WEEK 2.2
O R I G I N A L P A P E R
Is it Important to Examine Crime Trends at a Local ‘‘Micro’’ Level?: A Longitudinal Analysis of Street to Street Variability in Crime Trajectories
Elizabeth R. Groff • David Weisburd • Sue-Ming Yang
Published online: 1 January 2010 � Springer Science+Business Media, LLC 2010
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 con-
centration 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 accom-
plished 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.
E. R. Groff (&) Department of Criminal Justice, Temple University, 550 Gladfelter Hall, 1115 W. Berks Street, Philadelphia, PA 19122, USA e-mail: groff@temple.edu
D. Weisburd Hebrew University, Jerusalem, Israel
D. Weisburd Administration of Justice, George Mason University, Manassas, VA, USA
S.-M. Yang Criminal Justice, Georgia State University, Atlanta, GA, USA
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J Quant Criminol (2010) 26:7–32 DOI 10.1007/s10940-009-9081-y
Keywords Crime concentration � Hot spots � Micro � Spatio-temporal
[V]ariation in crime within communities is probably greater than variations across
communities. The very meaning of the concept of a bad neighborhood is an open
empirical question: whether the risk of crime is randomly or evenly distributed
throughout the neighborhood, or so concentrated in some parts of the neighborhood
that other parts are relatively safe (Sherman et al. 1989, p. 29).
Scholars have long been interested in how crime varies over space and the topic has
received increasing attention over the last 40 years. 1
Seminal studies examining crime
across larger geographic units such as states (Guerry 1833; Loftin and Hill 1974; Quetelet
1831[1984]), cities (Baumer et al. 1998), and even neighborhoods (Boggs 1965; Bursik and
Grasmick 1993; Bursik and Webb 1982; Byrne and Sampson 1986; Chilton 1964;
Kornhouser 1978; Reiss and Tonry 1986; Schuerman and Kobrin 1986; Skogan 1986;
Stark 1987) establish the foundation for the continuing interest. More recent studies point
to the potential theoretical and practical benefits of focusing research on micro crime
places (Eck and Weisburd 1995; Sherman 1995; Sherman and Weisburd 1995; Taylor
1997; Weisburd 2002). Cross-sectional micro level studies suggest that significant clus-
tering of crime at place exists, regardless of the specific micro unit of analysis defined
(Brantingham and Brantingham 1999; Crow and Bull 1975; Groff and LaVigne 2001;
Pierce et al. 1986; Potchak et al. 2002; Roncek 2000; Sherman et al. 1989; Weisburd and
Green 1994; Weisburd et al. 1992). Longitudinal work examining the developmental
trajectories at micro levels (Weisburd et al. 2004a, 2009b) has consistently identified
tremendous crime concentration at specific places. These micro level findings provide
evidence of significant intra-neighborhood variance in crime that is lost when neighbor-
hoods are examined as homogenous units.
A major question that has not been answered in prior research concerns the geography
(i.e., the form and the degree) of street to street variability of temporal crime trends at
micro levels of analysis. 2
The initial study identifying developmental crime trends across
micro places (Weisburd et al. 2004a, b) did not use full geographic tools. To more pre-
cisely answer this question, we examine both temporal and spatial variation in crime across
street segments in the city of Seattle Washington.
In this paper we explore the spatial distribution developmental trajectories. We begin by
using 16 years of crime incident data (1989–2004) to produce developmental trajectories
of crime in Seattle, WA (Weisburd et al. 2009a). We then use spatial statistics to examine
the geography of street segments in each group. Rigorous geographic analysis enables us to
more exactly describe the spatial relationships among temporal crime patterns and to
examine the open question of whether it is important to examine crime trends at a ‘micro’
level. We address the following research questions: (1) What is the distribution of temporal
crime trajectories across Seattle? (2) What is the spatial pattern of street segments within
the same temporal crime trajectory (i.e., clustered, dispersed or random)? and (3) Are street
segments of certain trajectories found near one another or are they spatially independent?
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.
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Finally, we discuss the implications of our findings both to the theoretical development of a
‘geography of crime concentration’ and to policy and practice.
Geography and the Criminology of Place
In the 1970s a group of scholars began to develop theories that focus on why crime
happens where it does. The ‘opportunity’ theories they developed have been instrumental
in guiding the investigation of place and crime. Routine activity theory (Cohen and Felson
1979) and crime pattern theory (Brantingham and Brantingham 1991 [1981], 1993) both
emphasize the context of crime events and importance of human activity in understanding
crime patterns. As a group, opportunity theories hold that crimes occur when the normal
everyday activities of offenders and victims intersect with no guardian present. Thus,
routine activities are the key dynamic element because they directly affect the convergence
of the other elements necessary for a crime to occur. Opportunity theories focus on how the
built and social environments shape human activity and thus provide the foundation for
understanding why crime happens where it does. After all, it is the interactions between
humans and their environment which serve as the source of explanation of observed spatial
patterns (Aitken et al. 1989; Gold 1980; Golledge and Timmermans 1990; Timmermans
and Golledge 1990; Walmsley and Lewis 1993). As software improved and data became
more widely available, the work of these theorists inspired many empirical studies of micro
places (Weisburd and McEwen 1997).
Just as Wolfgang et al.’s (1972, p. 89) identification of the ‘chronic six percenters’ (i.e.,
6% of study participants were responsible for over 50% of the offenses committed) gal-
vanized research into individual level criminality, Sherman et al.’s (1989) finding that 3%
of the addresses in Minneapolis produced 50% of the calls for service sparked renewed
interest in micro level crime patterns and how they develop over time. There is now a
significant body of literature demonstrating the existence of clustering of crime at place
regardless of the specific micro unit of analysis defined (Brantingham and Brantingham
1999; Crow and Bull 1975; Groff and LaVigne 2001; Pierce et al. 1986; Roncek 2000;
Sherman et al. 1989; Smith et al. 2000; Weisburd and Braga 2002; Weisburd et al. 2004a;
Weisburd and Green 1994). Taylor and Gottfredson (1986) took the examination of micro
places one step further by making the connection between the physical and social envi-
ronments of micro level places and their crime levels. This body of work suggests the
salience of Sherman et al.’s call for the development of a ‘‘criminology of places’’ (emphasis in original, Sherman et al. 1989, p. 30).
