Multiresolution Analyses of Neighborhood Correlates of Crime: Smaller Is Not Better

Abstract

Population analyses of the correlates of neighborhood crime implicitly assume that a single spatial unit can be used to assess neighborhood effects. However, no single spatial unit may be suitable for analyses of the many social determinants of crime. Instead, effects may appear at multiple spatial resolutions, with some determinants acting broadly, others locally, and still others as some function of both global and local conditions. We provide a multiresolution spatial analysis that simultaneously examines US Census block, block group, and tract effects of alcohol outlets and drug markets on violent crimes in Oakland, California, incorporating spatial lag effects at the 2 smaller spatial resolutions. Using call data from the Oakland Police Department from 2010–2015, we examine associations of assaults, burglaries, and robberies with multiple resolutions of alcohol outlet types and compare the performance of single (block-level) models with that of multiresolution models. Multiresolution models performed better than the block models, reflected in improved deviance and Watanabe-Akaike information criteria and well-supported multiresolution associations. By considering multiple spatial scales and spatial lags in a Bayesian framework, researchers can explore multiresolution processes, providing more detailed tests of expectations from theoretical models and leading the way to more effective intervention efforts.

Abbreviation

CAR

conditional autoregressive

In their efforts to assess the ecological correlates and determinants of crime across community neighborhoods, criminologists, social epidemiologists, and substance abuse researchers have typically selected some single spatial unit of analysis for their work (1–5). There appears to be no reason to believe that there is any single spatial scale suitable for ecological analyses of crime across community areas, but, with the limited and rare exception of studies which consider spatial lag effects (6–8), this assumption is common to all 3 literatures. Here we argue for the importance of examining multiresolution effects in these studies.

In considering issues of spatial scale, it is helpful to distinguish between the spatial scale of the underlying processes and the spatial scale observed and recorded in data sources. As researchers have developed more sophisticated theories and analytical methods for assessments of spatial patterns of crime and violence (9), many have made the claim that observed data from smaller spatial analysis units, like street segments and US Census blocks, can be used to provide better information about the ecological causes and correlates of crime than larger analysis units, like US Census tracts and zip codes (10). Certainly, in one respect, this “smaller is better” viewpoint must be true; data collected from smaller units, ideally point pattern data, can be aggregated to any larger area unit a researcher is inclined to use, and information loss due to aggregation is fully under the control of the investigator. Furthermore, many epidemiologists assume that aggregated data may be subject to the ecological fallacy and that less aggregation would reduce the amount of bias; this concern is substantially mitigated when ecological variables are the constructs of interest rather than acting as markers of individual-level covariates.

“Smaller is better” may also be a virtue from a theoretical perspective, as the spatial scales of underlying processes are often posited to be small. According to crime attractor theory (11), spatial crime patterns can be highly localized, potentially with some spillover (i.e., spatial lag) effects into immediately neighboring areas. This leads to the expectation that census blocks, or even smaller parcels of land, would be the “smaller” and “better” units by which to identify hot spots of violent criminal behavior. On the other hand, much theoretical work has focused on the impacts of larger “neighborhood” environments on health behaviors and outcomes (12, 13). In this case, collective efficacy and social disorganization theories suggest that “neighborhood” effects will be expressed across larger units like census block groups or tracts. Thus, in recent work examining spatial correlates of the geographic distributions of different types of crime, Quick et al. (14) found global crime patterns related to the larger social processes indicated by social disorganization theory and local crime patterns related to more local social processes reflecting routine activities of crime victims and perpetrators. With these thoughts in mind, it is not surprising to observe that some studies find health outcomes to be tightly related to locations of use (e.g., alcohol outlets and violence (15)) while others find distal associations with use (e.g., alcohol outlets and intimate partner violence and child physical abuse (16, 17)). Furthermore, the literature is somewhat muddied by studies that examine the same variables but identify significant associations at varying spatial scales (e.g., a link between alcohol outlets and violent crimes at the census block group (6, 18) vs. city (5) level).

