Quantitative Methods and Applications in GIS - Chapter 8

Part III Advanced Quantitative Methods and Applications 8 Geographic Approaches to Analysis of Rare Events in Small Population and Application in Examining Homicide Patterns When rates are used as estimates for an underlying risk of a rare event (e.g., cancer, AIDS, homicide), those with a small base population have high variance and are thus less reliable. The spatial smoothing techniques, such as the floating catchment area method and the empirical Bayesian smoothing method, as discussed in Chapter 2, can be used to mitigate the problem. This chapter begins with a survey of various approaches to the problem of analyzing rare events in a small population in Section 8.1. Two geographic approaches, namely, the ISD method and the spatial-order method, are fairly easy to implement and are introduced in Section 8.2. The spatial clustering method based on the scale-space theory requires some program-ming and is discussed in Section 8.3. In Section 8.4, the case study of analyzing homicide patterns in Chicago is presented to illustrate the scale-space melting method implemented in Visual Basic. The section also provides a brief review of the substantive issues: job access and crime patterns. The chapter is concluded in Section 8.5 with a brief summary. 8.1 THE ISSUE OF ANALYZING RARE EVENTS IN A SMALL POPULATION Researchers in criminology and health studies and others are often confronted with the task of analyzing rare events in a small population and have long sought solutions to the problem. For criminologists, the study of homicide rates across geographic units and for demographically specific groups often entails analysis of aggregate homicide rates in small populations. Several nongeographic strategies have been attempted by criminologists to mitigate the problem. For example, Morenoff and Sampson (1997) used homicide counts instead of per capita rates or simply deleted outliers or unreliable estimates in areas with a small population. Some used larger units of analysis (e.g., states, metropolitan areas, or large cities) or aggregated over more years to generate stable homicide rates. Land et al. (1996) and Osgood (2000) used 149 150 Quantitative Methods and Applications in GIS Poisson-based regressions to better capture the nonnormal error distribution pattern in regression analysis of homicide rates in small populations (see Appendix 8).1 On the other side, many researchers in health-related fields are well trained in geography and have used several spatial analytical or geographic methods to address the issue. Geographic approaches aim at constructing larger geographic areas, based on which more stable rate estimates may be obtained. The purpose of constructing larger geographic areas is similar to that of aggregating over a longer period of time: to achieve a greater degree of stability in homicide rates across areas. The technique has much common ground with the long tradition of regional classification (regionalization) in geography (Cliff et al., 1975). For instance, Black et al. (1996) developed the ISD method (after the Information and Statistics Division of the Health Service in Scotland, where it was devised) to group a large number of census enumeration districts (EDs) in the U.K. into larger analysis units of approximately equal population size. Lam and Liu (1996) used the spatial-order method to generate a national rural sampling frame for HIV/AIDS research, in which some rural counties with insufficient HIV cases were merged to form larger sample areas. Both approaches emphasize spatial proximity, but neither considers within-area homo-geneity of attribute. Haining et al. (1994) attempted to consolidate many EDs in the Sheffield Health Authority Metropolitan District in the U.K. to a manageable number of regions for health service delivery (hereafter referred to as the Sheffield method). The Sheffield method started by merging adjacent EDs sharing similar deprivation index scores (i.e., complying with within-area attribute homogeneity), and then used several subjective rules and local knowledge to adjust the regions for spatial com-pactness (i.e., accounting for spatial proximity). The method attempted to balance two criteria (attribute homogeneity and spatial proximity), a major challenge in regionalization analysis. In other words, only contiguous EDs can be clustered together, and these EDs must have similar attributes. The ISD method and the spatial-order method will be discussed in Section 8.2 in detail. The Sheffield method relies on subjective criteria and involves a substantial amount of manual work that requires one’s knowledge of the study area. Section 8.3 will introduce a new spatial clustering method based on the scale-space theory. The method melts adjacent polygons of similar attributes into clusters like the Sheffield method, but is an automated process based on objective criteria. Construct-ing geographic areas enables the analysis to be conducted at multiple geographic levels, and thus permits the test of the modifiable areal unit problem (MAUP). Table 8.1 summarizes all approaches to the problem of analysis of rates of rare events in a small population. 