Income Inequality and Homicide in the United States

Abstract

Choices of inequality measure and homicide type may account for mixed findings on the income inequality–homicide link. We aim to acquaint criminologists with several income inequality measures beyond the familiar Gini index and apply the different measures to general and specific homicide rates, noting the practical effect of these choices on results. The income inequality measures differ in their fidelity to relative deprivation ideas, but still correlated highly with each other in data from 208 large U.S. cities. Multivariate analysis also found that all measures of income inequality had significant and positive associations with both overall and specific homicide rates.

Keywords

inequality structural causes measurement income inequality relative deprivation economic homicide

Introduction

The relationship between income inequality and homicide has been extensively studied, and income inequality is a theoretically relevant correlate of crime in contemporary criminology. However, previous findings on the income inequality–homicide link in the United States are mixed (K. F. Parker, McCall, & Land, 1999; Pridemore, 2002). In response, some researchers have turned their focus to racial or gender, rather than overall, income inequality (e.g., Messner, Raffalovich, & McMillan, 2001). Although this focus on gender and race has yielded substantial insights, the mixed results of prior research and overall income inequality’s central role in theoretical perspectives such as relative deprivation still make it important for researchers to consider general income inequality. Specifically, critical examination of the inequality–homicide relationship needs to include exploration of the potential implications of different choices for the operationalization and measurement of both inequality and homicide. Most previous studies of income inequality and homicide in the United States have relied solely on the Gini index, generally neglecting alternative measures of overall income inequality despite their possible advantages over Gini for capturing the aspects of income inequality that may be most helpful in predicting homicide rates. Research in this direction would ask if estimation of the relationship is robust to different measures of income inequality, or if substantive conclusions depend on which measure is used. Another avenue for research is to investigate the possibly different relationships that different homicide types have with income inequality. Perhaps economically motivated homicides, as an extreme means by which individuals respond to unequal distribution of resources, are linked more closely to income inequality than are expressive homicides (Kovandzic, Vieraitis, & Yeisley, 1998; R. N. Parker, 1989).

The current study attempts to familiarize criminologists with a wider range of income inequality measures beyond the familiar Gini index, clarifying the different measures’ definitions and sensitivities. To explore the practical effect of the choice of measure when analyzing a real data set, we carry out a careful case study in which the different measures are applied to both general and specific homicide rates. We begin by reviewing the relative deprivation thesis, as a major theoretical framework for the income inequality–homicide link, as well as previous empirical findings on the relationship between income inequality and homicide in the United States. Then we review the oft-used Gini index in detail and discuss its limitations for addressing the specific dimensions of income inequality that are most relevant to relative deprivation and, in turn, homicide. We next introduce three alternative income inequality measures that might, by better capturing key elements of the relative deprivation argument, hold advantages over the Gini index as a predictor of homicide rates. Those measures include the inequality ratio, the Theil index, and the Atkinson index. The current study also considers an adjusted measure of income inequality that simultaneously addresses income inequality within a city as well as the city’s experience of central city–metropolitan (i.e., the surrounding area outside the central city) inequality.

Our analysis used 2005-2007 homicide data from large U.S. cities to pursue the two research directions above. First we examine similarities and differences among several alternatives to the familiar Gini index, and then consider patterns of statistical significance and direction of estimated effects when using these measures to predict city homicide rates. Second, we examine disaggregated measures of homicide that distinguish economic from non-economic homicides, to see whether these patterns of estimated relationships between income inequality and homicide differ between the two homicide types. We conclude with discussion of the results, limitations, and possibilities for future research.

Theoretical Explanations and Previous Findings on the Income Inequality–Homicide Link

The posited mechanism for the relationship between overall income inequality and homicide usually relies on the idea of relative deprivation, rooted in Marxist theory of capitalist exploitation and Merton’s (1968) social strain theory. According to the relative deprivation perspective, individuals evaluate their socio-economic standing in relation to others, and develop frustration, hostility, and resentment when they realize that they have fewer resources than others (e.g., Blau & Blau, 1982; Messner, 1982; Messner & Tardiff, 1986; Pridemore, 2002). As comparisons to those with more resources are most salient in the development of relative deprivation (e.g., Runciman, 1966), feelings of relative deprivation will be more likely in a society with a highly unequal income distribution, and the position of those in lower income groups means that they are most likely to develop such feelings (e.g., Schur, 1969). In extreme cases, individuals feeling relative deprivation may resort to criminal behavior, including homicide, to “redistribute economic resources” (e.g., through robbery-murders) “and satisfy their sense of injustice” (Fowles & Merva, 1996, p. 166).

Despite this theoretical foundation, empirical findings from multivariate studies on income inequality and homicide using U.S. data are mixed (Pridemore, 2002). Some studies found the expected positive relationship (e.g., Blau & Blau, 1982; Braithwaite, 1979; Fowles & Merva, 1996; McCall, Land, & Parker, 2011), whereas some others found a non-significant relationship (e.g., Bailey, 1984; Messner, 1982, 1983b; Messner & Tardiff, 1986; Williams, 1984). Some have even found a significant negative relationship (Kposowa, Breault, & Harrison, 1995; Messner, 1983a). Critics have pointed out problems related to statistical issues such as multicollinearity in this research (Land, McCall, & Cohen, 1990; K. F. Parker et al., 1999), and another explanation for the inconsistent findings is the use of possibly inappropriate units of analysis (Pridemore, 2002). The relative deprivation perspective is vague about the specific geographic proximity that is relevant for the assessment of economic standing relative to others (e.g., Deaton, 2003; Mangyo & Park, 2011), but researchers have argued that large units such as metropolitan areas and states are too large and heterogeneous (Bailey, 1984; Messner, 1982; Pridemore, 2002), whereas small units like neighborhoods are also inappropriate because they suggest that individuals assess their standing in a comparison of unrealistically limited scope (Messner & Tardiff, 1986). While neighborhood interactions are important, most people also frequently venture beyond their neighborhoods, having contact and sometimes in-depth interactions with individuals from outside. This contact yields awareness of income differences and permits the development of feelings of relative deprivation. For those reasons, cities seem to be preferable to larger or smaller units of analysis when studying the income inequality–homicide link (e.g., Kovandzic et al., 1998; Wadsworth & Kubrin, 2004). Some researchers also argue that the city focus is appropriate because homicide is mostly a central city phenomenon (e.g., Kovandzic et al., 1998; R. N. Parker, 1989).