Building a place-based criminology necessitates pursuing a deeper understanding of the
developmental aspects of crime at micro level places (Weisburd et al. 2004a, 2009a).
Weisburd et al. (2004a, b) used group-based trajectory analysis (Nagin 1999, 2005; Nagin
and Land 1993) to classify crime on street segments from 1989 to 2002 in Seattle,
Washington. They identified 18 trajectory groups that reflect distinct longitudinal crime
patterns. Echoing Wolfgang et al. (1972) and Sherman et al. (1989) they found approxi-
mately 4–5% of the street segments in each year account for 50% of all crime. At the other
end of the spectrum, each year between 47 and 52% of street segments had no crime at all.
In a related study of crimes committed by juveniles, Weisburd et al. (2009b) applied
trajectory analysis and once again found both concentration and stability in the patterns.
The results of Weisburd and colleagues in Seattle are consistent with other micro level
longitudinal examinations which focused on a subset of places. In Baltimore, Maryland,
Taylor and National Consortium On Violence Research (1999) reported a high degree of
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stability in both crime and fear of crime across ninety street blocks using data collected in
1981 and 1994 (see also Robinson et al. 2003; Taylor 2001). Spelman’s examination of
high schools, public housing communities, subway stations, and parks in Boston over
3 years found about 10% of locations account for 50% of the calls for service to the places
he examines. Importantly, those ‘worst’ places remain fairly stable over the 3 years of his
study.
While these longitudinal micro level studies examine the temporal distribution of crime
across places, they leave the question of the geographic distribution of those trajectories
largely unexplored. Weisburd et al. (2004a, b) took a first step in this direction. They
examined the distribution of trajectories via kernel density maps. The maps showed stable
trajectories tended to be found in all areas of the city but were concentrated in areas with
less residential density and higher income. Additionally, the maps revealed overlap in the
downtown area between areas with the highest concentrations of increasing trajectories and
those with decreasing trajectories; leading them to speculate that some of the same pro-
cesses may underlie both. In a later study of crimes for which a juvenile was arrested,
Weisburd et al. (2009b) used point maps of the locations of the streets in the highest rate
trajectories to show they were found all over Seattle. They once again discovered evidence
of strong street to street variability in that subset of crimes (Groff et al. 2008). Together
these studies suggest the importance of more fully describing the variation in crime across
micro places.
Toward a Geography of Crime Concentration
The previously outlined developments provide ample evidence for the existence of geo-
graphic concentration and spatio-temporal stability. What remains is the need for a more
in-depth examination of the structure of geographic concentration across micro places; specifically addressing the degree of block to block variability. Sherman et al. (1989, p. 28)
made the case for studying the geography of crime concentration by arguing that the study
of the ‘‘variation across space is one of the basic tools of science.’’ They noted that crime
concentration itself does not necessarily mean clustering. What if the high crime places are
randomly distributed across space? The theoretical and policy implications of such a
finding would be very different from a finding of clustering.
Sherman et al. (1989) tested for clustering in the distribution of calls for service by
comparing the observed distribution to the expected distribution under a Poisson model
and uncovered significant clustering. Other researchers have used spatial statistics to show
how concentration in the distribution of crime incidents exists in ‘hot spots of crime’
(Spring and Block 1988). Since then, the identification of geographic ‘hot spots’ of crime
has been the topic of many studies. 3
More recent work has expanded to include the
temporal structure of hotspot areas (Grubesic and Mack 2008; Johnson et al. 2008). These
authors call for more work to identify hot spots by the temporal trajectory of their crime;
‘‘upwards, downwards and time-stable’’ (Johnson et al. 2008, p. 43). 4
Other researchers have described the structure of concentration by measuring the dis-
tance between hot places. As part of the design of a patrol experiment, Sherman and
Weisburd (1995) identified 420 clusters of hot spot addresses with 20 or more calls for
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.
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service related to Part I offenses that were within one-half a block of one another. Other
work identified the mean distance between clusters of robbery hot spots in Minneapolis as
only 888 feet, which is approximately two blocks in most urban areas (Linnell 1988). The
same study revealed 90% of robbery hotspots in Minneapolis were on just seven arterial
roads. These studies provide ample evidence about the structure of crime concentration;
namely that concentration takes the form of hot spots (i.e., clusters) and tends to be found
near one another and in places with high amounts of human activity. They also begin to
describe the structure of the clustering.
Another open question about crime concentration pertains to whether areas such as hot
spots and neighborhoods are uniformly hot. Early evidence suggests these areas may be
heterogeneous rather than uniform. In other words, ‘bad neighborhoods’ may contain ‘good
streets’ and ‘good neighborhoods’ may be home to ‘bad streets’. Roughly 60 years ago,
Henry McKay noted the lack of offenders on some blocks within high crime neighborhoods
(Albert J. Reiss, Jr., personal communication as cited in Sherman and Weisburd 1995).
