Has there been a significant change?

As you may have guessed, our decision on whether to normalize the data for population changes will have a major influence on the question of significance. First, compare the normalized versus the unnormalized distributions for 2016 vs 2005-2015.

Viewing the distributions, it is pretty clear that 2016 was a unique year. Both the normalized and unnormalized distributions are right shifted from the 11 year average. The 2016 distribution is also narrower and higher peaked. These two facts seem to indicate two things– one, the number of thefts per day has increased regardless of whether or not you normalized the data and two, there are more high theft days occurring than before. The differing shape of these distributions is actually somewhat troubling for our analysis, but warrants a post of its own. For now, we are going to work with the assumption that the distributions are similar enough in shape to allow for the use of a t-test to compare means.

When we perform these t-tests, we unsurprisingly find that they indicate significant changes in the means of both the normalized and unnormalized data. Both data sets have p-values less than 2.2e-16(tiny!), meaning we reject the null hypothesis that the means are equal. When you look at the confidence intervals for both, you see the means of the 2005-2015 data both fall well under the lower bound.

## 
##  One Sample t-test
## 
## data:  sums_2016$count
## t = 12.39, df = 364, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 62.41937
## 95 percent confidence interval:
##  69.49977 72.17146
## sample estimates:
## mean of x 
##  70.83562

## 
##  One Sample t-test
## 
## data:  sums_2016$pop_adjusted
## t = 9.1167, df = 364, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 10.48949
## 95 percent confidence interval:
##  11.27765 11.71118
## sample estimates:
## mean of x 
##  11.49442

This set of tests is only a first, rough, step, but it does give us an idea of what we are dealing with. It appears 2016 saw a significant increase in thefts over the 11 year average. After normalization for population increases, we are seeing on average about 1 more theft per day per 100,000 people. This works out to about 6 more thefts per day when normalized. However, as usual, this sort of analysis needs some perspective. As mentioned above, the shapes of the distributions differ quite a bit. T-tests rely on distributions differing only in central tendency(i.e. the mean). If this assumption is violated, then our t-tests may not be accurate. There are some additional problems with using t-tests to compare central tendency in this crime example, but they are more detail than is wanted for this post. Perhaps I will address these in a later post. For now, it is reasonable safe to conclude that thefts in 2016 were higher by a statistically significant amount.

Analysis of Theft Dispersal

Since our map seems to indicate a wider dispersal of thefts, looking at theft counts by zip code should be informative.

Surprisingly, the highest crime zip codes are not in downtown Louisville at all, but are more in the south and west end of Louisville. There are still several downtown zip codes– particularly 40202 and 40203 which are located at the bright yellow area of our heat map– near the top of the plot, but we can see evidence that thefts are more evenly dispersed throughout the community. While just 6 of Louisville’s 33 zip codes accounted for 48.75% of all violent crime, it takes 9 to cover 48.52% of thefts.

But this sort of discussion is not particularly rigorous and, frankly, is a little unsatisfying. Luckily, there are entire fields devoted to the study of dispersion. In economics, it is common to measure income related inequality using the Gini coefficient and Lorenz curve(a good introduction can be found here and here). These techniques have also been applied to identify unequal distributions in crime frequencies. For our purposes, we will measure the inequality in the frequency distribution of thefts by zip code. A Gini coefficient of 1 indicates maximum inequality(in our case this would be all of the thefts taking place in one zip code) while a coefficient of 0 indicates complete equality (thefts are evenly dispersed throughout all zip codes).

Calculating Gini coefficients for 2007 and 2016, we get 0.36 and 0.32 respectively. This decrease indicates that theft offenses have become less unevenly distributed(i.e., thefts are becoming more evenly dispersed through all zip codes). We can visualize this with a Lorenz curve. If crimes were completely evenly distributed, they would fall on the diagonal ‘line of equality’. The further below the curved line bows, the more unequal the data. As you can see, the 2016 curve falls slightly inside of the 2007 curve, indicating lower inequality. So, for example, we can see that the bottom 50% of zip codes account for 28% or 25% of the thefts, depending on the year. This solidifies the idea we saw above where a fraction of zip codes accounted for a disproportionate percentage of the total thefts.

While this measure is a good start, it is a global measure of unequal distribution when what we would like is something more localized. A more crime specific measure that is a step in this direction is called the Offense Dispersion Index(ODI). When looking at the crime increase between two years, we first calculate the difference in each area—in our case between zip codes. Then, we order these differences from highest to lowest change. Finally, we remove the highest ranking area and recalculate the crime rate with the remaining areas. Then we take the second highest ranked area and remove it, again recalculating, but for the n-2 areas. This continues until only one area is left.

From this procedure we can calculate the ODI, which is just the proportion of areas that must be removed from the calculation before the increase in crime turns into no-change or a decrease in crime. So as you can see in the table below, it takes the removal of 14 zip codes worth of data to change the increase in crime rate from 2007 to 2016 into a decrease. So the ODI is just 14 divided by the total number of zip codes, 33, or 0.42. ODI’s range from 0 to 1 with values close to zero indicating a low crime increase dispersion factor. In other words, a value closer to 1 suggests suggests a problem across all areas, rather than something very localized. To compare, I calculated the ODI for drug crimes in Louisville to be 1. Since drug crimes were at nearly an all time high in 2007, we actually have an immediate decrease in crime with the first zip code. This suggests that, relative to drug crimes, theft remains relatively localized. Alarmingly, drug crimes appear to be widely dispersed throughout the community.

One additional measure worth mentioning is the non-contributory dispersion index(NCDI). The NCDI is the proportion of areas that showed crime increases divided by the total number of areas. Unlike the ODI ratio–which only uses areas that contributed to the overall crime increase–the NCDI looks at all areas that showed increases and can thus be used to show the spread of areas showing increases. For our theft and drug data, we see 0.82 and 0.36 NCDI ratios, respectively. The high NCDI on theft crimes indicates an increase in theft crimes in many areas. When combined with a mid-range ODI it seems that while theft is still relatively localized, it appears to be spreading to new areas. On the other hand, the lower NCDI of drug crimes combined with the extremely high ODI probably means that drugs are so evenly dispersed throughout the area already that further increases in these high drug crime areas are unlikely.

It must be noted that both ODI and NCDI measures are first, rudimentary, steps into quantifying crime dispersion. Much more advanced techniques which take full advantage of spatial analysis can provide significantly more detail on localized variation in crime patterns. These are far beyond the scope of this post, but with a little more work the ODI measures we calculated here can be combined with other techniques for more specific analysis. If you are interested, a good starting point is located here. This whitepaper was the inspiration behind much of this analysis and a full acknowledgement of the use of its ideas is essential.