Other possible explanations for inconsistent research findings are related to the measurement of income inequality and homicide. The Gini index is the most frequently used income inequality measure, but may be inadequate for capturing the elements of income inequality most relevant to relative deprivation (Bailey, 1984; Messner, 1983b; Messner & Tardiff, 1986). Furthermore, treatment of homicide as a monolithic crime category may be another reason for inconsistent findings, as some specific types of homicide might be expected to be more closely related to income inequality than others (Kovandzic et al., 1998; R. N. Parker, 1989; Pridemore, 2002). The current article focuses on these two measurement issues in its examination of income inequality and homicide.

Different Measures of Income Inequality

In this section, we first review the characteristics of the oft-used Gini index and its potential shortcomings in measuring some aspects of income inequality that are likely most important for capturing ideas of relative deprivation. Then we introduce several alternative income inequality measures—the inequality ratio, the Theil index, and the Atkinson index—reviewing their characteristics and potential advantages over the Gini index for representing the relative deprivation perspective. The section also discusses an adjusted measure of income inequality that simultaneously addresses income inequality within a city as well as the city’s experience of central city–metropolitan (outside the central city) inequality.

Gini Index

The Gini index is the most frequently used measure in studies of the relationship between homicide and income inequality. It can be understood through the Lorenz curve’s depiction of the income distribution. Figure 1 shows an example of a Lorenz curve plotting the cumulative proportion of the population (x-axis) and the cumulative proportion of total income (y-axis). The curve is produced by arranging incomes in the population from lowest to highest, so that for a particular x-axis value x the curve shows what proportion of total income is earned by the lowest earning (100 x)% of the population. If all incomes were equal, any (100 x)% of the population would earn (100 x)% of the total income, and so the Lorenz curve would simply be a 45° line. In a more realistic income distribution, the lowest earning (100 x)% will earn less than (100 x)% of the total income, and the Lorenz curve becomes more bowed away from the 45° line as total income is more concentrated among the highest earners. Noting the labels in Figure 1, the Gini index is calculated by the relative size of Area A between the Lorenz curve and the 45° line (compared with the whole area under the 45° line, A + B; Charles-Coll, 2011).

Figure 1.

Graphical presentation of the Gini index.

The Gini index ranges between 0 and 1. If all incomes are equal, the Lorenz curve is in fact the 45° line, so the Area A, and therefore the Gini index, equals 0. At the other extreme, if one individual (or household, depending on how the population is conceived) in a large population earned the entire total income, and all other incomes in the population were zero, the Lorenz curve essentially would be pushed all the way to the right-hand corner of the graph, that is, it would be flat until rising to 1 for the very last income in the population. In this case, Area B is essentially zero if the population is large, so the Gini index takes value 1 in this extreme inequality regime.

The Gini index can also be calculated from grouped income data. For example, Brown (1994) gave the following equation, where i = 1 to K indexes the K income groups from lowest to highest, X_i represents the cumulative proportion of the total income earned by Groups 1 to i, with X₀ = 0 by convention, and P_i represents the proportion of all individuals or households that fall into income group i:

G i n i = 1 - \sum_{i = 0}^{K - 1} (X_{i + 1} + X_{i}) P_{i + 1} .

For the special case in which the income groups are quintiles of the income distribution, P_i = 1 / 5 = .20, for all i.

The Gini index’s wide use probably stems from its easy-to-understand graphical representation and the simplicity of its calculation from grouped income data. Its main disadvantage in the context of crime is its relative insensitivity to changes in the top and bottom part of the income distribution (Allison, 1978; De Maio, 2007; Ellison, 2002). For most realistic income distributions, the size of the area between the 45° line and the Lorenz curve mostly reflects the shape of the middle portion of the Lorenz curve. This makes the Gini index more sensitive to changes in the middle of the income distribution than to changes in the top and bottom (Allison, 1978). This insensitivity to change in the economic standing of bottom income earners relative to others is especially problematic from the relative deprivation perspective. The idea that lower socio-economic status individuals are most prone to relative deprivation means that the nature of the lower end of the income distribution is especially important, and empirical work indicates that homicide offenders and victims are disproportionally drawn from lower socio-economic classes (e.g., Messner & Rosenfeld, 1998; Sampson & Lauritsen, 1994). The Gini index’s insensitivity to the nature of the upper end of the income distribution is also important for relative deprivation, as top earners are part of the larger comparison group of economically better-off people used by low earners to assess their own economic standing. Therefore, different patterns of income distribution among the upper groups will affect others’ conclusions. For these reasons, the Gini index may not be the best income inequality measure for capturing the idea of relative deprivation and predicting homicide (or crime in general). Indeed, some relative deprivation scholars (Messner & Tardiff, 1986) specifically note these shortcomings of the Gini index. Thus, it is important to explore existing alternative income inequality measures that may be preferable to the Gini index as predictors of homicide rates. Here we focus on measures that can be calculated from grouped income data, and examine the inequality ratio of top and bottom income groups, the Theil index, and the Atkinson index.

Inequality Ratio of Top to Bottom Income Groups

The ratio of the share of total income earned by high-income groups to the share earned by low-income groups is a simple measure of inequality. Within this framework, the most common choice is the 20/20 ratio, calculated as the ratio of the proportion of total income that is received by the highest earning 20% of households to the proportion of total income that is received by the lowest earning 20%. Variations such as the 20/40 ratio (using the bottom 40% instead of 20%) sometimes appear (e.g., Fiala & LaFree, 1988), and there are also even simpler measures that do not involve a ratio, such as the proportion of total income earned by the poor, or the proportion of total income earned by the rich (Kovandzic et al., 1998; Fiala & LaFree, 1988). Measures of this sort are easily calculated and can focus on the high income–low income difference. However, these crude measures ignore a good deal of information on the shape of the income distribution. Most obviously, the income inequality ratio does not directly consider the middle portion of the distribution. We noted that Gini’s oversensitivity to the middle (and less sensitivity to the top and bottom) of the income distribution makes it questionable for use in relation to relative deprivation and crime. However, inequality ratio measures’ lack of attention to the nature of the middle portion of the income distribution is also inappropriate, as it is likely that middle-income groups, not only the highest, are part of the comparisons that are important for feelings of relative deprivation among low earners.