More recently, Weisburd and Mazerolle (2000) found activity in drug hot spots to be more
related to the hot spots themselves, than the levels of crime and disorder in the surrounding
neighborhood. Taylor and Gottfredson (1986) uncovered street block level variation in
crime and found it to be related to the social and physical environment. Urban planners
have long pointed to street to street variation in characteristics of the physical environment
and their relationship to crime (Jacobs 1961). Crime prevention through environmental
design (Jeffery 1971) and ‘defensible space’ (Newman 1972) explained why such variation
may be related to the physical environment (For an opposing view see Merry 1981). Hillier
(1999) made a compelling case for focusing on the built environment directly and mea-
suring its affect on crime and other social variables of interest. By quantifying both the
accessibility and the type street pattern he demonstrated that traditional street patterns (i.e.,
grid) are safer (Hillier 2004). More recent studies uncovered evidence of extensive street to
street variation in both the characteristics of the physical environment and measures of
criminal activity such as burglary (Groff and LaVigne 2001) and auto thefts (Potchak et al.
2002). As a whole, these studies point toward the existence of bad places in good
neighborhoods as well as good places in bad neighborhoods. Our research takes the next
step and provides a direct, quantitative examination of that question.
Achieving a More Complete Description of Crime Variation Across Places
The concentration and stability of micro crime places suggest they are an important unit of
study for understanding crime at place. But those characteristics do not put to rest a key
concern in assessing the importance of such small geographic units in the crime equation.
If the focus on micro places adds to our study of crime, then it should represent a type of
‘reductionism’; the understanding of small parts will lead to an explanation of the whole
which is not provided by higher units of analysis. While prior studies have shown that
crime is concentrated at micro units of analysis, they have not examined whether this
variability is distinct from what would be observed had they focused on higher geographic
units such as communities or neighborhoods. For example, do macro level studies simply
mask concentrations of crime that are found in high crime communities? More specifically,
is there street to street variability in crime trends at micro places, or do examinations of
micro crime places simply divide up larger area trends?
The answers to these questions have implications for both theory and crime prevention
policy. If geographically proximal street segments have the same or similar temporal crime
patterns, it would suggest there is no need for micro-level examination of places. It would
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also provide support for neighborhood-level crime prevention initiatives rather than micro
level ones. If, however, street segments of the same trajectory are spread throughout the
city and/or street segments spatially adjacent to one another vary in their temporal crime
pattern, then further examination of micro level patterns is supported. In this case, more
narrowly focused efforts on individual street segments may provide more crime prevention
impact. Our focus in this study is on providing more comprehensive evidence of the
variation in crime across micro level places.
Analytic Strategy
The incorporation of spatial methods into criminological research has increased rapidly
since the 1990s (Messner and Anselin 2004). Researchers have taken seriously the error
introduced by failing to account for spatial effects when analyzing inherently spatial data
and have responded by incorporating a range of spatial data analysis techniques. 5
This
research continues that trend by using spatial statistics to describe geographic patterns of
crime trajectories across street segments.
Study Area
Our study focuses on Seattle Washington. Seattle was a logical choice because the city had
an extended and unbroken series of electronic crime incident data. Seattle is located on the
west coast of the United States. It is bounded on the west by the Puget Sound and on the
east by Lake Washington. This unusual geography has significant ramifications for human
activity patterns. The western border of the city is permeable only to water traffic (via a
ferry system). Automobile and bus traffic can enter Seattle from the east using one of two
bridges. It is only on the two shortest borders (on the north and the south) that typical levels
of porosity in boundaries are found. The lack of permeable boundaries has the benefit of
reducing concerns about spatial edge effects. Once inside Seattle, there are additional
natural barriers in the form of waterways. The southern section of the city is split northwest
to southeast by the Duwamish Waterway (three bridges cross it). The northern section of
the city is split from the central by a waterway consisting of Salmon Bay, Lake Union,
Portage Bay, and Union Bay. Seattle is a mature city. There was no additional development
of residential or commercial areas requiring changes to the street network between 1989
and 2004. There were two changes to the built environment of note over the study period; a
major baseball stadium was opened in 1999 (Safeco Field) and a new cultural center
opened in the Seattle Center downtown in 2003.
Unit of Analysis
We use the street segment, both sides of a street between two intersections, as our unit of
analysis ( n = 24,023).6 The average length of a street segment in Seattle is 387 feet. The
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).
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majority of the streets (roughly 64%) are between 200 and 600 feet. Using our definition,
very few streets (less than 2%) ended up longer than 1,000 feet. In addition to the theo-
retical and technical reasons discussed earlier, we have several other substantive reasons
and a technical one for using street segments as our unit of analysis including: (1) routine
activity theory is essentially a micro-level theory and thus informs patterns observed at the
micro level (Eck 1995; Sherman et al. 1989); (2) using micro places such as individual
addresses, intersections and street segments minimizes the aggregation in the analysis and
consequently, the risk of ecological fallacy (Brantingham et al. 1976); (3) when consid-
ering policing strategies as they relate to place, a key factor is how much of the variation in
crime involves factors the police are able to address (Taylor 1998) and whether the policy
implications from such will be immediately actionable. On the technical end, street
segments reduce spatial heterogeneity among the units of observation that has been
shown to exist when larger areal units are used (e.g., block groups and census tracts)
(Smith et al. 2000).
Data
We use computerized records of crime incident reports to represent crime for the period
from 1989 to 2004. Incident reports are generated by police officers or detectives after an
initial response to a request for police service. In this sense, they represent only those
events which were both reported to the police and deemed to be worthy of a crime report
by the responding officer and thus provide a measure of vetted crime. We include all crime
events for which a report was taken except those which occur at an intersection. 7
In
addition, we exclude records that lack a specific address, occur on the University of
Washington campus or at a police precinct or police headquarters, and those written for
crimes that occur outside city limits. We geocode the remaining records and are left with
1,697,212 incident reports over the time period. 8
Dependent Variable
We apply group-based trajectory analysis (Nagin and Land 1993; Nagin 1999, 2005) to
cluster street segments into groups with distinct developmental trends over the time period
studied (see ‘‘Appendix 1’’ for technical details). Our approach follows closely the
methodology of an earlier study using 14 years of data and hundred blocks (Weisburd et al.
