Section 3.1 Thematic maps: key terms and ideas
Subsection 3.1.1 Choropleth maps
Choropleth maps display variation in areas (postal codes, police districts, census administrative units, municipal boundaries, regional areas, etc.) through the use of the colour that fills each of these areas in the map. A simple case can be to use a light-to-dark colour to represent less to more of the quantitative variable. They are appropriate to compare values across areas.
Most thematic maps you encounter are classified choropleth maps. They group the values of the quantitative variable into a number of classes, typically between 5 and 7. We will return to this later in the chapter. It is one of the most common forms of statistical maps, and like pie charts, they are subject to ongoing criticism. [126] concluded they simply do not work. And there are indeed a number of known problems with choropleth maps:
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They may not display variation within the geographical units being employed. You are imposing some degree of distortion by assuming all parts of an area display the same value. Our crime map may show a neighbourhood as secure, when there is a part of this neighbourhood that has a high level of crime, and vice versa.
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Boundaries of geographical areas are to a large extent arbitrary (and unlikely to be associated with major discontinuities in your variable of interest). In crime analysis we very often use census administrative units, but these rarely represent natural neighbourhoods.
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They work better if areas are of similar size. Areas of greater size may be more heterogeneous internally than those of smaller size; that is, they potentially have the largest error of representation. Also, visual attention may be drawn by areas that are large (if size is not the variable used to create a ratio).
Subsection 3.1.2 What do we use choropleth maps for?
Subsubsection 3.1.2.1 Crime rates
In a choropleth map, you "can" show raw totals (absolute values), for example, number of crimes, or derived values (ratios), such as crime per 100,000 inhabitants. But as a general rule, you should restrict choropleth maps to show derived variables such as rates or percentages ([127]). This is because the areas we often use are of different size and this may introduce an element of confusion in the interpretation of choropleth maps. The size of, say, a province or a county, "has a big effect on the amount of colour shown on the map, but unit area may have little relationship, or even an inverse relationship, to base populations and related counts." ([128]: S30).
Mapping counts, that is, mapping where a lot of the crime incidents concentrate is helpful, but we may want to understand as well if this is simply a function of the spatial distribution of the population at risk. As [129] suggests, "practitioners often recognize that a substantial density of crime in a location is sufficient information to initiate a more detailed analysis of the problem", but equally we may want to know if "this clustering of crime is meaningfully non-random, and if the patterns observed are still present once the analysis has controlled for the population at risk" (pp. 11-12).
A map of rates essentially aims to provide information into geographic variation of crime risk, understood as the probability that a crime may occur. Maps of rates are ultimately about communicating the risk of crime, with greater rates suggesting a higher probability of becoming a victim of crime.
In social science, the denominator on mapped ratios is typically some form of population size, which typically we will want as current as possible. In other fields, some measure of area size may be a preferred choice for the denominator. However, a great deal of discussion in crime analysis has focused on the choice of the right denominator. Authors relate to this as the denominator dilemma: "the problem associated with identifying an appropriate target availability control that can overcome issues of spatial inequality in the areal units used to study crime" ([129]). The best measure for your denominator is one which captures opportunities. If for example you are interested in residential burglary, it makes sense to use number of inhabited households as your denominator (rather than population size). Whatever denominator you choose, you will usually want to make a case as to why that is the best representation of the opportunities for the crime type you’re interested in.
As noted, population is a common choice, but it is not always the one that best captures crime opportunities. Population is also highly mobile during the day. People do not stay in the areas where they live: they go to work, they go to school, they travel for tourism purposes, and in doing so they alter the population structure of any given area. As [129] highlights with an example "the residential population (as is usually available from the census) tells the researcher little about the real number of people outside nightclubs at 2 am". Geographers and criminologists, thus, distinguish between the standard measures of population that relate to people that live in an area provided by the census and government statistical authorities, and the so-called ambient population that relates to people that occupy an area at a given time and typically are a bit more difficult to source (see for example: [130]).
As we will see later, one of the key problems with mapping rates is that the estimated rates can be problematic if the enumeration areas we are studying have different population counts (or whatever it is we are counting in the denominator). When this happens, and those population counts produce small samples, we may have rates for some locations (those with more population) that are better estimated than others and, therefore, are less subject to noise.