Theil Index

The Theil index (Theil, 1967) tries to take advantage of the entire income distribution. It is not directly motivated by the Lorenz curve (though see Rohde, 2008), but instead relies on an analogy between income distributions and entropy in information theory (Conceição and Galbraith [2000] give a brief and especially clear explanation of this analogy). In the context of income distributions, the degree of entropy is the extent to which individuals or households are indistinguishable from each other in terms of their income. Higher entropy indicates higher equality (lower inequality), and is maximized when all incomes are equal. The Theil index is the most-often used member of a larger family of entropy-based measures. It shows how far the entropy in the observed distribution is from the maximum achieved under perfect equality, and therefore is larger when inequality is greater. For grouped data, with Y_i representing the share of total income earned by Group i, and P_i representing Group i’s share of the population, the Theil index may be written as

T = \sum_{i = 1}^{K} Y_{i} \ln (\frac{Y_{i}}{P_{i}}) .

The Theil index therefore involves a comparison of each group’s income and population shares. With perfect equality (corresponding to maximum entropy), Y_i = P_i for each group, so that T = 0. Note that the Theil index can exceed 1; in a population of H households, its maximum value of ln(H) is attained when a group consisting of one household earns all the income. Compared with the Gini index’s focus on the area between the Lorenz curve and the 45° line, or the 20/20 ratio, the entropy basis of the Theil index potentially better considers the whole income distribution (Allison, 1978).

Atkinson Index

Compared with the inequality ratio and the Gini and Theil indices, the Atkinson index (Atkinson, 1970) is notable for its flexibility in assessing income inequality. It uses the entire distribution, but can, through choice of a parameter, focus on the income group of greatest interest for relative deprivation and crime offending. For grouped data (with Y_i and P_i defined as above), Atkinson’s index may be written as follows, highlighting the comparison between income and population shares in the various income groups:

A_{ε} = 1 - {[\sum_{i = 1}^{K} {(\frac{Y_{i}}{P_{i}})}^{1 - ε} P_{i}]}^{\frac{1}{1 - ε}} .

The Atkinson index is motivated by consideration of a “social welfare function” rather than examination of the Lorenz curve, with Parameter ϵ expressing inequality “aversion” in this welfare function (Cowell, 1995). The index is flexible in the sense that different choices of Parameter ϵ change the measure’s focus on different aspects of the income distribution. Higher values of Parameter ϵ make the Atkinson index more responsive to changes in the standing of the lower groups, so that analysts can choose Parameter ϵ with theoretical aims in mind, or examine the impact of different choices for the parameter on research results (Allison, 1978). For studies of crime, researchers would likely choose ϵ > 1, to make the index sensitive to the lower part of the income distribution. In practice, ϵ = 2 or ϵ = 2.5 are reasonable choices of the parameter to produce an Atkinson index with a strong emphasis on the bottom of the income distribution; in analyses below, we used ϵ = 2. Note Cowell’s (1995) caution that, for grouped data, under higher values of Parameter ϵ, the Atkinson index becomes more influenced by the assumed distribution of income within the lowest income group. This means that very high values of Parameter ϵ are probably only sensible when the grouping of incomes provides substantial detail at the low end of the distribution. The ability to focus on the standing of the low-income groups relative to other groups in the income distribution, while still considering the middle-income and upper income groups’ positions in detail, gives the Atkinson index strong substantive relevance in capturing the idea of relative deprivation.

Adjusted Measure Addressing Within-City and Central City-Metropolitan Income Inequality

As presented above, the Gini index and the alternative income inequality measures assess only within-unit income inequality, in which the frame of reference for income comparison is others within the unit. This is appropriate if daily interactions and experiences with others in one’s immediate surroundings are most relevant to a person’s evaluation of his or her economic standing and development of feelings of relative deprivation (Bailey, 1984; Kovandzic et al., 1998; Messner, 1982; Williams, 1984). However, given relative deprivation’s vagueness about the geographical scope of one’s assessment of economic standing (Deaton, 2003; Mangyo & Park, 2011), it seems prudent to consider a variety of reasonable possibilities for the nature and scope of comparisons. Although many researchers have argued that the city is the preferred unit of analysis for studying the income inequality–homicide link, others have pointed out that individuals may also make comparisons in a larger social context beyond their immediate setting (A. Wilson, Hoshino-Browne, & Ross, 2001). This may be increasingly true as modern mass media and the ease of distant travel make it much easier to learn about the incomes and lifestyles of others outside of one’s immediate surroundings (Kovandzic et al., 1998; LaFree, 1998; Messner & Tardiff, 1986; Roberts & LaFree, 2004). In contemporary American cities, it is likely that most of the daily activities of inner city residents take place within the city, and such residents may have few personal interactions with commuters from wealthier suburbs. But inner city residents are still exposed to the relatively greater income of such commuters, whose visible symbols of wealth, such as clothes and vehicles, could amplify city residents’ feelings of relative deprivation that were already induced by within-city inequality. However, exposures that suggest that one’s city is better off than nearby places may attenuate the feelings of relative deprivation stemming from within-city inequality (Roberts & LaFree, 2004).¹

Roberts and LaFree (2004) developed a single measure to capture this amplification and attenuation of relative deprivation resulting from comparisons with a geographically broader social context. Their primary units were Japanese prefectures (analogous to cities in the current study), and the wider comparison was with the national standard of living (here we use the cities’ surrounding metropolitan area). They first calculated a ratio of national per capita income to prefectural per capita income as an indicator of a prefecture’s standing in a wider comparison with the whole country. Then they multiplied the within-prefecture Gini index by the square root of the national/prefectural income ratio to obtain an adjusted measure capturing within-prefecture as well as prefectural-national inequality. The within-prefecture Gini index was thus reduced or increased depending on the prefecture’s standing relative to the national average, so that the adjusted measure may better reflect inequality as perceived by a prefecture’s residents. Roberts and LaFree’s (2004) calculation was for the Gini index, but the adjustment could also be applied to other inequality measures. Applying this idea to large American cities reflects the intuition that both within-city and central city–metropolitan area inequality may be relevant to relative deprivation, and thus homicide.

