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Not ‘Islands, Entire of Themselves’: Exploring the Spatial Context of City-level Robbery Rates

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Abstract

The current study examines spatial dependence in robbery rates for a sample of 1,056 cities with 25,000 or more residents over the 2000–2003 period. Although commonly considered in some macro-level research, spatial processes have not been examined in relation to city-level variation in robbery. The results of our regression analyses suggest that city robbery rates are not spatially independent. We find that spatial dependence is better accounted for by spatial error models than by spatial lag models. Further exploration of various spatial weights matrices indicates that robbery rates of cities within the same state are related to robbery rates of other cities within the same state, regardless of their proximity. Our analyses illustrate how systematic inquiry into spatial processes can alert researchers to important omitted variable biases and identify intriguing problems for future research.

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Notes

  1. A few studies have used larger units of analysis, including a few county-level studies (Osgood 2000; Osgood and Chambers 2000; Worrall 2005).

  2. One concern that has been raised in deterrence research is the potential for artificially induced correlation due the common term (crime) in numerators (e.g. crime/population) and denominators (e.g. punishment/crime) of ratio variables, especially when there is error in the measurement of this variable (see e.g. Gibbs and Firebaugh 1990; Logan 1982). Recent research has shown that, even when measurement error is large as in the case of official crime measures, the practical impact is unlikely to influence substantive conclusions regarding deterrent effects (e.g. Levitt 1998; Pudney et al. 2000). This finding is consistent, more generally, with many of the concluding statements on the debate over the ratio variable problem (cf. Firebaugh 1988).

  3. A few studies have also examined the role of proactive or “broken windows” policing in the widely noted homicide decline in the 1990s. See Harcourt and Ludwig (2006), Kelling and Sousa (2001), Rosenfeld et al. (2005).

  4. Cliff and Ord ((1981); Anselin and Bera 1998) formally presented Moran’s I as \(I=\frac{N}{S_0 }\left( {\frac{{e}'We}{{e}'e}} \right)\) , where e is a vector of regression residuals, W is a spatial weights matrix, N is the number of observations, and S 0 is a standardization factor equal to the sum of the spatial weights. An alternative expression shows Moran’s I to be the two-dimensional (i.e., spatial) analog of the coefficient of autocovariance, ρ, for univariate time series correlation. This can be seen by comparing \(\hat{\rho}=\frac{\sum{\left({e_t \times e_{t-1}}\right)}}{\sum{e_t^2}}\) to the alternative expression of \(I=\frac{\sum{_i} \sum {_j w_{ij} \left( {e_i \times e_j } \right)}} {\sum {_i e_i^2}}\) . The most common specification of the spatial weights matrix W is to link every unit i to every other unit j by the inverse of distance between i and j, thus the elements of W are \(\frac{1}{d_{ij} }\) . Generally the W matrix is then row-standardized to unity across rows. We use the Moran’s I option in CrimeStat (Levine 2004) to calculate residual spatial autocorrelation under this specification.