Among social epidemiologists who study crime as a public health concern related to neighborhood environments, the relevant spatial scales of “neighborhood effects” are also uncertain and appear conditional upon the outcomes of interest. A few studies have conducted sensitivity analyses to find a single “best” spatial scale for a given location, outcome, and data source (19, 20), but the search for a single correct spatial resolution by which to examine any public health outcome may be misleading. In particular, there may be no single spatial unit of analysis suitable for analyses of the complex of social determinants of crime. Instead these effects may appear at multiple spatial resolutions, with some determinants acting globally, others locally, and still others by some function of both global and local conditions that support crime. This parallels concerns in geostatistics that typical geographic or multilevel spatial models that utilize 1 spatial scale, or that compare model fit between models with different (singular) spatial scales, may misspecify causal relationships (21).

Finally, given that many social epidemiologic data are available only from administrative units of one kind or another and these units often do not match (e.g., sociodemographic data from census block groups vs. health outcome data from zip code areas), reaggregation to a common unit has its own risks; the reaggregation itself may be a source of biased estimates of local neighborhood conditions (22). These problems can be ameliorated by defining neighborhoods more flexibly, in terms of the empirical range of covariate associations either expected by theory or empirically observed through multiresolution analyses. There may be no one spatial scale that is best for the assessment of a given covariate-outcome relationship. When a social process is localized, highly resolved spatial analyses can identify these local effects, while nonlocal, less resolved analyses may miss the mark.

We conducted a multiresolution spatial analysis that simultaneously examined US Census block, block group, and tract effects of alcohol outlets and drug markets on violent crimes within 1 West Coast city, while also incorporating spatial lag effects at the 2 smaller spatial resolutions. Using call data from the Oakland, California, Police Department from 2010–2015, we examined associations of assaults, burglaries, and robberies with multiple resolutions of alcohol outlet types. We inspected the performance of single- versus multiple-resolution models using a spatial analysis model that is widely used in the field, and we included conditional autoregressive (CAR) spatial random effects for census blocks and nonspatial random effects for blocks, block groups, and tracts. While the focus of these analyses was on spatially resolved crime data, these spatial models are commonly used in spatial epidemiology (23, 24), and issues of resolution and scale are a common theme (25). Our proposed solution is broadly applicable across many public health outcomes and is one method for addressing the important public health approach of multiresolution modeling.

METHODS

Study sample

We used point-located call data from the Oakland Police Department from 2010–2015, geocoded and aggregated to 6,282 census blocks, 336 census block groups, and 113 census tracts each year using 2010 US Census units (see Web Figure 1, available at https://academic.oup.com/aje, for example nesting of blocks, block groups, and tracts). Nationally, 4,871,270 of the 11,078,300 census blocks (approximately 44%) in the 2010 US Census had zero population (26). Because census blocks in our sample were often splinters that contained few (or no) individuals (e.g., median strips of roadways), we iteratively merged blocks with no population with those sharing the greatest contiguous border which did not also appear in another census block group (thus preserving the nested hierarchical structure of blocks within block groups). Splinters were defined as those blocks with a logged “compactness” ratio of 2.9 or greater, where “compactness” was defined as the ratio of the perimeter of each block area to the perimeter of an equal area circle. This value minimized the numbers of nonsplinters with no population (16% had populations under 10) and splinters with population (10% had population counts of 10 or more), based on the assumption that splinters should have no population and nonsplinters should have population. The resulting 3,768 synthetic blocks were relatively compact and preserved qualitative aspects of the local geographies of different areas of Oakland (see Web Appendix 1, Web Figure 2, and Web Table 1 for details). Spatial lags were calculated for census blocks and block groups as the unweighted average of all adjacent neighbor block and block group covariates, respectively.

Measures

All measures used in our analyses were geocoded to census block, block group, and/or tract levels, with spatial lags constructed around blocks and block groups of some covariates. Three crime outcomes were considered: assaults (California Penal Codes 240–248), burglaries (codes 458–464), and robberies (codes 211–215). The measure of each type of crime was the annual number of corresponding emergency phone calls for service to a police station in Oakland from 2010 through 2015. We were able to geocode 115,334 of 120,747 unique addresses (95.5% of all addresses). There were 153,190 geocoded crimes in these categories in the final sample, geocoded into 3,768 synthetic blocks for each year.