2004a, b). In our study we use 16 years of data and redefined the unit of analysis. Despite
the use of two additional years and a redefinition of the unit of analysis, the trajectory
results are roughly similar to the earlier study (Weisburd et al. 2004a). Twenty-two
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 sen- sitivity = 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%.
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distinct groups are predicted by the model (as opposed to 18 groups in the original study).
To simplify our description and focus our discussion more directly on patterns in the level
and direction of temporal changes in crime at place across time, we divide the initial
trajectories into eight patterns based on a visual inspection of the level of crime over the
time period and the overall direction of change (Table 1; Fig. 1a). These eight patterns
capture eight distinct types of temporal crime patterns and thus become our dependent
variable in the spatial analysis.
The first general pattern represents the street segments in our study that can be seen as
relatively crime free during this period (Fig. 1a). They experienced an average of less than two crimes per year over the entire time period. One trajectory started the period with three
crimes and declined to less than two and the other started low and increased to three crimes
per year. They account for about half of the street segments in Seattle. Approximately 30%
of the street segments are associated with what we have called the low stable pattern (Fig. 1b). As with the relatively crime free segments, these places evidence low and
essentially stable crime trends (between 3 and 12 crimes per year) and they reinforce the
simple descriptive finding that most places in the city have little or no crime. Two other
trajectory classifications represent places with much more serious crime problems, though
they also evidence strong stability in trends over time. What we term the moderate stable pattern, includes about 1.2% of the street segments (Fig. 1e). These street segments
average around 20 crime incidents throughout the study period and are basically stable
over the study period. Street segments in what we term the chronic high pattern can be defined as the most serious crime hot spots in the city (Fig. 1h). The average number of
crimes per segment is consistently more than 80 crime incidents per year. The difference in
level of crime is what made this trajectory group earn its own pattern. Only 1% of the street
segments ( n = 247) are found in this pattern. The trajectory patterns we have described so far all represent stable crime trends. The
four remaining trajectory patterns include only about one in five street segments in the city.
But nonetheless, they help recognize that crime trends at very micro levels of geography
are more complex than overall city trends would suggest. Two of the trajectory patterns
evidence decreasing crime trends during this period. We use the level of crime and the
trend to group these trajectories into the following patterns. The low decreasing pattern accounts for almost 10% of the street segments in the city and ranges from 8 to 18 crime
incidents on average (Fig. 1c). Importantly, by the end of the study period crime had
declined to less than half of the crime averages evidenced at the outset. Similarly the street
Table 1 Number of street seg- ments per temporal trajectory grouping
Group Number Percentage
1 Crime free 11,898 49.5
2 Low stable 7,688 32.0
3 Low decreasing 2,202 9.2
4 Low increasing 903 3.8
5 Moderate stable 292 1.2
6 High decreasing 572 2.4
7 High increasing 221 .9
8 Chronic high 247 1.0
Total 24,023 10.0
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segments in the crime pattern we term high decreasing evidence strong crime declines (Fig. 1f). Here the base levels of crime in 1989 are much higher ranging between 20 and 60
crime incidents on an average street segment. Only about 2.4% of the street segments in
our study are high decreasing. The last pattern of trajectories is particularly interesting in light of the overall crime
decline in Seattle. We term these trajectories low increasing (Fig. 1d). About 4% of the street segments fall in crime trajectories with a low increasing pattern. The high increasing pattern shows a similar trend but at a higher level (Fig. 1g). Roughly 1% of the street
segments fall in these groups.
Identifying Spatial Relationships in Group Distributions
Previous studies identifying spatial relationships among trajectory group members have
used a sequential approach, conducting a spatial analysis of the output of the develop-
mental statistic (Griffiths and Chavez 2004; Kubrin and Herting 2003; Weisburd et al.
2004a). We also follow a sequential approach but instead of descriptive analysis we
systematically apply a series of point pattern statistical techniques are used to analyze the
spatial patterns of street segments. 9
We use the Ripley’s K-function and a bivariate-K function to examine the second order effects (i.e., local relationships) related to spatial
dependence (Bailey and Gatrell 1995; Fotheringham et al. 2000). Together the two
Fig. 1 Temporal crime trajectories
9 In order to apply point pattern statistics to street segments we use the midpoint of each line/street to
represent the street segment.
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techniques provide a more nuanced picture of local variation than would be possible with
either alone. Unfortunately no single software package exists which combines the required
spatial statistics with a powerful cartographic display engine, so data analysis and display
were done using a variety of software packages including R, SPlancs, CrimeStat �
, GeoDa �
and ArcGIS �
9.3.
Ripley’s K describes the proximity of street segments in the same trajectory to one another. For each street segment, it counts the number of street segments of the same
trajectory that fall within a specified distance band and then repeats for subsequent distance
bands. In this way it characterizes spatial dependence among locations of street segments
with the same trajectory at a wide range of scales. In order to make more formal statements
about the point patterns, we compare the summary statistics calculated from the observed
distribution of street segments with those calculated from a model distribution (i.e.,
complete spatial randomness (CSR)). When used in this way the K-function is able to identify whether the observed pattern is significantly different than what would be
expected from a random distribution (Bailey and Gatrell 1995). Ripley’s K is calculated and then compared to a reference line that represents CSR: if K( h) [ pd2 then clustering is present (Bailey and Gatrell 1995, 90–95; Kaluzney et al. 1998, 162–163).