  5. We initially fit spatial lag and spatial error regressions in GeoDa (Anselin et al. 2005) using a threshold distance-based weights matrix. The threshold distance function in GeoDa finds the global minimal distance between cities’ geographic centers that will leave no city as an “island,” unconnected to any other city. We then use SAS’ PROC MIXED procedure with a spatial error covariance structure to replicate the spatial error model. In PROC MIXED, spatial autocorrelation is integrated into the general mixed model through a distance function in the error covariation. If we start by writing the general mixed model as y = Xβ + Zu + e, where everything is the same as in the general linear model except for the addition of the known design matrix, Z, and the vector of unknown random-effects parameters, u. Z can contain either continuous or dummy variables, just like X. The name “mixed model” comes from the fact that the model contains both fixed-effects parameters, β, and random-effects parameters, u. If we assume that u and e are normally distributed with \(E\left[ {\begin{array}{l} u\\ e\\ \end{array}} \right]=\left[ {\begin{array}{l} 0\\ 0\\ \end{array}} \right]\) and \(VAR\left[ {\begin{array}{l} u\\ e\\ \end{array}}\right]=\left[\begin{array}{ll} G&0\\ 0&R\\ \end{array}\right],\) the variance of y is V = ZG Z′ + Rand V can be modeled by specifying the random-effects design matrix Z and by specifying covariance structures for G and R (Littel et al. 2006, p. 438ff.). As with repeated measures, the spatial mixed model assumes that the errors (the elements of e) are correlated and the form of the spatial dependence can be reflected in R. In the general mixed model the covariance structure of R can be defined by letting Var(e i ) = σ 2 i and Cov(e i ,e j ) = σ ij . In the spatial mixed model the covariance is assumed to be a function of the distance (d ij ) between the locations s i and s j , thus the resulting models have the general form: Cov(e i ,e j ) = σ2[f(d ij )]. The MIXED procedure requires that observations be uniquely identified with a spatial coordinate and that the spatial (error) process that generated the data satisfy certain stationarity conditions, such as equal variance among the observations. Although the MIXED procedure in SAS supports many spatial functions, we use the power function \(f\left({d_{ij}}\right)=\rho^{d_{ij}},\) where d ij is distance between cities.

  6. Cities in Florida and Illinois did not report the arrest information used to generate the proactive policing measure to the UCR and are therefore excluded from the analyses. Additionally, cities in Alaska and Hawaii had to be excluded because the distance measures would have been unduly influenced by the fact that they are not contiguous with the rest of the states.

  7. The source is the National Archive of Criminal Justice Data located on the Inter-University Consortium for Political and Social Research (ICPSR) website (http://www.icpsr.umich.edu/NACJD/archive.html, as compiled in studies #9028 (1996–1997), #2904 (1998), #3158 (1999), #3447 (2000), #3723 (2001), and #4008 (2002).

  8. Census designated geographic entities do not recognize the popular conception of “cities” per se. Rather, the Census Bureau designates “places,” assigning them five-digit FIPS place codes, including “incorporated places” that generally match the entities we think of as cities. We use this summary level of aggregation to extract the socio-demographic characteristics that we link to UCR ORI robbery rates. In addition, the Census Bureau assigns a latitude and longitude (in decimal degrees) to an “internal point” in each designated geographic entity. A single point is identified for each entity that (usually) represents the geographic center of that entity. The source for the Census data and documentation is: Census 2000 Summary File 3 [United States]/prepared by the U.S. Census Bureau, 2002; Census 2000 Summary File 3 Technical Documentation/prepared by the U.S. Census Bureau, 2002.

  9. We are grateful to Debra K. Mack, Chief of Programs Support Section of the FBI’s Criminal Justice Information Services Division, for providing the arrest and employee data and for clarification of these data.

  10. Some other recent city-level research has found that a dummy variable for cities in southern states is unrelated to robbery rates (e.g. Shihadeh and Flynn 1996; Stretesky et al. 2004).

  11. The interpretation of the negative coefficient for proactive policing is problematic given potential endogeneity. High robbery rates may inhibit the capacity of police agencies to engage in proactive policing. However, partitioning potential reciprocal effects for proactive policing and robbery rates is not central to the purposes of the present analysis.

  12. Table 2 also reports BIC. We calculate BIC with d (the dimension of the model) as the rank of the X matrix + the effective number of estimated covariance parameters. Therefore, d = 10 (8 covariates + intercept + error variance) and OLS BIC  = 1949.5.

  13. Based on the observed I, the expectation of I, and the standard deviation generated by the randomization, the test statistic is \(Z_I =\frac{I-E\left( I \right)}{sd\left( I \right)}=\frac{0.0886-\left( {-0.0009} \right)}{0.0071}=12.61\) , with p ≤ 0.001. The Moran Scatterplot shown in Fig. 1 is produced in GeoDa based on a distance threshold spatial weights matrix. We also calculate the residual Moran’s I from Model 1 in CrimeStat utilizing a full distance weights matrix (see fn. 5). The result is almost identical: I = 0.087 and Z I  = 15.79.