Summary

The previous sections discussed the different income inequality measures and their potential advantages and disadvantages for capturing the idea of relative deprivation. Here we briefly summarize our main points about those measures before moving on to the next section. The relative deprivation perspective suggests that individuals in lower socio-economic positions are most likely to develop relative deprivation (which in turn could lead to criminal offending) when comparing themselves with economically better-off others (e.g., Runciman, 1966; Schur, 1969). These points suggest that to faithfully represent the idea of relative deprivation, an income inequality measure should reflect changes in the lowest income groups’ economic standing relative to others in the income distribution, but at the same time still consider data on economically better off groups (middle and top income groups) too, as the objects of potentially important comparisons made by the lower groups. This reasoning makes the Gini index and inequality ratio less suitable for representing ideas of relative deprivation, due to insensitivity to the bottom and top income groups (Gini index) and lack of direct attention to middle-income groups (inequality ratio).

The Theil and Atkinson indices seem to more faithfully convey ideas of relative deprivation, because they can consider the entire income distribution, so as to better reflect change in lower socio-economic groups’ situation relative to economically better-off comparison groups. The Atkinson index is particularly notable for its flexibility in emphasizing the lower groups’ standing in a measure that still addresses the entire distribution. Relative deprivation is mostly silent on the geographical scope of comparisons (e.g., Deaton, 2003; Mangyo & Park, 2011), but the theory’s principles are consistent with the adjustment of city income inequality measures for city–metro inequality, with this external comparison amplifying or attenuating feelings of relative deprivation indicated by within-city inequality measures (Roberts & LaFree, 2004).

Most previous studies on the income inequality–homicide link used the Gini index, but Kovandzic et al. (1998) compared different measures, including the Gini index, the inequality ratio, and the share of income received by the top 20% of families. They found high correlations among those three measures, and similarly significant and positive effects on homicide rates. Fowles and Merva (1996) also examined the Gini index and some less common wage inequality measures (relying on individual wage data in samples from 28 metropolitan areas) that are responsive to different parts of the Lorenz curve. They found that the Gini index and a different measure focusing on high incomes showed consistent positive effects on homicide across a variety of model specifications. The limited existing research on the income inequality–homicide relationship under different income inequality measures calls for more exploration of this topic.

Income Inequality and Specific Types of Homicide

Some researchers argue that the inconsistent findings of previous income inequality–homicide studies are due to an inappropriately unidimensional view of homicide, because not all types of homicides are equally apt to be associated with income inequality. Rates of economically motivated homicide such as robbery-murders could be more directly linked to income inequality than rates of other, more expressive types of homicide, with economically motivated homicides being an extreme response to unequal distributions of money and material goods (Bailey, 1984; Kovandzic et al., 1998; R. N. Parker, 1989). However, there are also arguments that income inequality has as strong or stronger an association with rates of non-economic homicide. Resentment and hostility flowing from relative deprivation could be channeled into expressive violence against readily available targets such as intimates, family, and friends (Danziger & Wheeler, 1975; Messner & Tardiff, 1986).

R. N. Parker’s (1989) analysis of large U.S. cities from 1973 to 1975 found that robbery-homicides were not significantly related to income inequality measured by the Gini index. On the other hand, Kovandzic et al. (1998) found partial support for an association between income inequality and economic homicide, as the Gini index was positively associated with stranger homicides, a large proportion of which are economically motivated. These differing views on the importance of inequality for specific types of homicides, and differing empirical findings, motivate our inquiry into income inequality and disaggregated homicide rates, using the Gini index and the other income inequality measures.

The Current Study

Using recent data from 208 large U.S. cities, the current study examines the income inequality–homicide link using various measures of income inequality. Along with the familiar Gini index, measures include the income inequality ratio of top and bottom income groups (20/20 ratio), the Theil and Atkinson indices, and adjusted Gini, Theil, and Atkinson indices capturing both within-city and central city–metropolitan income inequality. To explore contrasting views on the relationship between income inequality and rates of specific types of homicide (Danziger & Wheeler, 1975; Kovandzic et al., 1998; Messner & Tardiff, 1986; R. N. Parker, 1989), the current study also uses disaggregated homicide data distinguishing economic and non-economic homicides. After describing the relationship among the different income inequality measures in these data, we explore a series of multivariate models for city homicide rates. Income inequality and homicide measures differ across models, whereas model specifications otherwise remain the same. Our goal is to observe whether the patterns of statistical significance and direction of estimated coefficients for the income inequality–homicide relationship change under different income inequality measures, or for different homicide types.

Data and Methods

Data

We used the most recent 3-year (2005-2007) period from Fox and Swatt’s (2007) imputed Supplementary Homicide Reports (SHR) to obtain city-level data on economic and non-economic homicide. Fox and Swatt’s SHR data addressed missing data through multiple imputation; full details on the imputation procedure are in Fox and Swatt’s (2007, 2009) works. They provide five data sets in which missing information on homicide circumstance has been imputed, and we used this circumstance information to count each city’s economic and non-economic homicides; because different circumstance information could be imputed for an incident in the five different data sets, a city’s count of economic and non-economic can vary across the five data sets.² We linked the resulting five city-level homicide data sets to 2006 American Community Survey (ACS) data, which provided the income distribution information used to calculate the different city-level measures of income inequality and a variety of control variables. Total homicide counts were also obtained from Fox and Swatt’s (2007) data set. The city-level count of total homicides does not rely on any imputation, so for total homicides, only one data set was linked to ACS. We restricted our sample to large cities whose 2005-2007 average populations were greater than 100,000. We also omitted any cities that did not report homicides for these years because Fox and Swatt (2007) did not impute unit-missing data. The final sample included 208 cities with an average 2005-2007 population of 328,918 (minimum 100,340, and maximum 8,148,355). Appendix A lists the 208 cities included in the analyses.

