- Open Access
Hierarchical Bayesian spatial models for alcohol availability, drug "hot spots" and violent crime
© Zhu et al; licensee BioMed Central Ltd. 2006
- Received: 04 October 2006
- Accepted: 07 December 2006
- Published: 07 December 2006
Ecologic studies have shown a relationship between alcohol outlet densities, illicit drug use and violence. The present study examined this relationship in the City of Houston, Texas, using a sample of 439 census tracts. Neighborhood sociostructural covariates, alcohol outlet density, drug crime density and violent crime data were collected for the year 2000, and analyzed using hierarchical Bayesian models. Model selection was accomplished by applying the Deviance Information Criterion.
The counts of violent crime in each census tract were modelled as having a conditional Poisson distribution. Four neighbourhood explanatory variables were identified using principal component analysis. The best fitted model was selected as the one considering both unstructured and spatial dependence random effects. The results showed that drug-law violation explained a greater amount of variance in violent crime rates than alcohol outlet densities. The relative risk for drug-law violation was 2.49 and that for alcohol outlet density was 1.16. Of the neighbourhood sociostructural covariates, males of age 15 to 24 showed an effect on violence, with a 16% decrease in relative risk for each increase the size of its standard deviation. Both unstructured heterogeneity random effect and spatial dependence need to be included in the model.
The analysis presented suggests that activity around illicit drug markets is more strongly associated with violent crime than is alcohol outlet density. Unique among the ecological studies in this field, the present study not only shows the direction and magnitude of impact of neighbourhood sociostructural covariates as well as alcohol and illicit drug activities in a neighbourhood, it also reveals the importance of applying hierarchical Bayesian models in this research field as both spatial dependence and heterogeneity random effects need to be considered simultaneously.
- Census Tract
- Spatial Dependence
- Violent Crime
- Standard Incidence Ratio
- Deviance Information Criterion
With the ability to properly account for high variance of estimates in small geographic areas and clarify overall geographic trends and patterns, Bayesian methods are becoming popular tools for disease mapping. Besag et al.  described a Bayesian approach which separated spatial effects from heterogeneity. Waller et al.  developed a Bayesian hierarchical model that accommodated covariates and spatial structure which evolved over time. The widespread use of geographic information systems (GIS) and links to statistical packages have further encouraged spatial data analysis [3–5]. With the development of Markov Chain Monte Carlo (MCMC) methods and software such as WinBUGS [6, 7], Bayesian approaches are being applied to the analysis of many social and health problems in addition to disease mapping and modelling. In particular, Berry et al. , Cohen et al.  and Law and Haining  used Bayesian statistics for analyzing crime data. Random effects modeling is often used to deal with the problem of overdispersion in modeling count data [11, 12]. Poisson models require that mean and variance are equal, while in real data, overdispersion often results in larger variance than mean.
This paper reports the results of fitting random effect models to census tract-level high violence areas for the City of Houston, Texas, using WinBUGS and its spatial analysis extension GeoBUGS for model fitting. Model fit comparison is accomplished using the deviance information criterion . These analyses will inform the growing literature within public health and criminology that is focused on the geospatial relationship between aspects of the physical and social environment and the commission of violent crime [15–18].
The remainder of this paper is organized as follows. The Method section describes the datasets pertaining to alcohol availability, drug hot spots, violent crime and sociodemographic characteristics at the census-tract level in Houston, Texas. It also undertakes an analysis of these datasets in which the alcohol and the drug covariates are included with neighborhood sociostructural variables using four random effect models, and the models are compared based on Deviance Information Criterion. The Result section reports the results of the chosen Bayesian model. Map decomposition is applied to interpret the distribution of high-violence areas in Houston. The Discussion section provides a comparison of the findings to previous studies and presents some possible avenues for future research. The Conclusion section summarizes the major findings in this research.