Sociodemographic measures were obtained from GeoLytics, Inc. (Branchburg, New Jersey), providing annual estimates for census-derived variables at the block (population counts) and block group (all other covariates) levels (27). Population density was calculated at all 3 resolutions, since data on both population counts and area were available at the block level, and was scaled as 1,000 persons/mile². Several economic and demographic measures were included at the block group and block group-lag levels, since data on these variables were not available at the block level. Measures of economic conditions within each census area included median annual household income (per $10,000) and Census-defined percentage of families living in poverty. Racial/ethnic composition was represented by percent Hispanic and percent Black, nonexclusive categories of race (Black) and ethnicity (Hispanic).

Alcohol outlets were classified as off-premise (license types 20 and 21), restaurant (license types 41 and 47), or bar/pub (license types 23, 40, 42, 48, 61, and 75) establishments based on annual retail license data from the California Department of Alcoholic Beverage Control (28). Outlet densities were calculated as the number of outlets within each outlet type per square mile, with measures created at the block, block group, tract, block lag, and block group-lag levels.

Measures of overall retail parcel density were included in analyses at the block, block group, tract, block lag, and block group-lag levels. Oakland parcel data were accessed through openoakland.org, an online public-record repository for the city (29). A subset of 28 parcel use codes were identified that represent retail sales outlets (e.g., supermarkets, hotels) as of June 2013, and these were used to compute densities per square mile.

To control for effects related to illegal drug market activities, we calculated measures of drug sales and drug transportation crimes per square mile from the Oakland police call data. Drug-crime categories included marijuana sale and transport (California Health and Safety codes 11359, 11357.5, and 11360), loitering for drug activity (code 11532), narcotic sales and transport (code 11352), and dangerous drug sales and transport (codes 11378 and 11379). These crime market indicators were generated at the block, block group, tract, block lag, and block group-lag levels and were scaled as number of crimes per square mile × 10.

Estimating multiresolution effects

Based upon prior work examining drug and alcohol markets and their relationships to crime across community areas (30), we assessed multiresolution effects among 4 key exposures related to neighborhood crime: alcohol outlet densities, population density, retail parcel densities, and drug sale/transportation crimes. Aggregates of these measures at census block, block group, and tract levels were jointly related to crime counts measured within blocks. Additional covariates (median household income, percentage of families in poverty, and percent Black and Hispanic) were included at the block group- and block group-lag levels only. To ensure identifiability across similar measures at different spatial resolutions, block group data were calculated relative to tract averages (block group − tract), with block group spatial lags calculated as averages for surrounding block groups. Block data were calculated relative to block group averages (block − block group), with block spatial lags calculated relative to block averages (block spatial lag − block). Because the spatial lagged values were calculated relative to each spatial unit, estimated effects were centered with respect to those units. This enabled us to assess spatial lag effects while preserving a standard hierarchical framework with nested effects related to subordinate and superordinate measures.

Data analysis

Since crime data observed across spatially contiguous census tracts, block groups, and blocks are not independent, spatial analysis models that address failures of unit independence are required for crime analyses within cities. We used hierarchical Bayesian space-time Poisson models incorporating CAR random effects to account for spatial dependence at the block level (the smallest unit of analysis) and spatially unstructured random effects included to account for any additional loss of unit independence across all 3 spatial scales.

(1)

For each outcome, is the number of reported crimes (assaults, burglaries, or robberies) in block in year . is the expected number of crimes based on the assumption that crimes are distributed across blocks in direct proportion to block area, rather than population count. is the relative rate for each spatial unit at time . For the block models, the log(relative rate) is modeled as

(2)

Here, the matrix contains time-specific block-level covariates for each block at time , and is a vector of estimates of these effects. is a matrix of time-specific demographic covariates available only at the block group level for each block group at time . Similarly, the vector contains estimates for these effects. is the spatially unstructured random effect for each block at time , and is the same for each block group at time represents the spatially structured CAR random effect.

In the multiresolution models, the log(relative rate) is

(3)

is as described for the block model, now including block lag effects for each block. Similarly, contains block group–level effects, this time not confined to demographic effects, and including block group-lag effects. is an analogous matrix for tract-level effects for each tract at year . The are parameters describing the associations between each covariate and the log of the mean. Once the model is fitted, they become estimates. , and are as described for the block model. are spatially unstructured random effects for each tract at year . Since nonlinear time effects were apparent in the original scale of outcome counts, year was included as a categorical variable.