Dependent Variables

Dependent variables for the current analysis are city-level counts of total, economic, and non-economic homicides, calculated from Fox and Swatt’s (2007) SHR data. Economic homicides include homicides that occur in the context of economically motivated offenses including robbery, burglary, larceny, and auto theft, whereas non-economic homicides include more expressive and non-utilitarian forms of homicide.³ To introduce a desirable smoothing of the inherent variability in 1-year counts of disaggregated homicide, our analyses used 3-year (2005-2007) rather than 1-year homicide counts.

Independent Variables

Income inequality measures

The main independent variables for the current study are within-city income inequality measures, and adjusted measures simultaneously addressing both within-city and central city–metropolitan inequality. Within-city income inequality measures include the 20/20 ratio and the Gini, Theil, and Atkinson indices. Using income quintile data from 2006 ACS, the 20/20 ratio was calculated as the ratio of the proportion of income received by the city’s highest earning 20% of households to the proportion of income received by the lowest earning 20%.

For the other income inequality measures, we used the city’s 2006 ACS data for 16 income groups.⁴ In calculating Gini, Theil, and Atkinson indices from the equations in the previous section, we estimated total income within an income group from the number of households in the group and the midpoint of the group’s income range. For example $12,499.5 would be multiplied by the number of households in the $10,000 to $14,999 income range to obtain that group’s total income. The highest income category ($200,000 and greater) has no natural midpoint, and there are different practices for assigning a mean in this case. One approach is to assume a Pareto curve for income, and estimate the top income category’s mean via the relative proportions of households in the two highest categories (R. N. Parker & Fenwick, 1983). However, for smaller cities in our sample there may be few households in these highest categories, making this estimate of the top category’s mean rather suspect. Instead of estimating this top category mean in each city, we obtained a 2006 national estimate for the mean income in the $200,000 and greater category from Census reports based on the Current Population Survey. We applied this value ($347,279) in each city’s calculation.

We followed Roberts and LaFree (2004) in constructing adjusted measures that address within-city as well as central city–metropolitan inequality. First, we calculated a ratio of the outside central city–metropolitan area’s mean income to the city’s mean income. Then we constructed an adjusted measure by multiplying the within-city inequality measure by the square root of the metropolitan (outside of central city)/city income ratio, thereby adjusting for central city–metropolitan inequality. We executed this latter process for the Gini, Theil, and Atkinson indices, and thus obtained three measures of adjusted income inequality (adjusted Gini, adjusted Theil, and adjusted Atkinson).

Control variables

The current study used control variables that have been theoretically and empirically related to homicide in existing literature. To single out the impact of income inequality on homicide, previous research typically included absolute economic deprivation measures indicating the overall economic resource availability in the unit, in addition to the resource distribution indicated by income inequality (Messner, 1989). We initially included poverty, calculated as the percentage of population living under the poverty line, as an absolute economic deprivation measure. However, poverty was very highly correlated with most of the income inequality measures (as high as nearly .80) and the Variance Inflation Factor (VIF) score associated with poverty exceeded 4 in all regression models, indicating multicollinearity (Menard, 1995). We therefore used the percentage unemployed as our absolute measure of economic deprivation.⁵ Other control variables included percentage of households that were female headed and had children under 18 present, Black population, residential mobility (percentage of people living in the same dwelling as 1 year ago), percentage divorced, percentage young male population aged 15 to 24, and logged city population (as a predictor in addition to its role as offset). Descriptive statistics for all variables appear in Appendix B.

Negative Binomial Regression

Because homicide is relatively infrequent, especially when disaggregated by economic or non-economic nature, we viewed city homicide totals as count rather than continuous variables, so that Poisson or negative binomial regression was appropriate (Osgood, 2000). Comparison of equivalently specified Poisson and negative binomial regression models indicated significant overdispersion in analysis of the homicide count data (overall, economic, and non-economic), so that the negative binomial regression model, with an additional parameter accounting for overdispersion, was preferred (Osgood, 2000). We used the natural log of average 2005-2007 city population as the offset, or exposure, variable, to control for population at risk of homicide; substantively, this meant that the negative binomial regressions could be interpreted as log-rate models (Osgood, 2000). We conducted analyses in SAS PROC GENMOD. As described above, in analyzing homicide disaggregated by economic and non-economic motivation we constructed five city-level data sets for economic and non-economic homicides, from Fox and Swatt’s (2007) five imputed incident-level SHR files. We combined results from analysis of the five data sets in PROC GENMOD via PROC MIANALYZE. Reported estimates reflect averages across the five imputed data sets, and estimated standard errors used in significance tests reflect variability due to multiple imputation of the five data sets in addition to the usual uncertainty in parameter estimates. (Remember that there was no imputation needed for overall homicide.)

Results

Correlations Among Different Measures of Income Inequality

We first examined similarities and differences among the seven income inequality measures via their correlations, shown in Table 1. Focusing on within-unit (within-city in the current study) income inequality measures (the 20/20 ratio and Gini, Theil, and Atkinson indices), there were very high correlations—exceeding .90—among all within-city measures, and the .99 correlation between the Gini and Theil indices suggests that those two within-unit measures were virtually the same in the current data. The three adjusted measures that reflect both within-city and central city–metropolitan inequality (adjusted Gini, adjusted Theil, and adjusted Atkinson) were also very highly correlated (.95-.98) with each other, but notably less correlated with the within-city measures.

Table 1.

Correlations Among Different Income Inequality Measures (N = 208).

