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Bayesian Spatial Random Effect Modelling for Analysing Burglary Risks Controlling for Offender, Socioeconomic, and Unknown Risk Factors

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Abstract

This paper adopts a Bayesian spatial random effect modelling approach to analyse the risk of domestic burglary in Cambridgeshire, England, at the census output area level (OA). The model, in the form of Binomial spatial logistic regression, integrates offence and offender based theories and takes into account unknown local risk factors (represented as unexplained spatial autocorrelation in the model). A score of ‘proximity to offenders’ was calibrated for each OA based on the number of likely offenders in the county, the OAs they reside, and their proximities. Our results indicate that areas that have a score higher than the average score were at higher risks of being burgled. Household occupied by non-couple and economically inactivity are positively associated confounders. Household occupied by owner is a negatively associated confounder. These confounders diminish the effect of high score of proximity to offenders, which, however, remains positively associated with the risk of burglary. Bayesian spatial random effect modelling, which adds to the traditional (non-spatial) regression model a spatial random effect term, stabilizes estimated risks and remarkably improves model fit and causation inference. Mapping the results of spatial random effect reveals locations of high risk of burglary after controlling for offender and socioeconomic factors. Limitations of the study and strategies to deter burglaries based on the results of spatial random effect modelling are discussed.

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Notes

  1. This convolution model has been used to analyse spatial data that have clear spatial structure. Its spatially structured random effect term is an intrinsic conditional autoregressive model, which is a special case of the general conditional autoregressive model (see, for example, Law and Haining (2004), for further details).

  2. The number of dwellings refers to all dwellings in the area at the time of the 2001 census. A household’s accommodation (a household space) is defined as being in a shared dwelling if it has accommodation type ‘part of a converted or shared house’, not all the rooms (including bathroom and toilet, if any) are behind a door that only that household can use and there is at least one other such household space at the same address with which it can be combined to form the shared dwelling. If any of these conditions is not met, the household space forms an unshared dwelling. Therefore, a dwelling can consist of one household space (an unshared dwelling) or two or more household spaces (a shared dwelling). [Quoted from Dataset Title of Census 2001: Dwellings (UV55), column heading: All dwellings.]

  3. Classification of urban or rural for OA was based on the classification file for census wards from Neighbourhood Statistics (http://neighbourhood.statistics.gov.uk/dissemination/) and a point-in-polygon operation that matches OA to ward using a geographic information system.

  4. To calibrate the weighted function, we used data of all offenders of burglary between 1st January 2001 and 31st December 2003, a year before and after 2002, for a bigger sample of distances travelled by offenders. Offenders from urban and rural areas accounted for 1,947 and 627 burglaries, respectively.

  5. The two distance-weighted functions were found by testing different probability density functions to the distances travelled by the urban/rural offenders to their corresponding offence sites, and selecting the distribution that fits the distances best. The density functions tested are exponential, gamma, Weibull, and Pareto. They were selected for testing because they are capable of generating shapes similar to the shapes of the histograms of the distances travelled by the urban and rural offenders. Because the finest data that we have are at the output area level, distances travelled were estimated by the Euclidean distance between the population centroids of the output areas of the offender and burglary event. Outliers revealed by the histogram, which are distances greater than 10,000 m, were discarded before fitting the functions. The removal of outliers was also supported by chi-square tests from the fitting of the functions without discarding outliers. Based on chi-square tests, the distributions identified for urban and rural offenders are both gamma, \( f(d) = \frac{{{b^a}}}{{\Gamma (a)}}\,{(d)^{{a - 1}}}\,{e^{{ - bd}}},\,\,\,\,\,\,\,d\, > \,0 \), where d denotes distance, and a and b are the parameters of the gamma function. Testing the four functions using a Bayesian approach in WinBUGS also identified gamma as the best fit. The offenders’ journey-to-crime gamma distance functions fitted are similar for urban and rural areas. Both gamma functions are very similar to the inverse distance decay exponential function used typically in the literature.

  6. If a quantitative interpretation on an absolute value of the explanatory variable (SOFFENDER) were appropriate, then semiparametric binomial regression modelling with low-rank thin-plate splines (Crainiceanu et al. 2005; Wood 2003) might be useful. As an experiment, we were able to fit such models where the coefficients of the spline bases were modelled as random coefficients with different number of knots of spline of SOFFENDER and prior distributions of parameters using WinBUGS, although not all of the models tested converged satisfactorily. Further details of modelling and results are not reported here, but are available from the authors.

  7. In our initial analysis to identify explanatory variables that are significant, we have also fitted non-spatial univariate models using a Bayesian approach in WinBUGs. The non-spatial univariate model with “non-couple” gives the smallest DIC. This result also indicates that non-couple is the most prominent explanatory variable. Spatial models containing a couple or all of the explanatory variables, were very slow (the greater the degree of autocorrelation, the slower the fitting) and sometimes non straightforward to fit due to strong autocorrelation and thus convergence issue during MCMC simulation in WinBUGS. It was therefore desirable to identify a (parsimonious) non-spatial (multivariate) model using a non-Bayesian approach before fitting the spatial model using WinBUGS with the set of confounders identified in the (parsimonious) final non-spatial model.

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Acknowledgements

We thank Robert Haining for many useful discussions and directions on this research, and the reviewers for their thoughtful and helpful comments. In addition, we are grateful to UKBORDERS for the digital maps, National Statistics (UK) for the census data, and Cambridgeshire Police for the crime data, provided for this research.

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Authors and Affiliations

  1. School of Planning, University of Waterloo, Waterloo, ON, N2V 2S2, Canada

    Jane Law

  2. Department of Geography, University of Cambridge, Cambridge, UK

    Ping W. Chan

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  1. Jane Law
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  2. Ping W. Chan
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Correspondence to Jane Law.

Appendices

Appendix 1

Table 6 Descriptions of potential confounders for analysing the association between burglary and proximity to offendersx
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Appendix 2

WinBUGS code for the spatial random effect model.

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Law, J., Chan, P.W. Bayesian Spatial Random Effect Modelling for Analysing Burglary Risks Controlling for Offender, Socioeconomic, and Unknown Risk Factors. Appl. Spatial Analysis 5, 73–96 (2012). https://doi.org/10.1007/s12061-011-9060-1

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