We used a reparameterized version of the standard Besag, York, and Mollié model that allows the mixing hyperparameter to be interpreted as the proportion of marginal variance explained by the CAR spatial random effect at block at year (31). The hyperprior selected for was a penalized complexity (PC) prior, , and for , it was set to (31). In all models, noninformative priors were assigned to all other fixed and random effects.

Models were fitted using the R-INLA package (32). Integrated nested Laplace approximation is a fairly novel approach that yields approximations comparable to Markov chain Monte Carlo simulated solutions to complex Bayesian models. The main advantage of using integrated nested Laplace approximation is its greatly improved computational speed. Results from this Bayesian estimation procedure provide approximations of the posterior distributions of each of the parameters, providing incidence rate ratios that reflect associations of independent and dependent measures.

Remembering that crime counts were measured at the block level, we conducted 2 spatial analyses for each of the 3 outcomes: a standard block nested hierarchical model without census tract and spatial lag effects and a multiresolution model adding block group and tract effects and spatial lags at the census block and block group levels. Comparisons among the models used the deviance information criterion and the Watanabe-Akaike information criterion (33, 34), and we report standard 95% credible intervals of posterior incidence rate ratios for all parameter estimates. Finally, we visually compare the performance of the standard and multiresolution models in their prediction of crime outcomes by mapping ratios of estimated rates to relative rates for each outcome for 2015. R code for all models can be found in Web Appendix 2.

RESULTS

Oakland, California, is a region with large income disparities and overall rising incomes. From 2010 to 2015, the median annual household income in Oakland block groups was $58,500 (range, $8,357–$240,113; Table 1). During this period, median household income rose by $8,499. To give a sense of the relative sizes of the different area units, there were an average of 106 people in a block, 1,185 people in a block group, and 3,524 people in a tract, with block population densities ranging from zero to 16,817 persons/mile². There were annual averages of 2.62, 29.34, and 87.23 reported assaults per block, block group, and tract, respectively, with 3.00, 33.61, and 99.95 reported burglaries per year and 1.16, 13.04, and 38.77 reported robberies per year, respectively. The number of reported assaults and robberies decreased between 2010 and 2015, while the number of reported burglaries increased. As is often observed in these studies, there was a great range of variation in other measures; for example, the percentages of Black and Hispanic residents in a block group ranged from 0% to 100% and from 0% to 94.3%, respectively (mean values were 28.7% and 26.2%), and the number of drug sale/transportation crimes per year, where high values indicate places with very active drug markets, ranged from 0 to 56 (mean = 2.4).

Table 1.

Characteristics of US Census Tracts (n = 678 Tracts), Block Groups (n = 2,016 Groups), and Blocks (n = 22,608 Blocks) in a Study of the Associations of Alcohol Outlets and Drug Markets With Violent Crime, Oakland, California, 2010–2015

Covariate	Mean (SD)	Range	Mean Change (2015 vs. 2010)
No. of reported assaults per year
Block	2.62 (4.34)	0–73	−0.20
Block group	29.34 (25.88)	0–169	−2.20
Tract	87.23 (69.99)	1–380	−6.55
No. of reported burglaries
Block	3.00 (6.83)	0–511	0.64
Block group	33.61 (44.14)	0–722	7.17
Tract	99.95 (93.01)	1–1,007	21.32
No. of reported robberies
Block	1.16 (2.02)	0–44	−0.29
Block group	13.04 (12.22)	0–92	−3.31
Tract	38.77 (29.84)	0–198	−9.83
Population, no. of persons
Block	105.67 (114.69)	0–1,433.86	3.86
Block group	1,185.05 (452.25)	0–3,228	43.34
Tract	3,523.69 (1,408.21)	1–8,010	128.86
Population density, no. of persons/mile2 a
Block	13.38 (12.48)	0–168.17	0.28
Block group	14.94 (9.81)	0–76.53	0.16
Tract	13.66 (8.37)	0.00–42.68	0.24
Median household income, dollars/yearb,c	58,500 (38,485)	8,357–240,113	8,498.96
Proportion of families living in povertyb, %	16.48 (13.14)	0–65.58	−0.48
Race/ethnicityb, %
Black	28.68 (19.41)	0–100	−6.71
Hispanic	26.22 (23.25)	0–94.34	3.80
No. of off-premise outletsd
Block	113.62 (479.91)	0–5,780.14	2.20
Block group	116.26 (156.27)	0–1,365.47	−2.38
Tract	108.32 (94.82)	0–595.58	−2.51
No. of restaurantsd
Block	172.78 (934.74)	0–22,841.67	26.16
Block group	141.83 (474.86)	0–4,891.60	25.38
Tract	179.20 (421.74)	0–3,086.86	36.91
No. of barsd
Block	31.57 (291.48)	0–8,532.15	7.57
Block group	25.03 (91.95)	0–940.69	4.64
Tract	30.60 (77.17)	0–594.95	9.97
No. of retail parcelsd in 2013
Block	1,602.84 (3,659.38)	0–40,112.55	N/A
Block group	1,381.75 (1,598.21)	0–8,842.50	N/A
Tract	1,364.99 (1,301.86)	0–8,452.11	N/A
No. of drug sale/transportation reports
Block	0.22 (0.95)	0–39	−0.25
Block group	2.44 (4.61)	0–56	−2.79
Tract	7.25 (12.05)	0–91	−8.31
No. of drug sale/transportation reports per    mile2d
Block	310.15 (1,426.90)	0–44,345.17	−325.27
Block group	338.68 (707.15)	0–8,527.02	−355.72
Tract	312.68 (500.99)	0–3,549.66	−329.35

