N = 568) and 100 blacks for the black models (N = 540).

 

Acknowledgments

We would like to thank Darrell Steffensmeier, Stephen Matthews, Jeffrey Ulmer, Michael Massoglia, Luke Bonkiewicz and three anonymous reviewers for their helpful comments.

Copyright information

© Springer Science+Business Media, LLC 2012