The sample for this study comprised of 439 census tracts from Houston for which violent crime data were reported by the city police department. Houston is the largest city in Texas and the fourth largest in the USA. The boundaries for the tracts were those established for the 2000 US Census. Census tracts are not necessarily corresponding to neighborhoods in a socially meaningful sense, but they are considered to be the most appropriate boundary to use in assessing the relationship between neighborhood structure and violent crime [19, 20], because they are relatively homogeneous with regard to population and are the best local areas where the required data are available. Three archival datasets were employed in the present study. The first one included monthly police reports of four violent crime categories of murder, rape, robbery and aggravated assault in 2000, extracted from the City of Houston Police Department website . The violent crime data are based on first reports of offenses (that is, before investigation and final classification of crimes). The website also contained monthly reports of narcotic drug-law violations. Like the violent crime data, the drug-law violation data were also based on first reports of offenses. Available evidence suggests that police data represent a valid indicator of actual drug activity in a community . Such "call for assistance" data have been used in previous studies of alcohol availability and violent crime, and have strengths as well as limitations relative to official crime records [23–25]. Close to 98% of the data were successfully geocoded and aggregated to census tract level in the analysis. The second dataset pertained to alcohol outlets. A list of all alcohol outlets active in the year 2000 in Houston was obtained from the website of the Texas Alcoholic Beverage Commission (TABC) . There were a total of 6,609 alcohol outlets in the dataset, each of which included the name, geographic location, and type of permit or license of the outlet. Almost all of the outlets (99.5%) were successfully geocoded by street address using Centrus Desktop . TABC classified the outlets by type of consumption allowed, specifically, off-premise versus on-premise. 1,480 (22.4%) of all outlets were on-premise, 3,094 (46.8%) off- premise, and 2,035 (30.8%) combined on-/off- premise. Outlet density was defined as number of outlets per 100 persons in a census tract. Principal component analysis identified total alcohol outlet (including on-, off-, and on-/off- premise) density to be the major factor in this group of covariates.
The third dataset used in the study pertained to 12 neighborhood sociostructural factors. These variables, which were grouped under three broad categories, were selected as they had been used in previous ecologic studies of alcohol availability and crime [16, 25, 28]. We use these variables as potential covariates in our analysis to keep in line with the existing literature and for possible comparisons between different studies. We realized that there might be collinearity issue and our study area might have a unique set of covariates which may not be exactly the same as previous studies, so we conducted a variable reduction using principal component analysis described below. The data were extracted from Summary File 1 and Summary File 3 of the 2000 US Census . Of these variables, six were measures of concentrated disadvantage (% families below poverty line, % families receiving public assistance, % unemployed individuals in civilian workforce, % female-headed households with children, % Black, and % Latino), three were measures of residential instability (% of residents over age 5 who have lived in the same house for 5 or more years, % homes that are owner-occupied, and % vacant housing units), and three were sociodemographic measures of the resident population (adult to child ratio, population density, and % population that is male and aged 15–24). Since these variables were highly correlated with each other , principal component analysis was conducted to reduce the number of covariates. To retain the basic component structure of these neighbourhood sociostructural factors, principal component analysis was done separately for a) the six variables of concentrated disadvantage, b) the 3 measures of residential instability, and c) the three sociodemographic measures. In each of the three groups, components with eigenvalues greater than 1 were extracted and the variable with the highest factor loading for each extracted components was selected. Four variables were selected from the three groups, which were a) % of families below poverty, and % of Latino from the concentrated disadvantage measures, b) % owner-occupied housing units from the measure of residential instability, and c) % population that is male and aged 15–24 from the socidemographic measures.
Bayesian hierarchical modelling
The Bayesian approach has become standard in the epidemiology and environmental health literature, but has only recently been used in sociological applications. It is called "hierarchical" because it uses multiple levels of analysis in an iterative way. Unlike the conventional statistical inference which derives the average estimates of parameters, hierarchical Bayesian modelling produces parameter estimates for each individual analysis unit by borrowing information from all analysis units, the customary Bayesian "borrow of strength" effect. In a standard Poisson model, the variance is required to be equal to the mean. But in reality many Poisson models have more variances and these are called over-dispersed Poisson models. Hierarchical Bayesian modelling identifies these "extra variances". In the case of spatial statistics in non-Bayesian approach, if there is high uncertainty in the regression model, a regression that explains only a small amount of variance is obtained. In a hierarchical Bayesian model, on the other hand, the "unexplained variance" is usually identified as either spatially-correlated effects or heterogeneity effects. The ability to incorporate prior knowledge without the restriction of classical distributional assumptions makes Bayesian inference a potent forecasting tool in a wide variety of fields.