Aside from problems with the denominator, we may also have problems with the numerator. In crime analysis, the key variable we map is geocoded crime, that is crime for which we know its exact location. The source for this variable tends to be crime reported to and recorded by the police. Yet, we know this is an imperfect source of data. A very large proportion of crime is not reported to the police. In England and Wales, for example, it is estimated that around 60% of crime is unknown to the police. And this is an average! The unknown figure of crime is larger for certain types of crime (e.g., interpersonal violence, sexual abuse, fraud, etc.). What is more, we know that there are community-level attributes that are associated with the level of crime reporting to the police ([131]).
Equally, not all police forces or units are adept at properly recording all crimes reported to the police. An in-depth study conducted in England and Wales concluded that over 800,000 crimes reported to the police were not properly recorded (an under-recording of 19%, which is higher for violence and sexual abuse, 33% and 26% respectively) ([132]). There are, indeed, institutional, economic, cultural and political factors that may shape the quality of crime recording across different parts of the area we want to map out. Although all this has been known for a while, criminologists and crime analysts are only now beginning to appreciate how this can affect the quality of crime mapping and the decision making based on this mapping ([133]; [134]).
Finally, the quality of the geocoding process, which varies across crime type, also comes into the equation. Sometimes, there are issues with positional accuracy or the inability to geocode an address. Some authors suggest we need to be able to geocode at least 85% of the crime incidents to get accurate maps ([135]); otherwise, geocoded crime records may be spatially biased. More recent studies offer a less conservative estimate depending on the level of analysis and number of incidents ([136]); although some authors, on the contrary, argue for a higher hit rate and consider that hot spot detection techniques are very sensitive to the presence of non-geocoded data ([137]).
Although most of the time crime analysts are primarily concerned with mapping crime incidence and those factors associated with it, increasingly we see interest in the spatial representation of other variables of interest to criminologists such as fear of crime, trust in the police, and more. In these cases, the data may come from surveys and the problem that may arise is whether we have a sample size large enough to derive estimates at the geographical level that we may want to work with. When this is not the case, methods for small area estimation are required (for details and criminological examples see [138]). There has also been recent work mapping perception of place and fear of crime at micro-levels (see [139] and [140]).
Subsubsection 3.1.2.2 Prevalence vs. incidence
There are many concepts in science that acquire multiple and confusing meanings. Two you surely will come across when thinking about rates are: incidence and prevalence. These are well defined in epidemiology. Criminology and epidemiology often use similar tools and concepts, but not in this case. In criminological applications, these terms often are understood differently and in, at least, two possible ways. Confused? You should be!
In public health and epidemiology, prevalence refers to proportion of persons who have a condition at or during a particular time period; whereas incidence refers to the proportion or rate of persons who develop a condition during a particular time period. The numerator for prevalence is all cases during a given time period that have the condition; whereas the numerator for incidence is all new cases. What changes is the numerator and the key dimension in which it changes is time.
In criminology, on the other hand, you will find at least two ways of defining these terms. Those that focus on studying developmental criminology define prevalence as the percentage of a population that engages in crime during a specified period (number of offenders per population in a given time); while offending incidence refers to the frequency of offending among those criminally active during that period (number of offences per active offenders in a given time). For these criminologists the (total) crime rate in a population is the product of the prevalence and the incidence ([141]).
Confusingly, though, you will find authors and practitioners that consider incidence as equivalent to the (total) crime rate, as the number of crimes per population, and prevalence, as the number of victims per population. To make things more confusing, sometimes you see criminologists defining incidence as the number of crimes during a time period (say, a year) and prevalence as the number of victims during the lifetime. To avoid this confusion when producing maps, is probably best to avoid these terms and simply refer to crime rate or victimisation rate and be very clear in your legends about the time period covered for both.
Subsection 3.1.3 Proportional and graduate symbol maps: mapping crime counts
Proportional symbol maps, on the other hand, are used to represent quantitative variables for either areas or point locations. Each area gets a symbol and the size represents the intensity of the variable. It is expected with this type of map that the reader could estimate the different quantities mapped out and to be able to detect patterns across the map ([127]). The symbols typically will be squares or circles that are scaled "in proportion to the square root of each data value so that symbol areas visually represent the data values" ([128]: S29). Circles are generally used since they are considered to perform better in facilitating visual interpretation.
We often use proportional symbol maps to represent count data (e.g., number of crimes reported in a given area). A common problem with proportional symbol maps is symbol congestion/overlap, especially if there are large variations in the size of symbols or if numerous data locations are close together.
A similar type of map uses graduated symbol maps. As with choropleth classing, the symbol size may represent data ranges. They are sometimes used when data ranges are too great to practically represent the full range on a small map.
