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Multiscale spatiotemporal patterns of crime: a Bayesian cross-classified multilevel modelling approach

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Abstract

Characteristics of the urban environment influence where and when crime events occur; however, past studies often analyse cross-sectional data for one spatial scale and do not account for the processes and place-based policies that influence crime across multiple scales. This research applies a Bayesian cross-classified multilevel modelling approach to examine the spatiotemporal patterning of violent crime at the small-area, neighbourhood, electoral ward, and police patrol zone scales. Violent crime is measured at the small-area scale (lower-level units) and small areas are nested in neighbourhoods, electoral wards, and patrol zones (higher-level units). The cross-classified multilevel model accommodates multiple higher-level units that are non-hierarchical and have overlapping geographical boundaries. Results show that violent crime is positively associated with population size, residential instability, the central business district, and commercial, government-institutional, and recreational land uses within small areas and negatively associated with civic engagement within electoral wards. Combined, the three higher-level units explain approximately fifteen per cent of the total spatiotemporal variation of violent crime. Neighbourhoods are the most important source of variation among the higher-level units. This study advances understanding of the multiscale processes influencing spatiotemporal crime patterns and provides area-specific information within the geographical frameworks used by policymakers in urban planning, local government, and law enforcement.

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Notes

  1. Following Congdon (2011), the regression coefficients were standardized in order to compare the relative effects of the observed explanatory variables and the latent factors (see Appendix B). Table 1 reports the posterior medians and uncertainty intervals from the standardized regression coefficients.

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Acknowledgements

This work was supported by the Social Sciences and Humanities Research Council of Canada Grant Number [767-2013-1540]. All analyses and interpretation of this data are entirely that of the author.

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Appendices

Appendix A: Descriptive statistics for violent crime and explanatory variables

See Tables 2, 3.

Table 2 Total and annual violent crime counts at the dissemination area scale
Table 3 Descriptive statistics explanatory variables and the results of factor loadings. Posterior medians and 95% credible intervals of the factor loadings are shown

Appendix B: Bayesian factor analysis models

Let \(Y_{ik}\) denote the standardized rates of the ten explanatory variables associated with one of the four latent constructs, where i indexes small areas (i = 1, …, 656) and k indexes the variable type (k = 1, …, 10). \(Y_{ik}\) were assumed to follow a normal distribution with mean \(\eta_{ik}\) and with an unknown variance for each variable \(\sigma_{\eta k}^{2} ;\;Y_{ik}\) ~ Normal \(\left( {\eta_{ik} ,\;\sigma_{\eta k}^{2} } \right)\) (Congdon 2011). Model A1 estimates the four latent constructs representing residential instability, socioeconomic disadvantage, family disruption, and ethnic heterogeneity (n = 1, …, 4). Each of the ten census variables was modelled as the sum of a type-specific intercept \(\left( {\alpha_{\eta k} } \right)\) and a factor component \(\left( {\lambda_{nk} \cdot \psi_{ni} } \right)\), where \(\lambda_{nk}\) are the factor loadings and \(\psi_{ni}\) are the area-specific estimates of the four latent constructs. Following past research, each variable was a priori assigned to the latent constructs (Sampson et al. 1997; Sutherland et al. 2013). For example, \(\lambda_{1,1}\) and \(\lambda_{1,2}\) represent the factor loadings for the per cent of renters and the five-year mobility rate, respectively, and these variables were associated with the latent construct measuring residential instability, \(\psi_{1i}\) (see “Appendix A” and Sect. 4.1 for census variables and associated constructs).

$$\eta_{ik} = \alpha_{\eta k} + \left( {\lambda_{nk} \cdot \psi_{ni} } \right)$$
(3)