Abbreviations: N/A, not applicable; SD, standard deviation.

^a Number per square mile, divided by 100.

^b Block group level.

^c Mean (SD) of the median value.

^d Number per square mile, multiplied by 10.

Table 2 displays the multiresolution results for reported assaults, burglaries, and robberies. The deviance information criterion indicated that the multiresolution models performed better than the block models in every case: an improvement of 122.53 for reported assaults, 1,061.02 for reported burglaries, and a much more modest 7.24 for reported robberies. This shows that the much greater number of covariates in the multiresolution analyses contributed to improved explanation of crime counts across block areas, particularly for the assault and burglary models. These benefits are reflected in the multiresolution effects observed in each case. While in most cases well-supported effects (i.e., 95% credible intervals did not include a value of no association) observed for the block-level models were preserved in the multiresolution models, indicating that the effects observed using these models were not controverted in general by multiresolution effects, in every case additional covariates related to multiresolution effects were important to the explanation of the crime outcomes. Thus, tract-level associations relating outlet densities to assaults and robberies were well-supported in addition to those observed at the block level. Spatial lag effects were similarly important at the block group level (especially relating outlet densities to robberies and drug sales/transportation to assaults).

Table 2.

Incidence Rate Ratios for Reported Crimes by US Census Block (n = 22,608 Blocks) in Bayesian Hierarchical Spatial Models, Oakland, California, 2010–2015