In the first stage of the Bayesian hierarchical model, we specified a likelihood model for the vector of observed crime counts given the vector of relative risks of crimes, and then specified a prior model over the space of possible relative risks at the second stage. Using software packages such as WinBUGS/GeoBUGS [31, 13] or sophisticated computation algorithms could yield a set of posterior means for the relative risks given the observed crime counts. The set of posterior means or medians of the relative risks was then used to create a map to visualize the high- or low-risk census tracts. Crude maps were developed from the likelihood model (the first stage) only, and often feature large outlying relative risks in small areas (where the population is small). Hence, crude maps usually show high uncertainty. They also fail to catch similarity of relative risks in nearby or adjacent regions. An appropriately-tailored Bayesian approach will incorporate spatial assumptions and achieve spatial smoothing by borrowing information from all individuals.
The violent crime counts could be modelled as a conditional Binomial(n, p) variable with the population sizes to be n and a unknown rate p. An initial check into the data reveals that the population sizes in the majority of the census tracts are large (in thousands) and the violent crimes are rare, with an average rate of around 5%. So we use the Poisson approximation to binomial distribution at the first stage of likelihood specification. The model assumes that the number of crime counts in region i, Y i , has a conditional independent Poisson distribution with mean E i exp(μ i ). Here E i is the expected number of events, which is fixed and proportional to the corresponding known population n i . Specifically, we set E i = Rn i , where the proportional constant R is the grand rate (i.e., the total number of events divided by the total population). Hence exp(μ i ) is the relative risk: regions with exp(μ i ) > 1 generally have greater numbers of observed incidence than expected, and vice versa for regions with exp(μ i ) < 1. Thus μ i is log-relative risk, modelled linearly as
μ i = x' i β + θ i + φ i , i = 1, ..., I (1)
Here x' i are region-specific covariates, and β is a vector of fixed effects. θ i and φ i are the region-specific random effects capturing heterogeneity and spatial dependence, respectively. The typical way to impose this structure is to assume that θ i 's are i.i.d. Gaussian variables with mean 0 and variance 1/τ and φ i |φj≠i~ N( , ), i = 1,...,I, where
Weights w ij are fixed constants. With this structure, θ i 's capture heterogeneity among regions and φ i 's capture spatial dependence or autocorrelation. In practice, a common choice is to let w ij = 0 unless areas i and j are adjacent, in which case w ij = 1. Other forms of w ij (often using distance between centroids of regions i and j) are possible [32, 33]. This distribution for ϕ = (φ1,..., φ I ) is called an intrinsic conditionally autoregressive specification [2, 10, 13], which for brevity we typically write in vector notation as ϕ ~ ICAR(λ). A fully Bayesian model specification is completed by adding prior distributions on β, τ and λ. Without prior expectations about direction and magnitude of the covariate effects, a vague but proper prior distribution is put on the regression coefficients β. Prior distribution for τ and λ is Gamma(a, b) with mean a/b and variance a/b2. With no prior estimation for precisions of the random effects, small values of a and b are chosen to assume large variance. Here we have chosen Gamma(0.001,0.001), a vague prior for precision parameters of both effects.
Principal component analysis was conducted to obtain a subset of four neighborhood variables. The main effects of interest in this study are those of alcohol outlet densities and drug-law violation densities on violence. So the covariate matrix includes six explanatory variables, and the dependant variable is the occurrence of violence in each census tract which follows a Poisson distribution as specified earlier. We fitted four models with or without random effects, and assessed the performance of them using Bayesian approaches. Model I contained the fixed effects only. It assumes that the parameters were fixed for each region i and all relevant covariates had been correctly specified. An extension was to specify random effects which included either one or both of unstructured (θ) and spatial dependence effects (ϕ). These were called Model II (fixed effects and θ), Model III (fixed effects and ϕ), and Model IV (full model), respectively.
