## 1. Introduction

Geographers and criminologists have had an everlasting interest in the geography of crime [

1]. An empirical study of crime can investigate the geographical variations of crime risks and potential risk factors underling those variations. Clearly, a comprehensive understanding of this relationship contributes to better-informed crime intervention and prevention efforts. From the existing literature, researchers have adopted non-spatial regression models in ecological studies of crime [

2]. These models are not adequate to analyze crime at a small-area scale, as they do not take into account any spatial relationships between units of analysis. In practice, however, small-area crime data are usually spatially auto-correlated. Neglecting this spatial autocorrelation can result in biased inference [

3].

Methodologically, spatial autocorrelation has been accounted for by frequentist spatial statistical models. For instance, Morenoff et al. [

4] employed a spatial lag model to probe into the association between neighborhood-level characteristics and violent crime. Andresen [

5] employed a spatial error model to investigate the impact of socioeconomic features on crime rates at the census tract level. However, these frequentist statistical models have shortcomings. They cannot handle the small number problem [

6], possibly leading to unstable risk estimates and underestimated standard errors [

7]. Another methodological problem difficult to solve by frequentist statistical models is overdispersion in count data. It has been recognized as a particular problem in generalized linear modeling for the analysis of grouped data [

8]. Accounting for overdispersion is essential as it has an impact on the estimates for regression parameters [

9].

In addition to frequentist approaches, Bayesian methods have also been applied in spatial data analysis. The Bayesian approach, differing from the frequentist approach, views a parameter as a random variable with a probability distribution. Its basic principle is that prior knowledge is combined with data to produce posterior distributions of parameters on which the posterior estimates are based. Implementation of the Markov chain Monte Carlo (MCMC) simulations makes Bayesian statistics and modeling fashionable in a wide range of scientific research [

10]. By MCMC, it is feasible to fit complex models computationally intractable by frequentist statistics. In parallel, the rapid evolution of software such as WinBUGS has further promoted the popularity of Bayesian methods.

Bayesian approaches in spatial data analysis have grown rapidly since the 1990s. They have advantages in fitting complex spatial models and can overcome shortcomings of frequentist approaches. Through random effects terms, Bayesian models account for overdispersion and spatial autocorrelation. They can provide more stable risk estimates than comparable frequentist methods. In spite of these advantages, their application in small-area crime analysis is still not common. Nonetheless, there is a growing awareness of the strengths of the Bayesian spatial modeling approaches. An increasing number of studies have used Bayesian models. Specifically, the influence of socioeconomic variables on property crime [

11,

12,

13], suicide [

14], juvenile offenders [

15], and violent crime [

16], and the spatio-temporal patterns of crime [

17,

18] have been investigated. Bayesian spatial modeling approaches have gradually gained popularity among researches of crime analysis. Despite the growing awareness of and the increased literature on Bayesian spatial modeling, little work has focused on contextual effects in crime analysis.

Contextual effects generally denote the impact of the upper-level area on the lower-level units of analysis. Take the neighborhood and sub-district in China as an example to illustrate the contextual effects. The neighborhood set is a unique subdivision of the sub-district set in China. Each neighborhood falls uniquely within a sub-district and each sub-district consists of several neighborhoods. Neighborhoods are clustered within sub-districts. It is therefore likely that additional clustering of high (or low) crime risks common to neighborhoods falling within the same sub-district may exist due to urban construction, planning policy, urban administration, and public social control. Some grouping effect based on sub-district could be found for neighborhoods lying within a given sub-district. Specifically, the sub-district level should have a contribution from effects at the neighborhood level. As a consequence, there is multiscale information theoretically in the sense that the lower-level neighborhood units fall completely and uniquely within the upper-level sub-district units. Due to this clustered and hierarchical structure of data, local crime pattern is simultaneously affected by characteristics of the environment at multiple spatial scales [

19,

20]. It results from the joint effects of different layers of crime potential [

21]. Furthermore, the influence of crime risk factors is potentially conditional on context and can operate across multiple scales. Neglecting these information can lead to incorrect inferences on risk factors [

22]. One natural way using the multiscale information is to take into account the contextual effects. From the practical perspective, crime risks are influenced by both neighborhood characteristics themselves and contextual effects. Disentangling and identifying the contribution of the two sources of variation on risks are valuable for crime prevention and control. If most of the variation in crime risks is due to neighborhood characteristics, attention should be given to neighborhood-level factors. In contrast, if contextual effects account for most of the variation, a focus on upper-level sub-districts can contribute to a more effective crime prevention strategy.

