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Is it Important to Examine Crime Trends at a Local “Micro” Level?: A Longitudinal Analysis of Street to Street Variability in Crime Trajectories

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Abstract

Over the last 40 years, the question of how crime varies across places has gotten greater attention. At the same time, as data and computing power have increased, the definition of a ‘place’ has shifted farther down the geographic cone of resolution. This has led many researchers to consider places as small as single addresses, group of addresses, face blocks or street blocks. Both cross-sectional and longitudinal studies of the spatial distribution of crime have consistently found crime is strongly concentrated at a small group of ‘micro’ places. Recent longitudinal studies have also revealed crime concentration across micro places is relatively stable over time. A major question that has not been answered in prior research is the degree of block to block variability at this local ‘micro’ level for all crime. To answer this question, we examine both temporal and spatial variation in crime across street blocks in the city of Seattle Washington. This is accomplished by applying trajectory analysis to establish groups of places that follow similar crime trajectories over 16 years. Then, using quantitative spatial statistics, we establish whether streets having the same temporal trajectory are collocated spatially or whether there is street to street variation in the temporal patterns of crime. In a surprising number of cases we find that individual street segments have trajectories which are unrelated to their immediately adjacent streets. This finding of heterogeneity suggests it may be particularly important to examine crime trends at very local geographic levels. At a policy level, our research reinforces the importance of initiatives like ‘hot spots policing’ which address specific streets within relatively small areas.

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Notes

  1. Due to space constraints only selected references are mentioned here. More complete overviews of the history of place-based criminology can be found in the introductory chapters of Eck and Weisburd (1995) and Weisburd et al. (2008).

  2. For an exception see Groff et al. 2008. They use the same methodology but explore the patterns of crimes committed by juveniles.

  3. The literature around hot spots is immense. Two recent overviews provide the insight into the state of the art (Chainey et al. 2008; Eck et al.2005). See Weisburd et al. (1992) for a theoretical introduction.

  4. Weisburd et al. (2004a, b) came at this from the opposite direction. They first identified temporal trends in crime and then used kernel density maps to find hotspots of temporal trajectory patterns.

  5. The volume of research explicitly examining spatial dependence or spatial error in models is far too large to detail here (as examples see Baller et al. 2001; Chakravorty and Pelfrey 2000; Cohen and Tita 1999; Cork 1999; Jefferis 2004; Morenoff and Sampson 1997; Roman 2002).

  6. This unit of analysis is slightly different from the ‘hundred block’ measure used in the original Seattle study. See the final report for more information on the ‘hundred block’ definition (Weisburd et al. 2004b). More detailed information on the creation of geographically defined street segment is available (see Weisburd et al. 2009a).

  7. There are two main reasons for excluding intersection crime. First, since events at intersections could be considered ‘part of’ any one of the participating street segments, there is no satisfactory method for assigning them to one or another. However, it is also the case that incident reports at intersections differed dramatically from those at street segments. Traffic-related incidents accounted for only 3.77% of reports at street segments, but for 45.3% of reports at intersections.

  8. All geocoding was done in ArcGIS 9.1 using a geocoding locator service with an alias file of common place names to improve our hit rate. The geocoding locater used the following parameters: spelling sensitivity = 80, minimum candidate score = 30, minimum match score = 85, side offset = 0, end offset 3%, and Match if candidates tie = no. Manual geocoding was done on unmatched records in ArcGIS 9.1 and then in ArcView 3.x using the ‘MatchAddressToPoint’ tool (which allowed the operator to click on the map to indicate where an address was located) to improve the overall match rate. Research has suggested hit rates above 85% are reliable (Ratcliffe 2004). Our final geocoding percentage for crime incidents was 97.3%.

  9. In order to apply point pattern statistics to street segments we use the midpoint of each line/street to represent the street segment.