The standard deviations of type-specific variance parameters \(\left( {\sigma_{\eta k} } \right)\) were assigned positive half-normal prior distributions with means of zero and variances of 1000 (Gelman 2006). The type-specific intercepts \(\left( {\alpha_{\eta k} } \right)\) were assigned improper uniform prior distributions. To ensure model identifiability, one factor loading from each construct was set to positive (or negative) one, specifically for per cent of renters \(\left( {\lambda_{1,1} = 1} \right)\), median household income \(\left( {\lambda_{2,1} = - 1} \right)\), separated/divorced families \(\left( {\lambda_{3,1} = 1} \right)\), and the index of language heterogeneity \(\left( {\lambda_{4,1} = 1} \right)\). The remaining factor loadings were assigned positive half-Gaussian prior distributions with means of zero and variances of 1000 (Congdon 2011). The four sets of random effects terms representing the latent variables were assigned a multivariate normal distribution with means set to zero and with a four-by-four variance–covariance matrix \(\sum\). The multivariate normal distribution allows for correlation between the latent variables, where the diagonal elements of \(\sum\) are the conditional variances of the four sets of random effects and the off-diagonal elements are the covariances between the four constructs. The inverse of \(\sum\) was assigned a Wishart prior distribution with five degrees of freedom and diagonal and off-diagonal elements assigned values of 0.02 and 0, respectively (Thomas et al. 2004). Note that it is possible to impose spatial structure on the latent variables via a multivariate conditional autoregressive prior, as illustrated by Congdon (2008, 2011). Regression coefficients from analyses using spatially structured latent variables were virtually identical to the results shown in Table 1.

Following Congdon (2011), the equations to standardize the regression coefficients for the observed explanatory variables and latent explanatory variables are shown in Models A2, A3, A4, and A5 where \(\beta_{1}^{(s)}\) is the standardized regression coefficient for population size; \(\beta _{2:5}^{(s)}\) are the standardized regression coefficients for the binary variables (central business district, commercial land use, government-institutional land use, and recreational land use); \(\kappa_{n}^{(s)}\)’s are the standardized regression coefficients for the latent variables (residential instability, socioeconomic disadvantage, family disruption, and ethnic heterogeneity), and \(\lambda^{(s)}\) is the standardized regression coefficient for the per cent of active voters. The standard deviations of the observed and latent explanatory variables are denoted by \(\sigma_{{x_{j} }}\) and \(\sigma_{{\psi_{n} }}\), respectively, and \(\phi\) is the square root of the variance of the modelled violent crime counts \(\left( {\phi = \text{var} \left( {\log \left( {\mu_{it} } \right)} \right)^{0.5} } \right).\)

$$\beta_{1}^{\left( s \right)} = \left( {\beta_{1} \cdot \sigma_{x1} } \right)/\phi$$
(4)
$$\beta_{2:5}^{\left( s \right)} = \beta_{2} /\phi$$
(5)
$$\kappa_{n}^{\left( s \right)} = \left( {\kappa_{n} \cdot \sigma_{\varPsi n} } \right)/\phi$$
(6)
$$\lambda^{\left( s \right)} = \left( {\lambda \cdot \sigma_{\omega } } \right)/\phi$$
(7)

Appendix C: Posterior medians and 95% credible intervals (in parentheses) of the variance partition coefficients for the random effects terms in Model 1 and Model 2

 

Model 1

Model 2

Lower-level random effects terms

0.99 (0.98, 1.00)

0.85 (0.75, 0.92)

 Spatial

0.93 (0.91, 0.95)

0.79 (0.69, 0.86)

 Space–time

0.06 (0.04, 0.08)

0.06 (0.04, 0.08)

Higher-level random effects terms

NA

0.15 (0.08, 0.24)

 Neighbourhood

NA

0.08 (0.03, 0.16)

 Electoral ward

NA

0.02 (0.00, 0.07)

 Patrol zone

NA

0.04 (0.00, 0.10)

Temporal effects

0.01 (0.00, 0.02)

0.01 (0.00, 0.02)

Appendix D: WinBUGS code for Model 2

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Quick, M. Multiscale spatiotemporal patterns of crime: a Bayesian cross-classified multilevel modelling approach. J Geogr Syst 21, 339–365 (2019). https://doi.org/10.1007/s10109-019-00305-2

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