Maps and statistics: Exploratory spatial data analysis|
Chapter 2: What Crime Maps Do and How They Do It
Some statistical methods have been mentioned in the preceding discussion, and consideration of statistical concepts is unavoidable when considering how best to visualize numerical data. As noted earlier, because we can lie with statistics, we can also lie with statistical maps. Indeed, maps have been used throughout history as propaganda tools (Campbell, 1993, pp. 229-235), so potentially we can have honest error as well as pure cartographic deceit. Perhaps the greatest danger in the mapmaking process is that people tend to believe the information in maps (what MacEachren, 1995, p. 337) called the connotation of veracity), and they also believe that maps are unbiased (the connotation of integrity).
But mapmakers, like other elements of society, are culturally conditioned, selectively including and excluding data according to the values of the responsible parties. Given that maps can harbor many possible errors and biases, both intentional and accidental, it is incumbent on the crime analyst to be aware of possible sources of error and to work to avoid them. Nowhere is there more scope for distortion and misinterpretation than in the preparation of maps based on numerical data. This is due to the potential complexity of the information and the infinite set of display permutations, whether in raw form or as some derivative measure such as a rate or percentage.
Mapmakers can gain a preliminary understanding of what the numbers mean through the process of exploratory spatial data analysis. It is quite helpful to understand what the distribution of a set of numbers looks like when expressed graphically. Is this a normal (bell-shaped) distribution with most observations clustering around the mean, or average, and a few very low and a few very high values? Is it a skewed distribution with extreme values to the right (high values) or the left (low values)?
In the unlikely event of a normal, symmetrical, bell-shaped distribution, maps created by all of the classing methods look similar. Almost always, however, real-world data are somewhat skewed, and different classing methods produce maps that look different and convey different visual impressions to readers.
Consider in some detail what will happen when different methods are applied to a data set that has a strong positive skew (figures 2.16 and 2.17).
A Note on Skewness|
A normal distribution is the familiar bell-shaped curve that is seldom seen in crime data. Most crime data are positively skewed, meaning there is a long right "tail" representing a few high values. Hot spots (high crime areas are geographic expressions of skewness, which presents difficulties in mapping numerical data. See the box, How Much Exploration? and figures 2.15 and 2.16.
Let's review each histogram (or frequency curve) and map in figures 2.15 and 2.16, method by method.
- Equal count. On the histogram, the right tail (highest values of the distribution) is prominent because there are few extremely high values. Thus, the program has to seek the lower rank-ordered data values (farther to the left on the histogram) to come up with the 13 observations for the class. (Note that the number of observations per class is uneven, ranging from 13 to 17.) The resulting map tends to visually exaggerate the seriousness of the problem because color saturates more map areas.
Generally, natural breaks or equal intervals will be the best methods for creating area-type maps in crime analysis.
- Equal range. Because the distribution is right-skewed, equal range will tend to favor lower data values. The two lowest classes have 23 and 26 members, respectively, while the higher classes have 7 and 3. The map contrasts with the equal count version, now visually minimizing the problem.
- Natural break. This method appears to have struggled to come up with natural breaks, which is a problem, along with breaks in awkward places. The result here is quite similar to the equal range breakdown, with cuts between classes shifted to the left (lower values) as compared with the equal range. It comes as no surprise that the equal range and natural break maps are quite similar.
- Standard deviation. Here, the breakdown of class intervals is set with reference to the average, or mean, so that an interval of 1 SD is established to the left of (below) the mean (blue line, 0.47), and above the mean at the same distance. The effect of this on the right-skewed distribution is a symmetrical breakdown, with about as many observations in the lowest (10) and highest (9) classes and in the two middle classes (22 and 18). The visual impression conveyed by the associated map is close to the severity of the equal count method. This is due to the similar number of observations in the top category.
The basic point to be made from this discussion is that cases that may fall in a given class by one method may be in a different class by another. The only certainty is that the highest and lowest values will always be in the top and bottom map classes, respectively. What method is preferable? MacEachren (1994, p. 47) noted that, "for any skewed data, quantiles are a disaster for a presentation map!" In the above example, quantiles result in such a large data range in the highest class as to be almost meaningless. Standard deviation classes may be helpful in some situations where the distribution is not extremely skewed.
Note that a frequency curve shows skewness in the rank-ordered data values, but only a map can show skewness in geographic space. Are the high values distributed geographically in a random way or clustered? Either method yields useful information. If the high crime rates are clustered, it may indicate a hot spot. If the high rates are random, the net impact on the community may be about the same, but we are now unable to point to a hot spot.
We can see an empirical relationship between map scale and skewness, which is minimized in a small area (large scale) and maximized in a large area (small scale). Think of it this way: A very small area in the community, say 1 square yard, can have no spatial skewness because only one event can happen there. But as the spatial scope increases (smaller scale maps covering larger areas), the potential for skewness increases because there can be bimodal, or split, distributions in space (as well as time). A clump of events can occur in one small area with the rest emptyan extremely skewed pattern.
This is what the crime scene is like on a regional, national, or global scale. Clusters correspond to opportunities presented by the underlying controlling condition, population distribution. At the smallest scale (region or world), the crime map is for all practical purposes the same as the population map, but at larger scales (city or neighborhood) we refine the view and see that the presence of people actually means variations in rates conforming to varied social and physical environmental conditions. Also, at larger scales we will see different patterns depending on the denominators used to calculate crime rates.
Another way to visualize a distribution is the use of a box plot, which shows how data are spread in relationship to the mean, median, mode, and quartiles, with outliers symbolized in a special way. (Outliers are values more than 1.5 box lengths from either the 25th or 75th percentiles.) If we examine the HOMRATE data set using the box plot routine in the Statistical Package for the Social Sciences (SPSS), the result appears as shown in figure 2.17.5 Note that the box plot is an alternative way of visualizing the same data shown in figures 2.15 and 2.16. In the box plot in figure 2.17, the red box represents 50 percent of the data values, with the median shown by the bold line across the box. The 75th and 25th percentiles are the top and bottom of the box, respectively. The ends of the Ts represent the smallest and largest observed values that are not defined as outliers. Although the box plot seems to be repetitive, it provides a different perspective on the dataone that complements the more frequently used histogram. (For a detailed explanation, see SPSS documentation, such as SPSS for Windows Base System User's Guide Release 6.0, p. 186.)
Only the immediate objective and the available tools limit the amount of exploration and preprocessing crime mappers do. Perhaps the single most important exploratory step is the creation of a histogram, box plot, or comparable graphic with which to visualize the shape of the data and answer the fundamental question: Is it severely skewed or in some other way not normal (e.g., bimodal or double peaked)? How will this affect maps, and what type of map will permit a presentation that minimizes distortion and accurately portrays the data? Again, this examination of the data is the ideal. Not all analysts will have the tools or the time to go through this step. Nevertheless, these possibilities are outlined here to raise awareness of what constitutes the best practice.