  10. This is accomplished by using a series of random toroidal shifts on one set of points and comparing the cross K-function of the shifted points with another fixed set (Rowlingson and Diggle 1993). A toroidal shift provides a simulation of potential outcomes under the assumption of independence by repeatedly and randomly shifting the set of locations for one type of street segment and calculating the cross K-function for that iteration. The outcomes are used to create test statistics in the form of an upper and lower envelope. One thousand iterations are used for each simulation. In order to better explore micro level relationships, the bivariate - K analysis examines the distribution of the trajectory pairs at distances up to 2,800 feet (using 400 foot bins which approximate one street block). This strategy also allows us to more closely inspect the relationship of the bivariate k statistic to the upper bound of the simulation envelope. The null hypothesis of the bivariate- K test is independence (i.e., the spatial pattern of one trajectory group is unrelated to the pattern of the other group being compared).

  11. Readers should note slight scale changes among maps 1, 2 and 3. These were necessary to provide maximum enlargement of the three sections of Seattle.

  12. Ripley’s K also reveals whether the observed clustering is greater or less than would be expected under an assumption of Complete Spatial Randomness (CSR). CSR is of limited use when examining human-related distributions such as crime because the opportunity for a crime to occur is constrained to accessible areas adjacent to streets. By calculating the Ripley’s K for the street network, we can provide a more realistic metric with which to compare patterns in the trajectory group member ship of street segments (see ‘Street segments’ line in Fig. 4).

  13. These analyses produced 28 graphs. Space constraints do not allow the inclusion of the graphs in the paper; however, they are available from the authors.

  14. The group-based trajectory is often identified with typological theories of offending such as Moffitt (1993) because of its use of groups (see Nagin et al. 1995). But it is important to keep in mind that group assignments are made with error. In all likelihood, the groups only approximate a continuous distribution. The lack of homogeneity in the groups is the explicit trade off for the relaxation of the parametric assumptions about the random effects in the linear models (Bushway et al. 2003). For a different perspective on this issue, see Eggleston et al. (2004).

  15. Those interested in a more detailed description of the group-based trajectory approach should see Nagin (1999) or (2005).

  16. The procedure, with documentation, is available at www.ncovr.heinz.cmu.edu.

  17. Proc Traj also provides the option of estimating a Zero Inflated Poisson (ZIP) model. The ZIP model builds on a Poisson by accommodating more non-offenders in any given period than predicted by the standard Poisson distribution. The zero-inflation parameter can be allowed to vary over time, but cannot be estimated separately for each group. It is sometimes called an intermittency parameter, since it allows places to have “temporary” spells of no offenses without recording a change in their overall rate of offending. In this context, the ZIP model’s differentiation between short-term and long-term change is problematic. The Poisson model, on the other hand, tracks movement in the rate of offending in one parameter, allowing all relatively long-term changes to be reflected in one place. We believe this trait of the Poisson model makes it the better model for modeling trends, especially over relatively short panels, even though the ZIP model provides a better fit according to the BIC criteria used for model selection. For a similar argument see Bushway et al. (2003).

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Acknowledgments

This research was supported by grant 2005-IJ-CX-0006 from the National Institute of Justice (US Department of Justice). Points of view in this paper are those of the authors and do not necessarily represent those of the US Department of Justice. We would like to thank Dan Nagin for his thoughtful suggestions regarding trajectory analysis, Richard Heiberger for his assistance with R programming, and Breanne Cave and the anonymous reviewers whose comments were invaluable in strengthening the paper. We also want to express our gratitude for the cooperation of the Seattle Police Department, and especially to Lieutenant Ron Rasmussen for playing the crucial role of our main data contact and former Chief Gil Kerlikowske (now Director of the Office of National Drug Control Policy) for his interest in and support of our work.

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Correspondence to Elizabeth R. Groff.

Appendix 1: Technical Note of the Production of Developmental Trajectories for Street Segments

Appendix 1: Technical Note of the Production of Developmental Trajectories for Street Segments

The trajectory modeling reported here was developed for a larger study of crime and place in Seattle, WA (Weisburd et al. 2009a). The group-based trajectory model, first described by Nagin and Land (1993) and further elaborated in Nagin (1999, 2005), is specifically designed to identify clusters of individuals with similar developmental trajectories, and it has been utilized extensively to study patterns of change in offending and aggression as people age (see Nagin 1999). As such, we believe it is particularly well suited to our goal of exploring the patterns of change in the Seattle data.