	Model 1: Reported Assaults				Model 2: Reported Burglaries				Model 3: Reported Robberies
	Block		Multiresolution		Block		Multiresolution		Block		Multiresolution
Variable	IRR	95% CrI	IRR	95% CrI	IRR	95% CrI	IRR	95% CrI	IRR	95% CrI	IRR	95% CrI
Deviance information criteriona	72,260.02		72,137.49		87,757.36		86,696.34		53,674.11		53,666.87
Watanabe-Akaike information criteriona	73,838.89		73,717.09		90,154.71		88,980.29		54,491.63		54,498.39
Year
2011	0.888	0.861, 0.916b	0.948	0.908, 0.990b	1.018	0.987, 1.049	0.965	0.928, 1.003	0.883	0.844, 0.925b	0.901	0.849, 0.955b
2012	1.001	0.969, 1.033	1.085	1.036, 1.136b	1.395	1.354, 1.438b	1.311	1.257, 1.368b	1.067	1.020, 1.116b	1.101	1.034, 1.172b
2013	0.953	0.922, 0.985b	1.026	0.978, 1.076	1.334	1.294, 1.376b	1.245	1.193, 1.299b	1.185	1.132, 1.240b	1.227	1.153, 1.307b
2014	1.001	0.969, 1.034	1.076	1.029, 1.126b	1.260	1.221, 1.300b	1.124	1.078, 1.172b	0.830	0.791, 0.871b	0.861	0.809, 0.916b
2015	0.962	0.931, 0.995b	1.055	1.007, 1.106b	1.218	1.180, 1.257b	1.024	0.980, 1.070b	0.772	0.735, 0.811b	0.799	0.749, 0.853b
Population densityc
Block	1.014	1.012, 1.016b	1.016	1.013, 1.018b	1.012	1.010, 1.014b	1.011	1.009, 1.013b	1.010	1.008, 1.013b	1.009	1.006, 1.011b
Block group			1.012	1.007, 1.018b			1.012	1.005, 1.018b			1.014	1.008, 1.021b
Tract			1.001	0.995, 1.007			1.006	0.999, 1.013			1.015	1.008, 1.022b
Demographic effects (block group)
Median annual household income (in $10,000s)	0.985	0.973, 0.996b	0.971	0.956, 0.986b	1.000	0.991, 1.009	1.000	0.988, 1.011	0.991	0.976, 1.006	0.983	0.963, 1.003
Median household income lag			0.962	0.937, 0.988b			1.048	1.026, 1.072b			0.971	0.938, 1.005
Proportion of families in poverty, %	1.008	1.005, 1.012b	1.011	1.006, 1.016b	1.011	1.007, 1.014b	1.004	0.999, 1.009	1.011	1.006, 1.015b	1.009	1.003, 1.015b
Proportion of families in poverty, % lag			1.007	1.001, 1.013b			0.991	0.987, 0.995b			1.002	0.994, 1.011
Race/ethnicity
Black, %	1.003	1.001, 1.005b	1.008	1.004, 1.012b	0.994	0.992, 0.995b	0.981	0.978, 0.984b	1.001	0.998, 1.003	1.000	0.995, 1.005
Black, % lag			1.006	1.002, 1.009b			0.990	0.987, 0.993b			0.999	0.994, 1.004
Hispanic, %	1.001	0.999, 1.002	1.003	1.000, 1.006	0.990	0.988, 0.992b	0.987	0.984, 0.990b	1.002	0.999, 1.004	0.999	0.994, 1.003
Hispanic, % lag			1.002	0.999, 1.005			0.989	0.985, 0.992b			0.997	0.992, 1.001
Block alcohol outlet effectsd,e
Off-premise outlets	1.007	1.004, 1.010b	1.017	1.002, 1.032	1.004	1.001, 1.008b	1.014	0.998, 1.029	1.010	1.006, 1.013b	1.004	0.985, 1.022
Restaurants	1.002	1.000, 1.003	0.998	0.991, 1.005	1.005	1.004, 1.006b	1.009	1.003, 1.014b	1.000	0.998, 1.002	0.996	0.987, 1.005
Bars	1.004	1.000, 1.008	1.000	0.979, 1.022	1.005	1.001, 1.009b	1.012	0.995, 1.029	1.004	0.998, 1.010	0.987	0.960, 1.014
Block group alcohol outlet effectsd,e
Off-premise outlets			1.116	1.041, 1.197b			1.254	1.161, 1.354b			1.112	1.016, 1.218b
Restaurants			0.992	0.957, 1.029			1.067	1.035, 1.101b			0.926	0.887, 0.968b
Bars			0.973	0.869, 1.089			1.418	1.277, 1.574b			1.194	1.032, 1.381b
Tract alcohol outlet effectsd
Off-premise outlets			1.085	1.021, 1.153b			1.244	1.162, 1.331b			1.057	0.976, 1.145
Restaurants			0.958	0.928, 0.990b			1.037	1.008, 1.066b			0.930	0.895, 0.967b
Bars			1.158	1.052, 1.276b			1.786	1.631, 1.956b			1.221	1.074, 1.389b
Block lag alcohol outlet effectsd,e
Off-premise outlets			1.010	0.996, 1.023			1.007	0.993, 1.021			0.995	0.978, 1.011
Restaurants			0.997	0.990, 1.003			1.007	1.002, 1.012b			0.997	0.989, 1.005
Bars			0.998	0.979, 1.019			1.012	0.996, 1.027			0.985	0.960, 1.009
Block group-lag alcohol outlet effectsd,e
Off-premise outlets			1.076	1.014, 1.143b			1.215	1.137, 1.298b			1.085	1.005, 1.172b
Restaurants			0.984	0.953, 1.016			1.008	0.980, 1.037			0.931	0.896, 0.968b
Bars			0.982	0.887, 1.087			1.497	1.360, 1.648b			1.184	1.039, 1.351b
Retail parcel effectsd,e
Block	1.005	1.005, 1.006b	1.012	1.009, 1.014b	1.003	1.002, 1.004b	1.005	1.003, 1.007b	1.006	1.005, 1.006b	1.014	1.011, 1.016b
Block group			1.040	1.002, 1.081b			0.948	0.909, 0.988b			1.028	0.990, 1.068
Tract			1.044	1.007, 1.083b			0.937	0.901, 0.974b			1.029	0.992, 1.067
Block lag			1.007	1.004, 1.009b			1.002	1.000, 1.004			1.008	1.006, 1.011b
Block group lag			1.025	0.988, 1.064			0.944	0.906, 0.984b			1.011	0.975, 1.049
Drug sale/transportation crime effectsd,e
Block	1.001	1.001, 1.002b	1.004	1.002, 1.005b	1.002	1.001, 1.002b	1.000	1.000, 1.002	1.001	1.001, 1.002b	1.002	0.999, 1.005
Block group			1.018	1.011, 1.024b			1.014	1.006, 1.022b			1.008	0.997, 1.018
Tract			1.011	1.007, 1.016b			1.017	1.011, 1.022b			1.003	0.996, 1.010
Block lag			1.003	1.001, 1.004b			0.999	0.997, 1.001			1.000	0.998, 1.003
Block group lag			1.011	1.005, 1.016b			1.011	1.004, 1.018b			1.003	0.994, 1.012
Random effects
Nonspatial block-level (SD)	1.139	1.084, 1.189	1.097	1.047, 1.150	0.879	0.840, 0.918	0.882	0.841, 0.922	1.037	0.996, 1.078	1.031	0.976, 1.074
Nonspatial block group–level (SD)	0.149	0.084, 0.230	0.008	0.002, 0.025	0.274	0.227, 0.329	0.236	0.181, 0.307	0.146	0.076, 0.225	0.009	0.003, 0.031
Nonspatial tract-level (SD)			0.226	0.149, 0.332			0.233	0.142, 0.334
(spatial variance ratio)f	0.939	0.898, 0.966	0.906	0.857, 0.941	0.869	0.813, 0.907	0.861	0.800, 0.901	0.985	0.932, 0.996	0.983	0.951, 0.995

