Segregation measures based upon data gathered at different geographical levels cannot provide consistent results because of the scale effect under the Modifiable Areal Unit Problem (MAUP) umbrella. This paper proposes a framework, which decomposes segregation attributable to different geographical levels, to conceptually link segregation values obtained from multiple geographical levels together such that differences in segregation values among levels' are accounted for. Using two different indices, the dissimilarity index D and the diversity index H, this paper illustrates the decomposition methods specific to these indices. When these indices are decomposed, local measures of segregation pertaining to multiple geographical levels are computed. These local segregation measures indicate the levels of segregation contributed by the local units and the regional units to the entire study area.
The spatial scale effect, or the large umbrella of the modifiable areal unit problem (MAUP), has been a persistent issue in spatial analysis and geographical research (Tate and Atkinson 2001). The inconsistency of analytical results derived from data gathered at different scale levels and/or gathered from different spatial partitioning systems are found in almost all types of analysis involving spatial data. Measuring levels of segregation is no exception. In general, using data of higher spatial resolution or smaller enumeration units will yield a higher level of segregation reflected by measures such as the dissimilarity index D (Wong 1997).
Different approaches have been suggested to deal with the scale effect, including a call for multiscale analyses to obtain a more comprehensive understanding of the geographical issues involved (Fotheringham 1989). Using Geographic Information Systems (GIS), spatial analysis can be performed repeatedly with multiple scale data to assess the scale effect. Another appealing approach is to develop scale-insensitive spatial analytical techniques such that analyses conducted at different scale levels are relatively consistent. Unfortunately, few successes have emerged so far (e.g., Tobler 1991; Wong 2001).
This paper takes the approach between merely recognizing different results due to the scale effect and the quest for scale insensitive techniques to address measurement issues in segregation. Because segregation values based upon data gathered from different scales are different, the purpose of this paper is to develop a framework to relate the segregation values obtained from different scales such that these values can provide a consistent but more comprehensive depiction of the phenomenon or situation. In general, the levels of segregation found at the lower or local level (using smaller enumeration units) are higher than that at the higher or regional level (larger enumeration units). Similar to the approach used by Moellering and Tobler (1972) to decompose the total variance into different components attributable to various geographical levels, the approach adopted here decomposes segregation values obtained from the dissimilarity index at the local level (with higher observed segregation) to the pure local segregation and regional segregation, and decomposes the segregation values obtained from the diversity index at the regional level (with lower observed segregation) to the local segregation and the pure regional segregation. The purpose of this decomposition exercise is to account for the sources of segregation at different scale levels so that segregation values from different scale levels can be linked conceptually. This approach can provide a consistent accounting of segregation across multiple scales as long as the areal units at different scale levels are nested in a hierarchical manner. In addition, local measures can be derived in the process to reflect how much segregation from local units at a given geographical level is attributable to the overall segregation level.
The next section will provide a brief overview of the scale effect issue with reference to measuring segregation. Section 2 will introduce the framework to decompose the dissimilarity index D and the diversity index H. A simulation experiment and two empirical studies follow.
1. SCALE EFFECT AND MEASURING SEGREGATION
The scale effect is one of the two sub-problems (the other is the zoning effect) under the MAUP, which has been documented extensively. Because several comprehensive overview articles of the MAUP already exist (e.g., Fotheringham and Wong 1991; Wong and Amrheim 1996; Sui 2000), this paper will not duplicate these efforts. Nevertheless, it is important to point out that the impact of the MAUP is widespread and significant partly because the correlation of variables, which is an important component in statistical analysis, will change when data gathered at different scale levels are used (Openshaw and Taylor 1979). In general, data are spatially "smoothed" when they are aggregated to adjacent values, and thus less variation is preserved at the aggregated level (Fotheringham and Wong 1991). But if data have a strong positive spatial autocorrelation, the aggregation process will not remove much information as compared to negatively spatially autoeorrelated data. Amrhein and Reynolds (1997) and Reynolds and Amrhein (1997) evaluated the relationships between the changes in spatial autocorrelation and the changes in variance through the aggregation process, and provided a detailed explanation of the smoothing process.
Several research efforts have tried to correct or adjust for correlation among variables (e.g., Holt, Steel, and Tranmer 1996) in order to obtain relatively consistent statistical results across scale levels, but most of these procedures are either too computationally intensive or impractical. Focusing on the regression framework, Fotheringham, Brunsdon, and Charlton (2001) speculated that the spatially weighted regression could be a potential solution to the scale effect, yet it has to be demonstrated. Using a different approach to tackle the problem, King (1995) suggested an error-bound method to handle the MAUP. Some geographers are skeptical if this approach can provide a reasonable solution to the MAUP (e.g., Anselin 2000), but some see promise (Withers 2001). Besides these rather general "solutions" to the MAUP, more application-specific solutions have been proposed. Tobler (1989) suggested a "frame-independent" model specifically for modeling interregional migration. In analyzing count variables, Wong (2001) proposed a spatial correlation approach, which can yield results relatively consistent across scale levels.
In the area of segregation analysis, several papers have documented the scale effect on measuring segregation level. Wong (1997) provided a conceptual discussion on how segregation levels, as reflected by the index of dissimilarity D, changed with scale. Because the D index is purely a function of the homogeneity within an areal unit, therefore, in general, the smaller the areal unit, the more homogeneous the population mix and …