Contextual effects have been examined in past studies by hierarchical linear models. For example, Johnson and Bowers [

23] focused on the relationship between permeability of a street network and burglary risk while taking account of the nesting in the data. Davies and Johnson [

24] examined the role of road structure (betweenness and linearity) on the spatial variability of burglary risks by a quantitative network analysis. Steenbeek and Weisburd [

25] examined variability of crime for street segments, neighborhoods, and districts in Hague city. Deryol et al. [

26] illustrated the effects of nodes, paths, and the macro-environmental context on the formation and development of crime hotspots. Schnell et al. [

27] explored the impact of street segments, neighborhood clusters, and community areas on the distribution of violent crime in Chicago. The models these researchers adopted accounted for the hierarchical structure of the data and were fitted by frequentist approaches. More recently, Quick [

28] applied a Bayesian cross-classified multilevel modeling approach, accommodating the influence of non-hierarchical upper-level units, to analyze the spatiotemporal patterns of crime in the Region of Waterloo.

In the current research, we aim to shed light on contextual effects on burglary risks. We adopt a contextual effects model to investigate contextual effects on burglary risks. The model is built based on Bayesian spatial modeling strategy and accounts for the hierarchical structure of data. The research contributes to the increasing literature on modeling crime data by Bayesian spatial approaches, as contextual effects are generally overlooked.

The remainder of this paper consists of five parts. Following an introduction, the study area and data are briefly described. Then, we thoroughly introduce the research methodology. Subsequently, results of our analysis are provided. Afterwards, we discuss the results. Finally, we conclude our study.

## 5. Discussion

This article applied a contextual effects model to investigate contextual effects on burglary risks in Wuhan, China. The model was built based on Bayesian spatial modeling strategy. Crime risks are possibly due to a combination of both individual and its upper-level factors. To disentangle the impacts of the two factors on the overall crime risks, the conventional Bayesian spatial modeling approach should be adapted to suit the clustered and hierarchical structure of data. In our research, each neighborhood falls uniquely within a sub-district and each sub-district consists of several neighborhoods. Neighborhoods are clustered within sub-districts. This suggests that neighborhoods from within the same cluster have characteristics in common and may be more alike because of their shared environment. The sub-district level should have a contribution from effects at the neighborhood level. One way of identifying this contribution is to account for the contextual effects of sub-districts on neighborhoods.

We extended the conventional Bayesian spatial model by adding two random effects terms to account for contextual effects (Equation (2)). Both the conventional Bayesian model (Equation (3)) and the contextual effects model (Equation (2)) were used to fit the data. Our analyses showed the two models were nearly indistinguishable in terms of DIC (

Table 2). They had almost the same

$\overline{D}$ and

${p}_{d}$ values. Although the contextual effects model could not be considered superior to the conventional Bayesian spatial model based on their DICs, it was preferred in the sense that it provides insights into contextual effects on crime risks and can work out the relevant contribution.

The results of the study suggested that the conventional and the contextual effects models identified the same set of significant independent variables (

Table 3). To be specific, density of population, bar and department store had a strong association with burglary risks. Population density was inversely related to burglary risks. This is consistent with past research finding that the variable had a negative impact on crime rate [

2,

5,

34,

35]. This may be due to the fact that the Jianghan District is the most populous district of Wuhan and thus there are more opportunities for visual surveillance exit. The two land-use variables, i.e., bar density and department store density, both had a significant positive correlation with burglary risks. Our findings are in line with other studies suggesting that bars [

39] and alcohol outlets [

40] have strong positive impacts on crime. In addition to identifying the same set of significant independent variables, the conventional and the contextual effects models gave similar estimates for their regression coefficients.