Abbreviations: CrI, credible interval; IRR, incidence rate ratio; SD, standard deviation.

^a Measures comparing model performance, where lower values correspond to improved performance.

^b Well-supported 95% CrI.

^c Thousands of persons per square mile.

^d Per square mile, multiplied by 10.

^e Effects (except for tract) in the multiresolution models are centered; in the block-level models, they are not.

^f Calculated as the ratio of spatial effects to spatial and nonspatial random effects.

Web Figure 3 displays the ratios of fitted values of assaults, burglaries, and robberies in models with only block-level outlet effects to those from multiresolution models. Block-level estimated counts for each crime were obtained, and the ratios of these counts (multiresolution vs. block-only) were calculated for each block. Areas colored in blue represent locations where the multiresolution models predicted fewer crimes than the block-only models. Areas with red and orange coloring represent locations where the multiresolution models predicted a greater number of crimes than the block-only models. All 3 maps show many regions with dissimilar estimates. The burglary model shows an area of solid red, representing a 15%–45% greater number of predicted burglaries in the multiresolution model, which corresponds to the dense downtown area of Oakland. This highlights the spatial heterogeneity in the divergence of the predictions of the 2 models. To help visualize the magnitude of the overall associations of each outlet type with each type of crime, Web Figure 4 displays well-supported posterior predicted effects related to the geographic distributions of different types of alcohol outlets in 1 downtown area of Oakland. In some cases, the multiresolution models indicate tightly focused effects (e.g., burglaries associated with geographic distributions of restaurants). In other cases, the model indicates a smoother distribution of effects (e.g., assaults associated with bars).

DISCUSSION

These analyses demonstrate multiresolution associations relating measures of alcohol and drug environments to assaults, burglaries, and robberies across community areas. The observed multiresolution patterns, which varied by both crime and outlet type, highlight the importance of considering multiple spatial scales when assessing theoretical foundations of community-level effects. By considering multiple spatial scales and spatial lags in a Bayesian framework, researchers can explore multiresolution processes, providing more detailed tests of expectations from theoretical models.