	20/20	Gini	Theil	Atkinson	Adjusted Gini	Adjusted Theil	Adjusted Atkinson
20/20 ratio	1	.92	.91	.94	.65	.81	.74
Gini index		1	.99	.96	.66	.85	.72
Theil index			1	.94	.65	.85	.70
Atkinson index				1	.62	.79	.73
Adjusted Gini					1	.95	.98
Adjusted Theil						1	.95
Adjusted Atkinson							1

Multivariate Regression Results

Table 2 shows results of multivariate analyses investigating the relationship between within-city income inequality measures and overall homicide rates. Table 3 presents results for the relationship between adjusted measures incorporating both within-city and central city–metropolitan inequality and overall homicide rates. For all models in Tables 2 and 3, VIF scores for all independent variables were less than 4, indicating that there were no substantial multicollinearity problems among the independent variables (Menard, 1995). (Correlations among all variables included in the analyses are shown in Appendix C.) According to Table 2, estimated coefficients for all four within-city income inequality measures were statistically significant in the expected positive direction in the models for overall homicide rates. Table 3 shows that the adjusted measures also had statistically significant and positive associations with overall homicide rates. Control variables also gave very similar results across different models for overall homicide rates. Estimated coefficients for unemployment, female-headed households, Black population, and logged city population were statistically significant in the expected positive direction across all models, whereas divorced population and young male population were not statistically significant in any of the overall homicide models. Residential mobility’s coefficient was statistically significant (and positive) only in the model using the adjusted Gini index.

Table 2.

Coefficients for Negative Binomial Log-Rate Models: Impacts of Within-City Income Inequality Measures on Overall Homicide.

	20/20	VIF	Gini	Within-city income inequality				VIF
	20/20	VIF	Gini	VIF	Theil	VIF	Atkinson	VIF
Within-city income inequality
20/20 ratio	0.021* (0.010)	1.632
Gini index			2.282* (1.116)	1.612
Theil index					1.365* (0.671)	1.645
Atkinson index							1.456* (0.777)	1.490
Unemployment	0.043* (0.019)	2.110	.043* (0.019)	2.112	0.043* (0.019)	2.105	0.044* (0.019)	2.105
Female-headed households	0.077** (0.017)	2.301	.076** (0.017)	2.278	0.076** (0.017)	2.278	0.076** (0.017)	2.292
Black population	0.022** (0.003)	2.255	.022** (0.003)	2.114	0.022** (0.003)	2.148	0.022** (0.003)	2.113
Residential mobility	0.011 (0.009)	1.648	.011 (0.009)	1.656	0.010 (0.009)	1.659	0.011 (0.009)	1.653
Divorced population	0.005 (0.017)	1.349	.003 (0.017)	1.358	0.003 (0.017)	1.363	0.006 (0.017)	1.347
Males 15 to 24	−0.023 (0.025)	1.785	−.023 (0.025)	1.758	−0.023 (0.025)	1.767	−0.019 (0.025)	1.729
Logged population	0.224** (0.053)	1.187	.216** (0.055)	1.238	0.217** (0.054)	1.227	0.221** (0.054)	1.224

Note. N = 208. Standard errors are reported in parentheses; VIF = Variance Inflation Factor.

p < .05. **p < .01 (one-tailed).

Table 3.

Coefficients for Negative Binomial Log-Rate Models: Impacts of Within-City and Central City–Metro Income Inequality Measures on Overall Homicide.

	Adjusted Gini	VIF	Within-city and central city–metro income inequality
	Adjusted Gini	VIF	Adjusted Theil	VIF	Adjusted Atkinson	VIF
Within-city and central city–metro income inequality
Adjusted Gini	2.483** (0.667)	2.240
Adjusted Theil			1.854** (0.571)	2.232
Adjusted Atkinson					1.898** (0.553)	2.072
Unemployment	0.019 (0.020)	2.414	0.027 (0.019)	2.284	0.023 (0.020)	2.344
Female-headed households	0.067** (0.016)	2.279	0.074** (0.016)	2.257	0.071** (0.016)	2.262
Black population	0.022** (0.003)	1.976	0.021** (0.003)	2.097	0.022** (0.003)	2.012
Residential mobility	0.016* (0.008)	1.657	0.013 (0.008)	1.648	0.015 (0.008)	1.655
Divorced population	−0.006 (0.017)	1.383	−0.004 (0.017)	1.856	−0.002 (0.017)	1.365
Males 15 to 24	−0.041 (0.025)	1.829	−0.039 (0.025)	1.387	−0.036 (0.025)	1.807
Logged population	0.193** (0.053)	1.191	0.189** (0.054)	1.242	0.194** (0.053)	1.209

Note. N = 208. Standard errors are reported in parentheses.

p < .05. **p < .01 (one-tailed).

Table 4 shows multivariate analysis results for the inequality measures’ relationships with specific types of homicide (economic or non-economic), controlling for the same set of other variables as in the overall homicide analysis. (To simplify the presentation, results for the control variables are not shown in Table 4, but are noted below.) Because the same sets of independent variables were used as in the models reported in Tables 2 and 3, again there is no substantial multicollinearity issue among independent variables for Table 4’s models. Use of the specific homicide types did not change the main pattern of results for the within-city and the adjusted income inequality measures: both the within-city and the adjusted measures were significantly and positively associated with both economic and non-economic homicide rates. Results for control variables (not reported in Table 4) were again mostly consistent across analyses and like those reported above for overall homicide, except that the estimated relationship between unemployment and economic homicide was not significant.

Table 4.

Coefficients for Negative Binomial Log-Rate Models: Economic and Non-Economic Homicides.

	Economic homicide	Non-economic homicide
Within-city income inequality measures
20/20 ratio	0.030 (0.016)*	0.021 (0.010)*
Gini index	4.151 (1.837)*	2.152 (1.120)*
Theil index	2.532 (1.080)*	1.265 (0.672)*
Atkinson index	2.653 (1.297)*	1.382 (0.778)*
Within-city and central city–metro income inequality
Adjusted Gini	3.449 (1.042)**	2.411 (0.670)**
Adjusted Theil	3.033 (0.883)**	1.757 (0.572)**
Adjusted Atkinson	2.735 (0.857)**	1.838 (0.554)**

Note. N = 208. Standard errors are reported in parentheses.

p < .05. **p < .01 (one-tailed).