The rest of the independent variables, i.e., young males, unemployment, high education and policing, were not significant in both the conventional and the contextual effects models (

Table 3). Their signs in the regression estimates given by the two models were all about the same. Specifically, the percentage of young males did not clearly have a positive effect on burglary risk. This is in accordance with prior study indicating that the variable virtually has no effect on crime [

38]. High education and unemployment were negatively and positively correlated with burglary risks, respectively. This is somewhat contradictory to previous studies founding that unemployment had a significant positive [

5] and the percentage of residents that were well-educated had a significant negative impact on crime [

35]. One possible reason why they were not significant in our analyses may be that our measure of socioeconomic status using only these two variables was not adequate. In addition, issues surrounding the ecological fallacy [

52] are also worthy of attention. Although the magnitude was not significant, the direction of the effects of these two socioeconomic variables was expected. The positive coefficient for policing was not entirely unanticipated. Policing can prevent crime and enhance the perception of safety. However, a greater policing may also be a natural reaction to higher crime rates in the region. Gracia et al. [

16] also found that policing activity had a positive effect on intimate partner violence.

We reported the variance parameters of the random effects terms (

Table 3). Results suggested that risk variation due to neighborhood heterogeneity

${U}_{i}$ is greater than that due to spatial autocorrelation

${S}_{i}$. The contextual effects model further indicated variation due to local random effects is greater than that due to contextual effects.

We mapped the posterior means of burglary risks estimated the conventional and the contextual effects models (

Figure 2). The values demonstrated in

Figure 2a took account of the effects of the independent variables, overdispersion, and spatial autocorrelation. The risks shown in

Figure 2b further considered the contextual effects. It was implemented by extending the conventional Bayesian spatial model to include another two random effects term at the sub-district level. The conventional Bayesian spatial model and the contextual effects model gave almost the same estimate of burglary risk for each neighborhood, with a biggest difference of only 0.02. Both models indicated that there were 38 neighborhoods having a risk value lager than 1.0. Specifically, 21 neighborhoods had a value greater than 1.5, five neighborhoods greater than 3.0, and three neighborhoods greater than 10.0. By illustrating the spatial distribution of burglary risks, the maps demonstrated in

Figure 2 laid a foundation for crime prevention programs.

We adopted the map decomposition technique to work out the contribution of both neighborhood characteristics and contextual effects on burglary risks (Equation (16)). Map decomposition is particularly helpful in visualization and indicative analysis of individual area. It can disentangle the influence of neighborhood characteristics and contextual effects on crime risks. Based on

Figure 2b, by identifying and mapping the relative contribution of neighborhood characteristics (

Figure 3) and contextual effects (

Figure 4), and further demonstrating the posterior probabilities of contextual effects having positive impacts on crime risks (

Figure 5), crime prevention practitioners and police officers can assess their crime reduction measures.

## 6. Conclusions

This study demonstrated a contextual effects model to investigate contextual effects on crime risks. Contextual effects generally denote the impact of the upper-level area on the lower-level units of analysis. Research focusing on crime patterns is often conducted at one spatial scale or at different scales by separate models. Such studies neglect contextual effects and the complex spatial structure of the urban environment [

53]. They do not realize that local crime patterns result from the joint effects of different layers of crime potential [

21] and is simultaneously affected by characteristics of the environment at multiple spatial levels [

19,

20]. Due to the clustered and hierarchical structure of spatial data, crime risks are possibly due to a combination of both individual and its upper-level factors. The influence of crime risk factors is potentially conditional on context and can operate across multiple spatial levels [

22]. Lower-level crime patterns may vary with the characteristics of the upper-level spatial unit they fall within. The variation, to some degree, can be accounted for by including the upper-level factors. However, for some reason, these factors may be unavailable, leading their effects not be captured specifically.