Focusing on the current results, we saw well-supported associations at the census block level for off-premise outlets and assaults and for restaurants and burglaries. There were population associations at the block group and/or tract level for all outlet and crime types (with restaurants being associated with fewer crimes, while bars and off-premise outlets were associated with more crimes). This contrasts with the prevailing notion that smaller spatial scales are always “better” or more accurate, and that the processes linking alcohol environments and crimes primarily operate at micro- levels. Areas with greater numbers of restaurants often tend to have higher-income residents and/or visitors and lower levels of social and physical disorganization (while the opposite can be said of areas with greater densities of bars and off-premise outlets). Spatial lag effects indicated that the bar and off-premise outlet density effects continued in adjacent block groups, with increased risk for burglaries and robberies associated with greater densities in adjacent areas. This reinforces the idea that these effects may operate at a larger, population scale.

Empirical analyses utilizing a multiresolution approach made classical predictions from routine activity and crime potential theories more evident (11, 35, 36). Retail parcels and drug-related crimes both displayed multiresolution associations, particularly with assaults. Assaults were associated with both environmental characteristics at the block level, indicating that small areas may act as violent crime attractors or provide locations in which violent crimes are more likely to occur (“hot spots”). Simultaneously, neighborhoods with greater densities of retail parcels and drug-related crimes may also increase risks for assault via increased social disorganization and decreased collective efficacy and social capital. Higher block group–level median household income was associated with fewer assaults and robberies, while the spatial lags showed that a greater median household income in surrounding areas was related to fewer assaults but more robberies. While the social capital associated with higher-income areas tends in general to suppress crime rates, particularly violent crimes, higher rates of property crimes are to be observed in liminal areas where perpetrators and victims are more likely to come into contact with one another.

These results also remind us that interpretations of crime effects related to neighborhood conditions are conditional upon unit scale and resolution. Thus, the analyses presented here suggest that census block group units are good enough to represent some, but not all, of the effects associated with different covariates in the study. When choosing a different spatial scale, different results can be obtained; both of these results can be correct estimations of associations occurring at their respective scales of observation. This is, quite obviously, 1 reason the literature on outlets and crime has so much heterogeneity in its findings. As Web Figure 3 shows, reliance on a single scale (even if it is quite small) can lead to misestimation of associations in the highest-risk communities. Thus, identifying multiresolution processes has great potential benefit beyond the crime–alcohol environments arena, extending to many topics within spatial epidemiology.

Our approach to better understanding multiscale processes was not without limitations. Since we simultaneously considered the same covariates at multiple spatial resolutions, these covariates may have been confounded with spatial latent effects at the block level. These models do not account for the correlation of the spatial fields across time, as the random spatial effects are independent across time. There are other methods with which to better understand spatial scale, such as point pattern approaches (37). The CAR process approach we used assumes binary relationships (adjacent vs. nonadjacent). Future work could utilize a model-based geostatistics framework, using a distance-based correlation structure to provide a direct estimate of the spatial scale of correlation (38). Another possibility is to use the stochastic partial differential equation approach, which uses continuously defined spatial frameworks and assumes that spatial relationships within units arise from continuous spatial processes (39). Finally, since we are fitting independent models for 3 correlated outcomes, a multivariate modeling approach could be used in future work.

Many research questions and targets for change in public health are environmental and contextual. While public health research has identified many associations between neighborhood environments and health (40), translation of these statistical associations into the mechanisms underlying them has lagged far behind. Furthermore, the scales of these observed effects may be wrong or spatially confounded to the point where the significant covariates identified are not actually relevant to the spatial scale of the underlying processes. The results of our analyses are cause for significant concern about accurately interpreting “neighborhood effects” in the social epidemiologic research literature at large. More precise information about relevant spatial scales, and the identification of relevant mechanism(s), will lead the way to more effective intervention efforts. A third spatial scale, of the existing/potential intervention targets, must also be considered and best defined in order to better translate these spatial epidemiologic models to effective intervention efforts. Combined with appropriate theory and adequate data, multiresolution analyses allow for a clarity of mechanisms and development of interventions with great potential to improve the field of public health for a variety of outcomes.