8.2 THE ISD AND THE SPATIAL-ORDER METHODS The ISD method is illustrated in Figure 8.1 (based on Black et al., 1996, with modifications). A starting polygon (e.g., the southernmost one) is selected first, and its nearest and contiguous polygon is added. If the total population is equal to or more than the threshold population, the two polygons form an analysis area. Otherwise, the next nearest polygon (contiguous to either of the previous selected polygons) is added. The process continues until the total population of selected Geographic Approaches to Analysis of Rare Events and Homicide Patterns 151 TABLE 8.1 Approaches to Analysis of Rates of Rare Events in a Small Population Approach Examples Comments 1 Use homicide counts instead of per capita rates 2 Delete samples of small populations 3 Aggregate over more years or to a high geographic level 4 Poisson-based regressions 5 Construct geographic areas with large enough populations Morenoff and Sampson (1997) Harrell and Gouvis (1994); Morenoff and Sampson (1997) Messner et al. (1999); most studies surveyed by Land et al. (1990) Osgood (2000); Osgood and Chambers (2000) Haining et al. (1994); Black et al. (1996); Sampson et al. (1997) Not applicable for most studies that are interested in the offense or victimization rate relative to population size Deleted observations may contain valuable information Impossible to analyze variations within the time period or within the large areal unit Effective remedy for OLS regressions; not applicable to nonregression studies Generate reliable rates for statistical reports, mapping, regression analysis, and others Select starting tract from pool of unallocated tracts Add to analysis areas; remove from pool Is population of analysis No area ³ threshold Yes Select the tract contiguous & nearest The analysis area completed Are all tracts allocated? No Yes Stop FIGURE 8.1 The ISD method. polygons reaches the threshold value and a new analysis area is formed. The whole procedure is repeated until all polygons are allocated to new analysis areas. One may use ArcGIS to generate a matrix of distances between polygons and another matrix of polygon adjacency, and then write a simple computer program to imple-ment the method outside of GIS (e.g., Wang and O’Brien, 2005). The method is primitive and does not account for spatial compactness. Some analysis areas 152 Quantitative Methods and Applications in GIS 2 1 0.202 0.157 6 0.080 3 0.361 4 0.582 5 0.371 7 0.688 8 9 0.687 0.880 10 0.656 6 1 2 3, 5 Node ID 4 10, 8, 7 9 0.0 0.5 1.0 Spatial order value FIGURE 8.2 An example for assigning spatial-order values to polygons. generated by the method may exhibit odd shapes, and some (particularly those near the boundaries) may require manual adjustment. The spatial-order method follows a rationale similar to that of the ISD method. It uses space-filling curves to determine the nearness or spatial order of polygons. Space-filling curves traverse space in a continuous and recursive manner to visit all polygons, and assign a spatial order (from 0 to 1) to each polygon based on its relative positions in a two-dimensional space. The procedure, currently available in ArcInfo Workstation, is SPATIALORDER, based on one of the algorithms developed by Bartholdi and Platzman (1988). In general, polygons that are close together have similar spatial-order values and polygons that are far apart have dissimilar spatial-order values. See Figure 8.2 for an example. The method provides a first-cut measure of closeness. The SPATIALORDER command is available in the ArcPlot module through the ArcInfo Workstation command interface. Once the spatial-order value of each polygon is determined, the COLLOCATE command in ArcInfo follows by assigning nearby polygons one group number and accounting for the capacity of each group formed by polygons. Finally, polygons are dissolved based on the group numbers. 8.3 THE SCALE-SPACE CLUSTERING METHOD The ISD and the spatial-order method only consider spatial proximity, but not within-area attribute homogeneity. The spatial clustering method based on the scale-space theory accounts for both criteria. Development of the scale-space theory has bene-fited from the advancement of computer image processing technologies, and most of its applications are in analysis of remote sensing data. Here we use the method for addressing the issue of analyzing rare events in small populations. ... - tailieumienphi.vn