To work out the contribution of the lower-level and upper-level factors on the overall crime risks, conventional regression approaches must be adapted. Our contextual effects model was built based on Bayesian spatial modeling strategy. It was implemented by extending the conventional Bayesian spatial model (Equation (3)) to include two random effects terms at the upper-level area (Equation (2)). The upper-level random effects terms are used to capture the effects of the upper-level factors. The contextual effects model accounts for the effects of independent variables, overdispersion, spatial autocorrelation, and contextual effects. By the model, we can not only identify the relative contribution of the lower-level independent variables and contextual effects on crime risks, but can also obtain the probabilities of contextual effects having positive impacts on lower-level crime risks. Our research contributes to the increasing literature on modeling crime data by Bayesian spatial approaches, as contextual effects are generally neglected.

The analyses presented in this paper have theoretical and practical implications. Theoretically, data are often clustered among hierarchies. This can lead to the emergence of clusters. In our research, neighborhoods are clustered within sub-districts. Neighborhoods within the same sub-district may share the environment and thus have characteristics in common. It is possible that some grouping effect based on sub-district could be found for neighborhoods lying within a certain sub-district. Therefore, the sub-district level should have a contribution from effects at the neighborhood level. The neighborhood’s crime risk may be pulled towards the sub-district level expected risk. Analyzing crime patterns while accounting for contextual effects provides insight regarding the crime-generating process arising across multiple spatial scales and facilitates the identification of the role of each spatial scale on the variation of crime risks. From the practical perspective, police focusing on burglaries in Jianghan ought to focus on neighborhoods having a high bar or department store density. Furthermore, careful thought must be given to neighborhoods having a high burglary risk. An in-depth investigation to identify the distinctive local characteristics of these neighborhoods is necessary. In addition, by illustrating the spatial distribution of neighborhood burglary risks (

Figure 2b), the relative contribution of neighborhood characteristics (

Figure 3) and contextual effects on the risks (

Figure 4), and the posterior probabilities of contextual effects having positive impacts on burglary risks (

Figure 5), our analyses provide the evidential base essential to underpin the decision making process. It can facilitate practitioners, such as police managers and crime analysts, to evaluate crime intervention and prevention efforts that could be strategically targeted towards the characteristics of the problem.

There are also several limitations to our research. The first limitation concerns the contextual effects model itself. As our main aim was to present and illustrate a contextual effects model, we gave the full form of the model but did not further discuss the necessity of incorporating both the two contextual effects terms in the model. Future researchers can specify their contextual effects model according to specific objectives and the actual situation based on our work. The second limitation regards the crime data used in the study. Our burglary data was from the city’s 110-reporting system. Such data could suffer from the problems of entry error and underreporting [

54], and may not accurately reflect the scope of victimization [

55]. The third limitation of the paper involves the independent variables. We used the residential population instead of other measures of population, such as the ambient population [

56] and the commuting population [

57]. This may influence the results of our analyses [

56,

57]. In addition, variables acting as proxies for socioeconomic status, i.e., high education and unemployment, are not comprehensive and adequate. Other commonly used variables, including immigration [

12] and the percentage of families receiving public assistance [

58], were not incorporated into our model due to the unavailability of data. Further, we did not assess how the processes hypothesized by social disorganization and routine activity theories to be applied in multiple spatial scales as data at the sub-district level were not available. The fourth limitation relates to the measure of model complexity. We used

${p}_{d}$ to capture model complexity in our research and did not consider the complexity of the geographical space examined. However, the wide range of spatial and functional complexities involved in studying geographical problems can affect the complexity of the Bayesian models [

59], which in turn has an impact on the computational performance of the models themselves, particularly when used in networks [

60] or trees [

61]. Another limitation concerns the modifiable areal unit problem (MAUP) [

62]. Our data were aggregated to neighborhoods and the results of our analysis might have been influenced by the MAUP.

This research could be expanded in the near future in several ways. First, the applicability of the contextual effects model should be further validated. Further application of the model at other spatial scales with other types of crime is recommended. Second, the contextual effects model can be utilized for a joint analysis at multiple spatial scales. If data are available at different aggregation levels, the model can be used to model them jointly within an analysis. This joint multiscale analysis is worth conducting. Third, through a spatio-temporal analysis, future studies should investigate the temporal dynamics of crime patterns as they change over time [

63].