These findings indicate that (a) the choice of inequality measure did not influence the pattern of statistical significance and direction of estimated coefficients, despite the Atkinson index’s greater relevance for relative deprivation and the potential importance of incorporating city–metro inequality, and (b) main results did not change no matter which outcome measure—overall homicide, economic homicide, or non-economic homicide—was used. To check the robustness of results for the income inequality measures, we also explored slightly different measurements for control variables, and various model specifications. For example, we included the White–Black racial segregation index from 2005-2009 ACS instead of the percentage Black, and included squared percentage Black, to consider the possible non-linearity of the relationship between minority population and homicide discussed in some previous studies (e.g., Kovandzic et al., 1998). However, under all these variations, the main patterns of results for the different income inequality measures remained the same, so findings of the income inequality measures’ statistically significant and positive associations with homicide appear robust across different model specifications and control variable definitions.

Discussion and Conclusion

The current study used data from 208 large U.S. cities to examine the income inequality–homicide link using different measures of income inequality and homicide, and explored whether the choice of these measures influences the main analytic results. This work was motivated by concerns that the commonly used Gini index might not capture the elements of income inequality that the relative deprivation perspective holds to be most relevant to crime (Bailey, 1984; Messner, 1983; Messner & Tardiff, 1986). Of course classic relative deprivation arguments do not explicitly call for one income inequality measure over another. However, the key implications of relative deprivation theory indicate that, with respect to crime, inequality measures should be sensitive to changes in the distribution of income among lower income groups, but that the wide-ranging nature of comparisons implicated in the formation of relative deprivation requires that measures also be affected by changes across the entire income distribution. We therefore explored a spectrum of inequality measures that varied in their fit with relative deprivation theory and their emphases on different areas of the income distribution. From that investigation, the less common Atkinson index seems, in principle, to be a better measurement choice than the Gini index when examining inequality’s relationship with crime, due to its consideration of the entire income distribution while focusing on the lower income groups. Furthermore, relative deprivation theory’s vagueness about the geographical scope of comparisons of economic standing suggests that it may be helpful to adjust within-city income inequality measures by the degree of city–metropolitan inequality, to reflect a likely additional element of comparison in the modern age of widespread mass media exposure and feasible travel to distant places.

Regarding the measurement of homicide when studying the income inequality–homicide link, overall homicide rates obscure the multi-faceted nature of homicide, and some types of homicide could be more strongly associated with income inequality than other. One view argues that income inequality’s association should be stronger with rates of economically motivated homicides, as such homicide represents an extreme form of income redistribution (Bailey, 1984; Kovandzic et al., 1998; R. N. Parker, 1989). Another argues that the relative deprivation resulting from income inequality could lead to expressive forms of violence against the most available targets (Danziger & Wheeler, 1975; Messner & Tardiff, 1986).

Despite our belief in the potential advantage of the Atkinson index, and the substantive importance of incorporating city–metro inequality in capturing the idea of relative deprivation, the income inequality measures were generally all very highly correlated in our data, though correlations between the within-city and adjusted measures (incorporating city–metropolitan inequality) were somewhat lower. Also, all of the income inequality measures showed a pattern of statistically significant and positive association in multivariate analyses of all types of homicides. So, contrary to our expectation, results indicated that the choice of income inequality or homicide measure does not influence main findings on the income inequality–homicide link. Although it is surprising that the different measures, with their quite different justifications and mathematical forms, are so highly correlated and produce such similar regression results, this finding is an important contribution of our research. And when viewed from the perspective of the field as a whole, it is encouraging that our results suggest that the main conclusions of research on inequality and crime are likely at least fairly robust to the specification of the inequality measure. It seems that the noted inconsistency of the literature’s findings on the relationship between income inequality and crime probably does not stem from the use of different inequality measures, so our work may help direct researchers’ attention to other aspects of this puzzle. The consistency here among the different income inequality measures’ associations with homicide may also be a function of there being relatively insubstantial differences among American cities’ income distributions. In other contexts, such as cross-national data, qualitatively greater differences among different units’ income distributions may allow the measures’ differing emphases to emerge more vividly. The similarity of patterns of results for economic and non-economic homicide also indicates that greater income inequality may result in not only in a greater number of homicides with economic motivation, but of the more common expressive homicides too. As for the inequality measures, the use of different homicide types or a monolithic homicide measure does not seem to be the key to inconsistency in the previous literature.

The statistically significant association that we found between income inequality and homicide when considering city–metro inequality calls attention to the breadth of comparison inherent in the development of relative deprivation. This association is consistent with the idea that mixing of central city and suburban residents through commuting, leisure activities, and other elements of contemporary urban life affects the development of relative deprivation among city residents. This could be further explored through data on patterns of city–metro social interaction, for instance, on the extent of leisure or commuting traffic between central cities and outlying areas. Such data could help determine where central city residents might reasonably make income comparisons, as this may differ in metropolitan areas of different size. More research is also needed to develop clear geographical boundaries of comparisons.

One obvious and important limitation of the current study is the exclusion, due to multicollinearity, of poverty as an absolute deprivation measure. Because poverty is theoretically and empirically a strong predictor of homicide (e.g., Bailey, 1984; Messner & Tardiff, 1986; Pridemore, 2008), our exclusion of poverty may lead to model misspecification and improper estimation of the association between income inequality and homicide. When we repeated the analyses with poverty included, the estimated coefficients for all of the within-unit income inequality measures became non-significant, with marginal significance (and the expected positive direction) for adjusted measures capturing both within-city and city–metropolitan income inequality. However, we do not know whether the non-significance for within-unit measures reflects a genuine absence of meaningful association, or instead that multicollinearity prevented the analysis from distinguishing within-unit income inequality’s (relative deprivation) association with homicide from poverty’s (absolute deprivation) in the current sample of cities. Nevertheless, the pattern of associations for different within-unit income inequality measures was very consistent across different types of homicide (all significant without poverty in the model, all non-significant with poverty included). Although results for within-unit measures differed from those for adjusted measures, choices among the within-unit income inequality measures, or among the adjusted measures, or among the homicide types, did not seem to matter much empirically.