Formally, the model specifies that the population is comprised of a finite number of groups of individuals who follow distinctive developmental trajectories. Each such group is allowed to have its own offending trajectory (a map of offending rates throughout the time period) described by a distinct set of parameters that are permitted to vary freely across groups. This type of model has three key outputs: the parameters describing the trajectory for each group, the estimated proportion of the population belonging to each group, and the posterior probability of belonging to a given group for each individual in the sample. The posterior probability, which is the probability of group membership after the model is estimated, can be used to assign individuals to a group based on their highest probability.Footnote 14

This approach is less efficient than linear growth models but allows for qualitatively different patterns of behavior over time. There is broad agreement that delinquency and crime are cases where this group-based trajectory approach might be justified, in large part because not everyone participates in crime, and people appear to start and stop at very different ages (Nagin 1999, 2005; Raudenbush 2001). Given that we have no strong expectation about the basic pattern of change, the group-based trajectory approach appears to be an excellent choice for identifying major patterns of change in our data set.Footnote 15

There are two software packages available that can estimate group-based trajectories: Mplus, a proprietary software package, and Proc Traj, a special procedure for use in SAS, made available at no cost by the National Consortium on Violence Research (for a detailed discussion of Proc Traj, see Jones et al. 2001).Footnote 16 In using Proc Traj, we had three choices when estimating trajectories of count data: parametric form (Poisson vs. normal vs. logit), functional form of the trajectory over time (linear vs. quadratic vs. cubic), and number of groups.

The Poisson distribution is a standard distribution used to estimate the frequency distribution of offending that we would expect given a certain unobserved offending rate (Lehoczky 1986; Maltz 1996; Osgood 2000).Footnote 17 We found that the quadratic was uniformly a better fit than the linear model, and that the cubic model did not improve the fit over the quadratic in the case of a small number of groups. In choosing the number of groups we relied upon the Bayesian Information Criteria (BIC) because conventional likelihood ratio tests are not appropriate for defining whether the addition of a group improves the explanatory power of the model (D’Unger et al. 1998). BIC = log( L) − .5 × log( n) × ( k); where “ L” is the value of the model’s maximized likelihood estimates, “ n” is the sample size, and “ k” is the number of parameters estimated in a given model. Because more sophisticated models almost always improve the fit of a given analysis, the BIC encourages a parsimonious solution by penalizing models that increase the number of trajectories unless they substantially improve fit. In addition to the BIC, trajectory analysis requires that researchers also consider posterior probabilities of trajectory assignments, odds of correct classification, estimated group probabilities, and whether meaningful groups are revealed (for a more detailed discussion, see Nagin 2005).

These models are highly complex, and researchers run the risk of arriving at a local maximum, or peak in the likelihood function, which represents a sub-optimal solution. The stability of the answer when providing multiple sets of starting values should be considered in any model choice. In the final analysis, the utility of the groups is determined by their ability to identify distinct trajectories, the number of units in each group, and their relative homogeneity (Nagin 2005).

We began our modeling exercise by fitting the data to three trajectories. We then fit the data to four trajectories and compared this fit with the three-group solution. When the four-group model proved better than the three-group, we then estimated the five-group model and compared it to the four-group solution. We continued adding groups, each time finding an improved BIC, until we arrived at 24 groups. Models for 23 and 24 groups were not stable and could not be replicated consistently. After reviewing the Bayesian Information Criteria and the patterns observed in each solution, it was determined that a 22 group model was the most optimal model for the crime data. We therefore chose the 22 group model.

The validity of the model was also confirmed by conducting the posterior probability analysis. The majorities of the within-group posterior probabilities in the model are above .90, and the lowest posterior probability is .77. The lowest value of the odds of correction classification (OCC) is 26.58. Nagin (2005) suggests that when average posterior probability is higher than .7 and OCC values are higher than 5, the group assignment represents a high level of accuracy. Judging by these standards, the 22-group model performs satisfactorily in classifying the various crime patterns into separate trajectories.

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Groff, E.R., Weisburd, D. & Yang, SM. Is it Important to Examine Crime Trends at a Local “Micro” Level?: A Longitudinal Analysis of Street to Street Variability in Crime Trajectories. J Quant Criminol 26, 7–32 (2010). https://doi.org/10.1007/s10940-009-9081-y

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