Statisticians have long paid attention to the model selection problem. For years, Bayesian statisticians were advised to use only Bayes factors  for this purpose. However, Bayes factor becomes quite difficult both to compute and interpret for high-dimensional hierarchical models, and is not well-defined for models having improper prior distributions. The difficulties with Bayes factor have led to a host of alternative model choice criteria. Most recently, Spiegelhater et al.  proposed a generalization of the Akaike Information Criterion  that was based on the posterior distribution of the deviance statistic
D(ϑ) = -2log p(y|θ) + 2log f(y).
Here p(y|θ) is the likelihood function for the observed data vector y given the parameter vector θ, and f(y) is a standardizing function of the data alone. In this approach the model fit is summarized by the posterior expectation of the deviance, = Eθ|y(D), while model complexity is captured by the number of effective parameters p D , which is defined as expected deviance minus deviance evaluated at the posterior expectations, i.e.
p D = Eθ|y(D) - D(Eθ|y(θ)) = - D ( )
Deviance information criterion (DIC) is then defined as the summation of fit and complexity, i.e.
DIC = + p D = 2 - D( )
Deviance summaries for the four hierarchical models
I Fixed effects only
II Fixed and heterogeneity
III Fixed and dependence
IV Full model
where sd(.) is empirical marginal standard deviation. Hence α is the proportion of variability in the random effects that is due to spatial dependence. Larger values (near 1) suggest a dominating spatial dependence, while smaller values (near 0) suggest a negligible one. Recall that we specified the same gamma prior distribution for τ and λ, i.e., α = 1/2. Fitting data with the full model obtains the posterior distribution of α with mean 0.584, median 0.582, and a 95% credible interval (0.450, 0.733). This indicates that approximately 60% of excess variability is due to spatial dependence, while 40% is due to unstructured random noise. This confirms that Model IV (full model with both spatial dependence and spatial heterogeneity) is the best among the candidate models.
Two parallel sampling chains were run with overdispersed initial values. Convergence was assessed by checking the trace plots of the samples, autocorrelation functions, the Gelman-Rubin convergence statistic , and Monte Carlo standard errors . The four models described above had different "burn-in" (pre-convergence) periods, with slower convergence for the more complex models. Convergence was detected at 20,000 iterations in the full model. For each model, the first 20,000 pre-convergence samples were discarded and each chain was run for a further 25,000 iterations, giving 50,000 samples with acceptable Monte Carlo errors (<5% of the posterior standard deviation).
Sensitivity analysis was conducted to investigate whether results in the analysis remained essentially unchanged in the presence of different prior information. Vague but proper prior distributions Normal(0, .000001) and Gamma (0.001,0.001)  were first specified for the covariate coefficients and precision parameters, respectively. Then we made modifications to the prior distributions, recomputed the posterior quantities of interest and checked whether they imposed a practical impact on interpretations or decisions. We adapted an informative prior distribution Gamma (0.1, 0.1) and another commonly used Gamma (0.5, 0.0005)  for this purpose. The three distributions gave almost identical results for all the parameters, indicating the results are robust to changes in prior information.
Posterior summaries for regression and precision coefficients
Monte Carlo error
% Population of male aged 15–24
Alcohol outlet density
Drug-law violation density
Unstructured random effect
While data transformation makes interpretation somewhat difficult, it is instructive to examine in detail a typical (and real) inner-city census tract with a population the size of 1191, 12 alcohol outlets, and 116 drug-law violation reports in the year 2000. Assuming all other covariates remain the same, if the number of alcohol outlets increases to 48 (with a standard deviation of 36), then, on average, the relative risk of violence will increase 16%; if the number of drug-law violation reports increases to 168 (with a standard deviation of 52), then the relative risk of violence increases 1.5 times. It is apparent that the effects of alcohol outlets are quite modest compared to the effects of drug-law violations. It is hard to imagine any real-word community would increase the alcohol outlets such dramatically, but increase of drug-law violations from 116 to 168 would be possible. Precision for spatial random effects is lower than the unstructured heterogeneity. Later analysis of the maps shows the same tendency.
