Abstract
The primary objectives of this research are (1) to introduce summary measures of concentration that are relatively new to our field; (2) compare four concentration measures to determine whether there are reasons to use one in favor of the others; and (3) apply the measures to a real-case data to further understand the concentration phenomenon. Using the crime data of Cincinnati, we compare four commonly used social science measures of concentration: Gini, Simpson, Shannon, and Decile indices. For some purposes, the measures are interchangeable, while for other purposes the measures may suggest different interpretations for the same set of data. This paper is the first quantitative comparison of multiple measures of crime place concentration. We describe the benefits and limitations of each index and the circumstances for which each is most useful. We also answer the question: is crime within street segments spread along the segment or is it concentrated at a few addresses, as most place-based crime studies have overlooked the interior variability of crime on street segments.
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Source: modified the Gini–Lorenz curve from Lorenz (1905)
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Notes
By definition of the Decile index, 0 percent (or 0) Decile is not possible. Ten percent Decile would be the least value that any street segment could achieve.
However, species diversity should not be confused with species richness and evenness. Species richness is based on the number of different species while evenness refers to homogeneity (repeat appearance) of the species in an environment. That is, the more equal the proportions for each of the groups, the more homogeneous they are. Though these indices are useful for comparing different environmental systems, simply counting the number of species or kinds of species will not allow comparison of diversity.
For those who are interested in using the Shannon index to measure the concentration of crime across places, be careful when transforming \(p_{i}\) into \(\ln p_{i}\). It is a very common mistake to use \({ \log }_{ 1 0}\) instead \({ \ln }\) (natural log). Using \({ \log }_{ 1 0}\) will render totally different values for both the Shannon index and the Shannon Effective Number of Crime Places explained in the following paragraph.
As street segments with no addresses inside never experienced crime (Cincinnati Police Department assign every crime to its corresponding address), we include these segments into the analysis with other segments for consistency.
We also provide the distributions of all four indices across the population of street segment in “Appendix 1.”
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Appendices
Appendix 1
See Fig. 12.
Distributions of four indices across the population of street segment in Cincinnati, Ohio
Appendix 2
See Fig. 13.
Comparison between Simpson Original versus Simpson Adjusted
Appendix 3
Calculating the Spatial Gini using Excel spreadsheet.
In the Excel spreadsheet, we assigned all addresses to their nested street segment ID. During this procedure, the number of crimes for each address is also joined to the corresponding street segment, in ascending order of address in value of crime. Then, we created another column with the cumulative number of crimes from the least crime address to the highest crime address. Using the total number of crimes for the street segment as a denominator, we created another column where each cell shows the proportion of cumulative crimes per address. Each of these values was used to identify two base lines (both lower base and upper base) in the trapezoid calculation. We created another column where each cell shows the proportion of addresses, so that if a segment has n addresses, each address is assigned \({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 n}}\right.\kern-0pt} \!\lower0.7ex\hbox{$n$}}\). This value is used as height in the trapezoid calculation. Finally, subtracting the aggregated areas of each trapezoid beneath the Lorenz curve from 0.5, and then dividing it by 0.5 gave the Gini coefficient for the street segment. The aggregation of trapezoid areas within a segment can be calculated using the “sum” function in the Microsoft Access. Exactly the same logic can be applied to the neighborhood–segment or city–neighborhood levels to calculate their Gini coefficients.
Appendix 4
Application of indices to a real case.
We chose one of the high-crime street segments in downtown area because this segment has the maximum value of Shannon index among the population of the street segments. Figure 14 shows this area in detail: Nottingham Road where 98 out of 217 addresses were victimized in 2009. The small dots beneath the street segment are the locations of crime assigned to the Nottingham road street segment. The mapping software places the crime on the road centerlines, instead their actual locations. The streets in this area, including Nottingham Road, are all part of a single housing project. The uniformity of buildings may account for the high dispersion and low concentration, relative to other segments (Fig. 14).
Nottingham Road in Cincinnati, Ohio, with corresponding statistics
When we look at the table on the right pane, all four indices are provided. Decile index of 0.41 indicates modest concentration, as does the Gini. The raw Shannon index is 4.45, and the Effective Number of Crime Places is 86. This means that the distribution of crime among the addresses in this street segment is equivalent to a street segment with 86 addresses experiencing a single crime for each address. The Simpson ENCP is 12 places lower. Since the Shannon Effective Number of Crime Places is greater than Simpson Effective Number of Crime Places, it is likely that a few addresses dominate the others, with regard to crime, despite the overall dispersion of crime along the segment.
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Lee, Y., Eck, J.E. Comparing measures of the concentration of crime at places. Crime Prev Community Saf 21, 269–294 (2019). https://doi.org/10.1057/s41300-019-00078-2
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DOI: https://doi.org/10.1057/s41300-019-00078-2