  14. Formally, the LM statistic against spatial error autocorrelation takes the form LME = [e′We/s 2]2/T, with e as a vector of OLS residuals, s 2 as its estimated standard error, and T = [tr(W + W′)W]. This statistic, LME, is asymptotically distributed as χ2(1) under the null. The LM statistic for spatial lag dependence is somewhat more complex: LML = [e′Wy/s 2]2/[(WXβ)′M(WXβ)/s 2 + T], with M = I − X(X′X)−1 X′, βas the OLS estimate, and T and the distributional form of the statistic as before. Anselin et al. (1996) further developed a robust form of the LM statistics wherein the robust LME is robust to the presence of spatial lag when quantifying spatial error, while the robust LML is robust to the presence of spatial error. In this application, LML = 45.76 and LML Robust  = 4.35; LME = 154.96 and LME Robust  = 113.31. Although LML Robust remains statistically significant at p < 0.04, given the relatively large N, the robust quantification of spatial error that is over 26 times larger than the robust quantification of spatial lag, and the statistically significant residual spatial autocorrelation remaining in the spatial lag model (I = 0.05, Z I  = 7.44, p < 0.001), the evidence points unequivocally to a spatial error model as the preferred spatial econometric model.

  15. Given the strong preference for spatial error model estimation over the spatial lag model, we do not report parameter estimates from the spatial lag model among the models in Table 2 because our diagnostics confirm that this model is misspecified.

  16. The indicator of politicization of judge selection is a scale from 1 to 4 based on how local judges are selected. The scale has a value of 1 for appointed judges; 2 for judges that are initially appointed and then renewed through popular election (Missouri plan); 3 for nonpartisan popular elections; and 4 for partisan popular elections. The source of this data is The Book of the States (Council of State Governments 2000). We are grateful to Paige Harrison, Bureau of Justice Statistics, for providing the incarceration data in a personal communication.

  17. We note that this finding is the opposite to what one might predict from the standpoint of recent research on social support or institutional anomie theory. These perspectives imply that conservative policies leave citizens more susceptible to the vicissitudes of the market and thus increase crime (see e.g. Colvin et al. 2002; Messner and Rosenfeld 1997; Savolainen 2000).

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Authors and Affiliations

  1. Department of Sociology, University at Albany, State University of New York, Albany, NY, USA

    Glenn Deane, Steven F. Messner & Kelly McGeever

  2. Department of Criminal Justice, Indiana University-Purdue University Indianapolis, Indianapolis, IN, USA

    Thomas D. Stucky

  3. Department of Sociology, George Washington University, Washington, DC, USA

    Charis E. Kubrin

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  1. Glenn Deane
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  2. Steven F. Messner
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  4. Kelly McGeever
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  5. Charis E. Kubrin
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Correspondence to Glenn Deane.

Additional information

This material is based upon work supported by the National Science Foundation under Grant No. SES-0215551 and SBR-9513040 to the National Consortium on Violence Research (NCOVR). An earlier draft of this paper was presented at Seventy-sixth Annual Meeting of the Eastern Sociological Society, Boston, MA, February 23–26, 2006. The paper evolved from research originally presented at a workshop sponsored by NCOVR. We are grateful to the participants at the workshop for comments on the paper, with special thanks to Robert J. Bursik, Jr., Thomas Bernard, and Paul Nieuwbeerta.

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Deane, G., Messner, S.F., Stucky, T.D. et al. Not ‘Islands, Entire of Themselves’: Exploring the Spatial Context of City-level Robbery Rates. J Quant Criminol 24, 363–380 (2008). https://doi.org/10.1007/s10940-008-9049-3

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