The analysis presented suggests that activity around illicit drug markets is more strongly associated with violent crime than is alcohol outlet density. This supports previous studies that have shown a spatial link between drug "hot spots" and violence . However, it is possible that some of the association between drug crime density and violent crime rates found in the present study is due to the fact that the data used to assess each of these came from a common source, namely the Houston police department. So, while available evidence suggests that police data provide a meaningful indicator of drug activity in a community , it would have been preferable to have some additional measure of this variable that was independent of law enforcement records (e.g., neighborhood surveys). It also needs to be noted here that an association has been identified between violent crimes and neighborhood covariates as well as alcohol and drug activity, it requires further investigation as to whether the exposure (alcohol and drug activity) and the outcome (violence) are both consequences of the same unmeasured phenomenon. There is not necessarily a causal relation between the exposure and the outcome.
A major contribution of the current study is to reveal the importance of spatial analysis in the research into the interaction of alcohol availability, illicit drug market, and violent crime. The Bayesian hierarchical modelling approach provides the methodology that incorporates complex data and model levels, with spatial dependence structure. Freely available software such as WinBUGS/GeoBUGS enables wider use of the developed methods. A useful extension to the methods described here would be to assess spatially heterogeneous interactions between disparate data sources . In the current study, the impact of alcohol availability and illicit drug activity on violent crime is assumed to be constant over the whole study area. There are theoretical reasons to believe that the associations may be different in different geographic locations. The challenge for ecological research programs is to develop models of the spatial interactions of people and places to predict violence across community areas.
In this paper we have presented a three-level Bayesian hierarchical modelling approach to model the occurrence of violence in Houston, Texas. The first level was the likelihood of the occurrence of violence that follows a Poisson distribution. Level 2 modelled the log relative risk as a linear combination of three components which accounted for fixed effects of possible covariates, random effects of unstructured heterogeneity and spatial dependence. At level 3, non-informative hyperprior distributions were assigned to the precision parameters for the random effects. Eight explanatory variables were identified via principal component analysis. Four models that included/excluded the two random terms were compared based on Deviance Information Criterion and proportion of the variability in the random effects that is due to spatial autocorrelation. The full model which considered both unstructured and spatial dependence random effects was selected as the best fitted model. Sensitivity analysis was performed to check whether the prior assumptions on the precision parameters had an undue effect on the results. Of the three fixed effects that contributed to the relative risk of violent crime, drug-law violation explained a greater amount of variance in violent crime rates than alcohol outlet densities and neighbourhood sociostructural variables. The census tract level random effects included both a spatial dependence and an unstructured heterogeneity effect, suggesting that it is necessary to carry out this research in a Bayesian hierarchical framework where different formats of random effects and fixed effects are considered. The residual map indicated that the quality of model fit was satisfactory.
This research was supported by a grant from the Harry Frank Guggenheim Foundation.
