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

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.

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.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.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).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).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.