A bivariate- K (also called a cross K) function is used to test for independence between movement patterns. The bivariate- K answers whether the pattern of street segments belonging to one trajectory is related to the pattern of street segments in another trajectory
(Bailey and Gatrell 1995; Rowlingson and Diggle 1993). As described by Rowlingson and
Diggle (1993) and applied here, the bivariate- K function expresses the expected number of street segments of a particular trajectory (e.g., high decreasing) within a distance of an arbitrary point of a second type of street segment (e.g., high increasing), divided by the overall density of high increasing street segments. As with Ripley’s K, simulation is used to test whether two patterns are independent.
10 The output is a graph representing three
possible relationships: independence, attraction, and repulsion for the tested distance range
(400 foot bins from 0 to 2,800 feet). If the two patterns are independent of one another;
they are most likely the result of different processes (Bailey and Gatrell 1995). A finding of
spatial interaction between them can take two forms, attraction or repulsion. Since street
segments are stationary, attraction in this context refers to a tendency for street segments of
one trajectory to be found in closer proximity to street segments of another trajectory than
would be expected under independence (i.e., their patterns are similar). Repulsion refers to
a tendency for street segments of one particular trajectory to be found at longer distances
from another.
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).
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Findings
We begin with a simple visualization of the distribution of all the temporal trajectories
across Seattle. Seattle is divided into three sections: northern (Fig. 2), middle (Fig. 3) and
southern (Fig. 4). Lower crime street segments are represented by thinner lines and the
higher crime street segments by thicker lines. For example, crime free groups are the thinnest and lightest grey lines. Low stable street segments are darker but still very thin. Street segments which are low increasing are symbolized using thin dark lines and those that are high increasing are thicker and darker in color.
At first glance the impression is one of large areas in which streets are the same color
and thickness broken up by linear patterns. Closer inspection reveals the variety in the
pattern. While there are large areas consisting of predominantly crime free and low stable groups (not surprising given their overwhelming numbers, 12,033 and 7,696 street seg-
ments, respectively), street segments from higher rate trajectory groups are interspersed
within those areas. Street segments that are thicker (designating a high rate group) are most
often arterial roads (i.e., roads which have higher speed limits and collect traffic from
residential streets). A closer examination of the pattern of temporal trajectories reveals
differences by section of Seattle.
In the northern part of the city (Fig. 2), we see most places are crime free or in another low crime trajectory. However, there are thicker/darker lines interspersed throughout.
There is a definite linear arrangement to the patterns related to connected streets with
similar temporal patterns. The area with the most concentrated heterogeneity is a
Fig. 2 Spatial distribution of temporal trajectories (northern Seattle)
J Quant Criminol (2010) 26:7–32 17
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commercial area near the University of Washington campus (solid gray area). In this one
small area (approximately 50 street segments) we see tremendous street segment by street
segment variation in high rate temporal trajectory groups; all high rate trajectory groups are
represented there. There is also a sharp transition at the commercial area’s edge into
predominantly crime free and low stable trajectory pattern street segments. Turning to the middle section of the city we immediately notice the influence of the
downtown area of Seattle on the western portion (Fig. 3). It is here we observe the greatest
magnitude of variation of the temporal trajectories from street to street and the greatest
concentration of streets with high crime. 11
A high degree of variability is also present in the
older residential sections east of downtown. Finally, in the southwestern and southeastern
sections of the city a slightly different pattern emerges (Fig. 4). Once again large areas of
predominantly low crime and crime free street segments are interspersed with the other trajectory types. However, the southern section seems to have the greatest number of linear
patterns of high rate street segments often following arterial roads.
While these maps do not provide quantitative substantiation for the significance of the
spatial associations revealed, they do offer a strong indication of heterogeneity as well as
homogeneity in crime patterns at the street segment level. Thus, it is not the case that we
can understand the action of crime by simply extrapolating from large area trends.
Something seems to be going on at the micro level that needs explanation.
Fig. 3 Spatial distribution of temporal trajectories (central)
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.
18 J Quant Criminol (2010) 26:7–32
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Next, we use Ripley’s K to examine whether street segments of the same temporal trajectory are clustered in space.
12 The statistic reported from Ripley’s K in CrimeStat
(Levine 2005) is the L value. This is a rescaled Ripley’s K where CSR is represented by a horizontal zero line. Figure 5 shows the L( t) line for each of the trajectories; the higher the line, the greater the degree of clustering among the places which are members of the
trajectory group. The X-axis provides information as to the scales at which the clustering occurs. Since we are interested in comparing neighborhood level clustering with street
segment level clustering only distances up to about one-half mile (2,640 feet) are
discussed.
In general, we find the degree of clustering increases with the rate of crime exhibited by
the temporal trajectory (Fig. 5). More specifically, chronic high street segments have the greatest degree of local clustering. There is essentially no difference in the lines for the
other three higher trajectory groups until between 1.5 and 2 blocks (.12 and .15 miles)
where they begin to diverge. At this point, high increasing street segments are most likely to be near other one another followed by high decreasing and moderate stable groups. At the other end of the spectrum, among low rate street segments, it is the low increasing
street segments which are most likely to be found near one another followed by the low
Fig. 4 Spatial distribution of temporal trajectories (southern 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).
J Quant Criminol (2010) 26:7–32 19
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decreasing street segments. The crime free and low stable street segments are the least clustered which reflects their ubiquitousness. All are more clustered than the reference line
representing the level of intrinsic clustering in the street network. In sum, street segments
of the same temporal trajectory group to be generally more clustered at distances of less
than a half-mile than would be expected as compared to either the intrinsic clustering in the
street network or under CSR. Only the degree of clustering varies by the temporal pattern
of crime. Street segments exhibiting high crime trajectory patterns show the strongest
clustering.