- Besag J, York J, Mollie A: Bayesian image restoration, with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics. 1991, 43: 1-59. 10.1007/BF00116466.View ArticleGoogle Scholar
- Waller LA, Carlin BP, Xia H, Gelfand AE: Hierarchical spatio-temporal mapping of disease rates. Journal of the American Statistical Association. 1997, 92: 607-617. 10.2307/2965708.View ArticleGoogle Scholar
- Lawson A: Statistical methods in spatial epidemiology. 2001, New York: John Wiley & SonsGoogle Scholar
- Waller LA, Gotway CA: Applied spatial statistics for public health data. 2004, New York: John Wiley & SonsView ArticleGoogle Scholar
- Banerjee S, Carlin BP, Gelfand AE: Hierarchical modeling and analysis for spatial data. 2004, New York: Chapman & Hall/CRCGoogle Scholar
- Spiegelhalter DJ, Thomas A, Best N, Gilks WR: BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.50. 1995, Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge UniversityGoogle Scholar
- Carlin BP, Louis TA: Bayes and empirical Bayes methods for data analysis. 2000, Boca Raton: Chapman & Hall/CRC, 2View ArticleGoogle Scholar
- Berry D, Evett I, Pinchin R: Statistical inference in crime investigations using deoxyribonucleic acid profiling. Journal of the Royal Statistical Society C. 1992, 41: 499-531.Google Scholar
- Cohen J, Nagin D, Wallstrom G, Wasserman L: Hierarchical Bayesian analysis of arrest rates. Journal of the American Statistical Association. 1998, 93: 1260-1270. 10.2307/2670041.View ArticleGoogle Scholar
- Law J, Haining R: A Bayesian approach to modeling binary data: the case of high-intensity crime areas. Geographical Analysis. 2004, 36: 197-215. 10.1353/geo.2004.0010.View ArticleGoogle Scholar
- Mollie A: Bayesian mapping of disease. Markov Chain Monte Carlo in Practice. Edited by: Gilks W, Richardson S, Spiegelhalter D. 1996, London: Chapman and Hall, 359-379.Google Scholar
- Wakefield J, Best N, Waller L: Bayesian approaches to disease mapping. Spatial Epidemiology: Methods and and Applications. Edited by: Elliott P, Wakefiled J, Best N, Briggs D. 2000, Oxford: Oxford University PressGoogle Scholar
- Thomas A, Best N, Lunn D, Arnold R, Spiegelhalter DJ: GeoBUGS user manual Version 1.2. 2004, Medical Research Council Biostatistics Unit, Cambridge UniversityGoogle Scholar
- Spiegelhalter DJ, Best N, Carlin BP, van der Linde A: Bayesian measures of model complexity and fit (with discussion). Journal of Royal Statistical Society B. 2002, 64: 583-639. 10.1111/1467-9868.00353.View ArticleGoogle Scholar
- Lipton R, Gorman DM, Wieczorek WF, Gruenewald P: The application of spatial analysis to the public health understanding of alcohol and alcohol-related problems. Geographic Information Systems and Health Applications. Edited by: Khan OA. 2003, Hershey, PA: Idea Group Publishing, 57-79.View ArticleGoogle Scholar
- Morenoff JD, Sampson R, Raudenbush S: Neighborhood inequality, collective efficacy, and the spatial dynamics of urban violence. Criminology. 2001, 39: 517-559. 10.1111/j.1745-9125.2001.tb00932.x.View ArticleGoogle Scholar
- Skogan WG: Disorder and decline: crime and the spiral of decay in American neighborhoods. 1990, Berkeley: University of California PressGoogle Scholar
- Cohen DA, Ghosh-Dastidar B, Scribner R, Miu A, Scott M, Robinson P, Farley TA, Bluthenthal RN, Brown-Taylor D: Alcohol outlets, gonorrhea, and the Los Angeles civil unrest: a longitudinal analysis. Social Science & Medicine. 2006, 62: 3062-3071. 10.1016/j.socscimed.2005.11.060.View ArticleGoogle Scholar
- Krivo LJ, Peterson RD: Extremely disadvantaged neighborhoods and urban crimes. Social Forces. 1996, 75: 619-650. 10.2307/2580416.View ArticleGoogle Scholar
- Scribner RA, Cohen DA, Kaplan S, Allen SH: Alcohol availability and homicide in New Orleans: conceptual considerations for small area analysis of the effect of alcohol outlet density. Journal of Studies on Alcohol. 1999, 60: 310-316.PubMedView ArticleGoogle Scholar