Infant mortality may be a desirable alternative to traditional poverty measures, and Pridemore (2008) argues that analysis with infant mortality may better represent contributions of absolute and relative deprivation to homicide (though also note Messner, Raffalovich, & Sutton, 2010). Because infant mortality data are not publicly available for many cities in our sample, we obtained county-level (for the county containing the city) infant mortality rates per 1,000 births from the Centers for Disease Control and the National Center for Health Statistics, and conducted analyses with this alternative absolute deprivation measure. Using the county-level infant mortality rate instead of the city poverty measure resulted in patterns of statistical significance and direction of association between the various income inequality measures and the different homicide types that were like those in the main analyses (i.e., the analyses that did not include poverty as a control). Of course the county-level data are only an approximation, and many counties may be too heterogeneous for those figures to accurately depict conditions in the central city (and some counties contained more than one of the cities in our sample). More investigation of poverty’s role in understanding income inequality and homicide is needed.

Even though our main analyses showed a positive and significant association between income inequality and homicide rates, the aggregate nature of the analysis cannot determine whether the supposed micro-level process of development of feelings of relative deprivation actually occurred (e.g., Krahn, Hartnagel, & Gartrell, 1986; Messner, 1989; Stack, 1984). Some researchers have argued that income inequality could lead to crime and violence even without the presence of relative deprivation (Stolzenberg, Eitle, & D’Alessio, 2006). For example, income inequality may promote increased resource competition, leading more individuals to seek out illegitimate economic opportunities, but out of necessity rather than a feeling of relative deprivation. This increased participation in the illegitimate economy, such as the illegal drug trade, could in turn lead to greater rates of violence, including homicide (Kovandzic et al., 1998). Furthermore, the association between homicide and adjusted measures incorporating city–metropolitan differences could be explained by the social isolation of the poor and minorities in inner cities. This isolation, due to structural forces such as deindustrialization, disinvestment, municipal policies that degrade social networks of inner city neighborhoods, and class-based migration from the central cities, is discussed in the works by W. J. Wilson (1987, 1996, 2009), Sampson and Wilson (1995), and other scholars as an alternative to relative deprivation as a cause of crime. Note too that we explored different income inequality measures based on the assumption that lower income groups are most prone to the development of feelings of relative deprivation, but relative deprivation could also stem from other, non-economic forms of inequality, based on factors such as age, gender, and race/ethnicity (Kovandzic et al., 1998).

The current study took a cross-sectional approach to income inequality and homicide, but these inequality measures could also be used in analyzing city homicide data over time. Increasing income inequality in the United States likely has not affected all cities in the same way, and also may have changed central cities’ standing relative to wider metropolitan areas. Time-series cross-section data will thus have particular significance for understanding how inequality trends affect homicide trends, and for anticipating how national-level changes in economic inequality may or may not be reflected in changing homicide rates. This requires that research like our evaluation of income inequality measures also be conducted in that dynamic context. As Fowles and Merva (1996) noted, work on inequality and homicide must be part of the policy conversation on whether (and, if so, how) to ameliorate inequality. Continued inquiry into how to best assess income inequality, and its relationship with homicide, is therefore of utmost importance. We hope that our work on the measurement issues associated with inequality and homicide will help stimulate more research.

Footnotes

Appendix

Appendix C

Correlation Matrix of All Variables (N = 208).

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
1. Homicides	1.00
2. Economic homicides	.92**	1.00
3. Non-economic homicide	.99**	.90**	1.00
4. 20/20 ratio	.37**	.37**	.36**	1.00
5. Gini index	.36**	.38**	.35**	.92**	1.00
6. Thiel index	.36**	.39**	.35**	.91**	.99**	1.00
7. Atkinson index	.35**	.35**	.34**	.94**	.96**	.94**	1.00
8. Adjusted Gini	.41**	.40**	.40**	.65**	.66**	.65**	.62**	1.00
9. Adjusted Theil	.43**	.44**	.42**	.81**	.85**	.85**	.79**	.95**	1.00
10. Adjusted Atkinson	.42**	.41**	.41**	.74**	.72**	.70**	.73**	.98**	.95**	1.00
11. % Unemployed	.35**	.33**	.35**	.36**	.36**	.36**	.31**	.65**	.59**	.61**	1.00
12. Female headed	.23**	.21**	.23**	.25**	.25**	.25**	.20**	.54**	.47**	.49**	.67**	1.00
13. Black population	.39**	.41**	.38**	.50**	.46**	.47**	.43**	.56**	.58**	.55**	.60**	.62**	1.00
14. Res. mobility	−.09	−.03	−.10**	.20**	.25**	.27**	.21**	.12	.20**	.12	.06	−.10	.06	1.00
15. Males 15 to 24	−.09	−.08**	−.09**	.26**	.24**	.25**	.21**	.25**	.28**	.25**	.12	.08	.05	.48**	1.00
16. Divorced population	−.01	.01	−.02	.12	.16*	.18**	.11	.20**	.22**	.17*	.21**	.13	.24**	.23**	−.20**	1.00
17. Log population	.77**	.72**	.77**	.29**	.34**	.32**	.33**	.27**	.31**	.30**	.14*	.09	.18*	−.02	−.16*	.02	1.00

p < .05. **p < .01.

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.

Notes

Author Biographies

Aki Roberts is an assistant professor in the Department of Sociology, University of Wisconsin–Milwaukee. Her interests include quantitative methods, crime clearance, National Incident-Based Reporting System (NIBRS) data, Japanese crime, police networks, and motor vehicle theft.

Dale Willits is an assistant professor in the Department of Sociology, California State University, Bakersfield. His interests include quantitative methods, situational and contextual predictors of violence, and police organizations.

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