While Ripley’s K provides important information about the degree and scale of clus-
tering of street segments within the same temporal trajectory, we must use a variation on
Ripley’s K, the bivariate- K statistic, to answer our question about whether there are specific temporal trajectory pairs that tend to be physically proximal (e.g., low increasing streets near high increasing streets). We conduct a series of pairwise comparisons to evaluate the patterns of each group as compared to those of every other group (i.e., group 1
to group 2, group 1 to group 3 etc.). The results indicate whether the patterns of two groups
tend to systematically ‘hang together’ (attraction) or be found in different places (repul-
sion) or whether their locations are independent of one another. 13
We find no evidence for repulsion; a pair of temporal trajectory patterns which are
consistently far from one another (Fig. 6). We did find evidence for independence (i.e.,
members of two trajectory patterns which are neither consistently near to nor far from one
another) and for spatial interaction in the form of attraction. To simplify our discussion, we
discuss independence first and then attraction. The patterns of streets in the following
trajectory pairs are not related (independent) at some distances up to about a half-mile:
• Crime free as compared to moderate stable (at distances less than or equal to about 1,200 feet), high decreasing, high increasing, and chronic high.
• Low stable as compared to chronic high (at distances less than or equal to about 800 feet).
• Low decreasing as compared to high increasing (at distances less than or equal to about 1,200 feet) and chronic high (at distances less than or equal to 400 feet).
Fig. 5 Ripley’s K of all trajectories
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.
20 J Quant Criminol (2010) 26:7–32
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• Moderate stable as compared to crime free (at distances less than or equal to about 1,200 feet).
• High decreasing as compared to crime free. • High increasing as compared to crime free and low decreasing (at distances less than
or equal to about 1,200 feet).
• Chronic high as compared to crime free, low stable (at distances less than or equal to about 800 feet), and low decreasing (at distances less than or equal to about 400 feet).
These results indicate the process which is producing crime free trajectory patterns is different from the one producing the high crime trajectory patterns but not independent
from the process or processes producing low rate crime patterns. Crime free street seg- ments are the most independent overall; they are unrelated to the patterns three of the other
crime trajectories at all distances and to one other (moderate stable) up to about 1,200 feet. Low stable street segments are independent from chronic street segments at distances two blocks and under (about 800 feet) before showing evidence of attraction at larger distances.
Low decreasing street segments are independent from both high increasing and chronic high at short distances (1,200 feet and 400 feet, respectively). These results make intuitive sense since low crime places are typically very different from very high crime places.
Overall, the degree of independence between crime free, low stable, and low decreasing as compared to high crime temporal patterns suggests the existence of distinct processes
underlying the level of crime observed across places.
The predominant type of spatial interaction in these pairwise relationships at micro
distances is attraction. The members of the following trajectory pattern pairs exhibit
attraction at most distances up to one-half mile (i.e., they tend to ‘hang together’):
• Crime free with low stable, low decreasing, low increasing, and moderate stable (at distances greater than 1,200 feet).
2 – Low Stable
3 – Low Decreasing
4 – Low Increasing
5 – Moderate Stable
6 – High Decreasing
7 – High Increasing
8 – Chronic
1 – Crime Free <=
1,200
2 – Low Stable
<= 800 ft
3 – Low Decreasing <=
1,200 ft <= 400 ft
4 – Low Increasing
5 –Moderate Stable
6 – High Decreasing
7 – High Increasing
Attraction Independence
Fig. 6 Graphical representation of pairwise comparisons
J Quant Criminol (2010) 26:7–32 21
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• Low stable with low decreasing, low increasing, moderate stable, high decreasing, and high increasing and chronic high (after 800 feet).
• Low decreasing with low increasing, moderate stable, high decreasing, and high increasing (after 1,200 feet), and chronic high (after 400 feet).
• Low increasing with moderate stable, high decreasing, and high increasing. • Moderate stable with crime free (after 1,200 feet), low stable, low decreasing, low
increasing, high decreasing, high increasing, and chronic high. • High decreasing with low stable, low decreasing, low increasing, moderate stable,
high increasing, and chronic high. • High increasing with low stable, low decreasing (after 1,200 feet), low increasing,
moderate stable, high decreasing, and chronic high. • Chronic high with low stable (after 800 feet), low decreasing (after 400 feet), low
increasing, moderate stable, high decreasing, and high increasing.
The finding of attraction indicates common processes may be underlying these pairs of
temporal crime patterns. Turning first to the low rate groups, we find spatial attraction
between street segments with crime free, low stable, low decreasing, and low increasing trajectory patterns at all distances (Fig. 6). The attraction between moderate stable and crime free street segments is only present at distances of greater than 1,200 feet. Thus moderate stable street segments tend to be within a half mile of crime free street segments
but not within approximately three blocks of them.
Low stable trajectory patterns are attracted to all other trajectory patterns except chronic high (but only at distances of greater than 800 feet). Here again, we uncover a pattern of independence related to adjacent streets and those within two blocks but general attraction at
distances up to one-half mile. One explanation for this finding is that the processes under-
lying the realization of low stable places are similar to those underlying the places with both higher and lower crime trajectories. However, there may be place-specific differences that
occur at nearby streets which cause their temporal crime patterns to change. Alternatively, it
could indicate the role low stable places play as ‘jumping off points’ for changes in crime
rates. At the same time, the finding confirms the significant geographic dispersion of crime free and low stable places. They are found all across Seattle, not just in ‘good’ areas.
The spatial distribution of low rate street segments that are experiencing decreasing or
increasing temporal crime patterns is especially interesting. Low decreasing street seg- ments tend to be found near higher rate crime trajectory members such as moderate stable and high decreasing at all distances. However, similar to the finding just discussed for low stable street segments, they are only attracted to high increasing (at distances greater than 1,200 feet) and chronic high (at distances of greater than 400 feet) at distances greater than one block. On the other hand, low increasing street segments are also near moderate stable, high decreasing, high increasing, and chronic high. It may be the processes sustaining or increasing crime on streets nearby are also influencing the low increasing street segments.