- City of Houston Police Department Crime StatisticsWebsite. (Accessed 1/15/2005), [http://www.houstontx.gov/police/cs/beatpages/beat_stats.htm]
- Warner BD, Coomer BW: Neighborhood drug arrest rates: are they a meaningful indicator of drug activity? A research note. Journal of Research in Crime and Delinquency. 2003, 40: 123-138. 10.1177/0022427802251018.View ArticleGoogle Scholar
- Nelson AL, Bromley RDF, Thomas CJ: Identifying micro and temporal patterns of violent crime and disorder in the British city center. Applied Geography. 2001, 21: 249-274. 10.1016/S0143-6228(01)00008-X.View ArticleGoogle Scholar
- Weisburd D, Mazerolle LG: Crime and disorder in drug hot spots: implications for theory and practice in policing. Police Quarterly. 2000, 3: 331-349.View ArticleGoogle Scholar
- Zhu L, Gorman DM, Horel S: Alcohol outlet density and violence: a geospatial analysis. Alcohol & Alcoholism. 2004, 39: 369-375. 10.1093/alcalc/agh062.View ArticleGoogle Scholar
- TABC License/Permit Public Information. (Accessed 1/15/2005), [http://www.tabc.state.tx.us/pubinfo/rosters/default.htm]
- Group 1 Software: Centrus Desktop, Version 4.01. 2003, Boulder: Group 1 SoftwareGoogle Scholar
- Lipton R, Gruenewald PJ: The spatial dynamics of violence and alcohol outlets. Journal of Studies on Alcohol. 2002, 63: 187-195.PubMedView ArticleGoogle Scholar
- US Census Bureau: 2000 United States Decentennial Census (Summary File 1 and Summary File 3). 2003, Washington: US Department of CommerceGoogle Scholar
- Gorman DM, Zhu L, Horel S: Drug "Hot Spot", alcohol availability and violence. Drug and Alcohol Review. 2005, 24: 507-513. 10.1080/09595230500292946.PubMedView ArticleGoogle Scholar
- Spiegelhalter DJ, Thomas A, Best N, Lunn D: WinBUGS user manual Version 1.4.1. 2003, Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge UniversityGoogle Scholar
- Cressie N, Chan NH: Spatial modeling of regional variables. Journal of the American Statistical Association. 1989, 84: 393-401. 10.2307/2289922.View ArticleGoogle Scholar
- Conlon EM, Waller LA: Flexible spatial hierarchical models for mapping disease rates. American Statistical Association Proceedings of the Section on Statistics and the Environment: Aug 13–17. 2000, 82-87.Google Scholar
- Jeffreys H: Theory of probability. 1961, Oxford: University Press, 3Google Scholar
- Akaike H: Information theory and an extension of the maximum likelihood principle. 2nd International symposium on information theory. Edited by: Petrov BN, Csaki F. 1973, Budapest: Akademiai Kiadol, 267-281.Google Scholar
- Zhu L, Carlin BP: Comparing hierarchical models for spatio-temporally misaligned data using the deviance information criterion. Statistics in Medicine. 2000, 19: 2265-2278. 10.1002/1097-0258(20000915/30)19:17/18<2265::AID-SIM568>3.0.CO;2-6.PubMedView ArticleGoogle Scholar
- Gelman A, Rubin DB: Inference from iterative simulation using multiple -sequences (with discussion). Statistical Science. 1992, 7: 457-511.View ArticleGoogle Scholar
- Best NG, Waller LA, Thomas A, Conlon EM, Arnold RA: Bayesian models for spatially correlated diseases and exposure data (with discussion). Bayesian Statistics 6. Edited by: Bernardo JM, Berger JO, Dawid AP, Smith AFM. 1999, Oxford: Oxford University Press, 131-156.Google Scholar
- Weisburd D, Mazerolle LG: Crime and disorder in drug hotspots: implications for theory and practice in policing. Police Quarterly. 2000, 3: 331-349.View ArticleGoogle Scholar
- Gelfand AE, Kim HJ, Sirmans CF, Banerjee S: Spatial modeling with spatially varying coefficient processes. Journal of the American Statistical Association. 2003, 98: 387-408. 10.1198/016214503000170.View ArticleGoogle Scholar
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