We find much more heterogeneity among temporal trajectory groups representing high
rate crime places. The pattern of chronic high street segments is related to moderate stable, high decreasing, and high increasing places at all distances. High increasing street seg- ments tend to be found near low increasing, moderate stable, high decreasing, and chronic high at all distances. However, high increasing street segments are independent of low decreasing streets up to 1,200 feet. This spatial separation hints at the existence of sig- nificant differences between the processes underlying low decreasing and high increasing street segments. Streets which are part of high decreasing temporal patterns tend to found near street segments of both low and high rate trajectory patterns (i.e., low stable, low
22 J Quant Criminol (2010) 26:7–32
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increasing, high increasing, and chronic high) at most distances. Thus, in general, streets with higher crime or changing temporal trajectories tend to be more frequently associated
with other streets which are also higher crime or undergoing a change in crime rates.
In sum, our descriptive and statistical analysis reveals temporal crime trajectory pattern
membership often varies from street segment to street segment. This is apparent in both the
descriptive maps and in our finding of widespread spatial attraction among different
temporal trajectory groups. Overall, these results are consistent with the spatial pattern we
saw on the descriptive maps showing proximal places can have very divergent temporal
crime trajectories. While certain areas of the city consist of predominantly crime free and low stable trajectory patterns, other areas are characterized by extreme spatial heteroge- neity with street to street variation in both the level and pattern of temporal crime pattern
membership. A set of any four nearby streets might include one crime free, one moderate stable, one high decreasing and one chronic street segment.
The outcomes of the pairwise comparisons are consistent with earlier analyses. In
general, high rate groups tend to attract other high rate groups. This may be a reflection of
the built environment. Place characteristics such as land use patterns, disorder, housing
quality etc. tend to be shared among nearby street segments. Thus it is not surprising that
high level and low level streets might generally attract. Our results clearly indicate inde-
pendence between the pattern of crime free streets and all high rate street segments
(although moderate stable is only up to three blocks away). Thus the processes which
create crime free streets are most likely very different from those which result in streets
with significant amounts of crime.
The finding of attraction between low stable and low decreasing and the high rate groups is more puzzling. It may be the geographic patterns of street segments in low stable and low decreasing temporal patterns are a reflection of their proximity to high crime places. Low stable street segments are found near both low and high crime street segments and thus are influenced by processes operating to produce both low crime and high crime
places. Similarly, low decreasing street segments may be ones which have experienced changes in their characteristics which run counter to the general high crime environment in
which they are situated and have resulted in a decreasing crime trajectory pattern. Together
these results demonstrate the micro level of analysis is capturing important variability in
trends; variability that would be masked at a more macro level.
At the same time, we do find two phenomena that demand we explain and understand
clustering at different scales. First, we find that there is a dependence of street segments
within the same temporal crime trajectories at distances up to one-half mile. Second, we
find evidence for significant attraction between specific types of temporal crime trajecto-
ries. This is especially true with low or no crime segments, where we observe large areas in
which these trajectories prevail. However, it is also true of chronic segments and other high
crime segments for example in the city center.
Discussion
But what do these findings mean to our understanding of the spatial pattern of temporal
crime trajectories and of the structure of crime concentration at places more generally?
What does the existence of varying degrees of heterogeneity in street to street temporal
crime trajectories indicate? How do we interpret our finding that street segments within a
single trajectory group are more likely to be found near one another? What can we learn
from the knowing certain pairs of temporal crime patterns tend to be found in the same
J Quant Criminol (2010) 26:7–32 23
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general areas? Since these observations are not mutually exclusive, how do we reconcile
places where more than one of the above relationships exist? Absent data on the nature of
places we can of course only speculate on what might be driving the observed patterns.
One interpretation is that there are simply two sets of influences operating on crime in
the city. One set of influences may reflect the existence of ecological labels. Ecological
labels provide a shared perception of places that is generally homogeneous across larger
areas of a city (Brantingham and Brantingham 1991 [1981]). Assuming perceptions
influence actions, shared perceptions can lead to shared actions which in turn serve to
reinforce the existing crime situations. While these perceptions apply to an area they are
‘played out’ at the micro level of street segments. This cycle may explain the existence of
generally high crime rate areas as well as generally low crime rate areas and thus our
finding that streets with similar trajectories are generally clustered.
A second interpretation focuses on micro level influences of opportunity factors on
crime and the importance of local trends. Theories such as routine activity theory (Cohen
and Felson 1979; Felson 1986, 2002) and crime pattern theory (Brantingham and Bran-
tingham 1984, 1991 [1981], 1993) focus on the spatio-temporal characteristics of crime
events in an attempt to explain why some places have more crime than others (Eck 1995).
The urban backcloth provides a local baseline susceptibility to crime through a shared set
of crime-related characteristics (Brantingham and Brantingham 1991 [1981]; Jacobs 1961).
In this way, place characteristics also play a significant role in shaping the differential
distribution of potential targets and motivated offenders across space and time.
At the micro level, opportunity theories would predict variation in micro level crime if
there was significant variation in characteristics of places that facilitate the convergence of
the elements necessary for a crime to occur. However, variation in crime across larger
areas and even neighborhoods could also be consistent with the perspective. If there is no
significant variation in characteristics which affect the frequency of convergence, then we
would expect relatively homogenous and stable crime patterns over time. In this way,
opportunity theories can explain both heterogeneity and homogeneity in the geographic
pattern of temporal crime rates.
In the same vein, our finding of street to street variability in temporal crime trajectories
may simply be a reflection of the street to street variability in the urban environment.
Activity generators such as employment centers and retail stores attract both residents and
non-residents, law abiding and those open to criminal opportunity, to specific places and in
doing so produce local variation above and beyond the baseline for a place (Brantingham
and Brantingham 1995; Felson 1987; Kinney et al. 2009). Additionally, the influence of
activity generators/attractors is not confined to the street segment on which they are
located. As individuals travel to and from the activity/crime generator they become
familiar with the places along their route. Their awareness of opportunities is highest along
their route and at the places they visit; it declines with distance from each.
In this way, micro level patterns can also explain the pair-wise clustering of street
segments with different temporal crime patterns. While there is much street to street
variability in the urban backcloth, there is also general continuity. The general continuity
explains why low rate temporal crime trajectories tend to be found near other low rate
street segments. At the same time, the character of streets is influenced by the mixture of
land uses present and by unique characteristics of places that may influence the amount of
crime experienced.
Together these assumptions lead us to believe that crime attractors/generators have a
differential impact. Levels of traffic along blocks provide an indicator of the number of
people for whom that block is part of their activity space. The more activity spaces of
24 J Quant Criminol (2010) 26:7–32
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which a block is a part, the greater the potential that an offender and a target will converge
without a capable guardian present and thus supply the necessary elements for a crime to
occur (Cohen and Felson 1979). These mechanisms are at work regardless of neighborhood
level influences. As we saw in our results, what varies by neighborhood is the degree to
which increased opportunity translates into increased crime (e.g., is the change from crime free to low stable or from moderate stable to chronic high). The exact form of the change may also be a function of variability in the temporal crime patterns of nearby places.
Given the above, our work would propose the importance of recognizing commonalities
among micro level places that stem from consistency in the built environment. However,
while our data suggest the importance of recognizing crime trends at micro places, they do
not necessarily reinforce the importance of community level influences. This is especially
the case in our finding of clustering of similar trajectories in certain areas. For example, the
clustering of chronic segments near one another may reflect important similarities in social and built environment characteristics. In this context it would be the street segment
characteristics that are important to understanding the concentration in crime at place.
Other researchers have advocated for starting small (Brantingham and Brantingham
2008; Groff et al. 2008; Oberwittler and Wikstrom 2008; Rengert and Lockwood 2008; van
Wilselm 2008). Even if some of the variability present at micro levels does work its way up
to more macro levels, we will still have wanted to know that. Otherwise, we will be unable
to separate observed macro level patterns which are being driven by variability at micro
levels from those that are not.
We think it also important to recognize that there may not be a simple dichotomy
between social disorganization and opportunity theories as they have often been linked to
units of geographic analysis. Since both social disorganization and opportunity variables
are related to the urban backcloth, it may be too simplistic a model to attribute area trends
to social disorganization factors and local trends to opportunity factors. The story of crime
at place may be more complex.
At a policy level, our research reinforces the importance of initiatives like ‘hot spots
policing which address specific streets within relatively small areas (Braga 2001; Sherman
and Weisburd 1995; Weisburd and Green 1995). If police become better at recognizing the
‘good streets’ in the bad areas, they can take a more holistic approach to addressing crime
problems. For example, they can more precisely target community building and law
enforcement operations to maximize efficiency and effectiveness. More broadly, they can
work with other city agencies to change the physical and social environment of problem
places (Johnson et al. 2008). Alterations to the built environment which improve sur-
veillability, control access, and increase the capacity for territoriality among legitimate
users can reduce crime (Lab 2007).
Before concluding we want to note some specific limitations of our work and make
suggestions for future study. While our findings break new ground in describing the spatio-
temporal relationships among street blocks, there are two limitations to our micro-level
examination of spatio-temporal crime patterns. Although our study examined a longer time
period than has been available to other scholars at the micro level of analysis, it remains
only a snapshot in the development of places. Accordingly, our analysis might have
underestimated some dynamic elements while overestimating others. Another limitation
involves the low numbers of incidents at street segments. This feature of street segment
level research necessitated we use total crime events rather than exploring specific types of
crime as is recommended in place-based research (Clarke 1983; Clarke and Felson 1993).
Future research may want to follow Braga et al. (2009) and isolate high crime street
segments for more in-depth study.
J Quant Criminol (2010) 26:7–32 25
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This paper establishes the importance of looking at the micro level. Of course, our data
only allow us to identify the importance of temporal crime patterns. To advance the field
future work should collect street segment level characteristics which would allow the
identification of characteristics which are associated with particular temporal crime pat-
terns. Only by conducting research that uses street blocks as the unit of analysis and by
thoroughly describing their characteristics will we be able to shed additional light on these
questions. Future work should utilize prospective data collection to take advantage of the
more robust information systems available today. In addition, prospective data collection
allows for the inclusion of systemic social observation (SSO) (Sampson and Raudenbush
1999) and ethnographic components that are crucial to ‘adding flesh to the bones’ of
quantitative research. Neither of which are possible with retrospective studies.
Conclusions
While there is a growing body of evidence regarding the value of examining micro level
places, the predominant paradigms of place and crime focus on large area trends. Firmly
grounded in the ecological tradition, no one doubts the importance or salience of these
investigations to understanding crime and delinquency. However, the work herein rein-
forces the idea that we need to focus on small area trends as well.
Our systematic examination of the spatial patterns within temporal crime trajectory
groups allows us to enrich our understanding of the structure of concentration in crime
patterns at micro level places. In doing so our study reveals the significant geographic
variability in temporal trends from street segment to street segment which suggests
something is going on at the micro level that requires explanation. Since our study lacks
any information on the characteristics of places we are unable to explain why these patterns
are occurring or to conjecture about the underlying processes at work. We leave those
explorations to future work.
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 pro- gramming, 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.
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.
26 J Quant Criminol (2010) 26:7–32
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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 uni-
formly 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
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).
J Quant Criminol (2010) 26:7–32 27
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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 9 log( n) 9 ( 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 proba-
bility 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 satis-
factorily in classifying the various crime patterns into separate trajectories.
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