Focal Location Quotients: Specification and Applications

Transcription

1 Geographical Analysis (2012), Focal Location Quotients: Specification and Applications Robert G. Cromley, Dean M. Hanink Department of Geography, University of Connecticut, Storrs, CT, USA Location quotients (LQs) are commonly used descriptive statistics in spatial analysis. They are not directly affected by neighbor relationships, and their calculation ignores spatial information. This article computes LQs as focal, rather than strictly local, functions by incorporating the spatial structure of the observations in the reference group in their computation. Focal LQs, tested for significance by event-based Monte Carlo simulations, are applied both to North Carolina sudden infant death syndrome data for comparison with the G i *spatial statistic and to sectoral employment data in that state in a more typical context for LQ analysis. The significance tests show that the focal LQ is far more sensitive to the size of the denominator for spatially intensive data than is the G i *statistic. Introduction While the location quotient (LQ) can be used as a foundation for global measures (Mulligan and Schmidt 2005), it is perhaps the most commonly used local descriptive statistic in spatial analysis. Its use was widely established some time ago (Isard 1960), and it was widely employed in spatial research well before recent advances in geographic information science (GIScience) induced an emphasis on analysis at the local scale (Fotheringham 1997). Its use has been marked especially in the geographical analysis of economic activity; typical examples include the impact of geographical clustering on foreign investment (Fernhaber, Gilbert, and McDougall 2008), the location of manufacturing sales offices across the U.S. city system (Holmes 2005), the calculation of economic base multipliers (Riddington, Gibson, and Anderson 2006), and concentrations of U.S. automobile production (Carroll, Reid, and Smith 2008). Among noneconomic applications, LQs have been used to assess population distributions geographically (Brown and Chung 2006) and the spatial arrangement of health outcomes (Dira, Schaarschmidt, and Fayissa 2010). The usefulness of LQs as local descriptors is enhanced by their standardized form, which facilitates comparison not only to the reference set of observations in an aggregate but also across the individual units of observation. The mappability of LQs also facilitates cross-unit comparison and regionalization based on perceived patterns in their spatial distribution. While the utility of LQs in spatial analysis has been widely demonstrated, two of their limitations render them somewhat incomplete measures for spatial analysis. One is that they are Correspondence: Robert G. Cromley, Department of Geography, University of Connecticut, Campus Unit 4148, Storrs, CT robert.cromley@uconn.edu Submitted: June 27, Revised version accepted: April 3, doi: / x 2012 The Ohio State University 1

2 Geographical Analysis rarely subected to statistical significance testing. For example, a typical LQ with reference to a given sector s employment at observation (or location) i is a ratio of ratios. The ratio for the local unit of observation can be written as e i/e i, where e i is employment in the given sector at location i, and E i is total employment at location i; the ratio for the aggregate reference economy can be written as e/e where e and E are total employment in the given sector and overall employment in the reference economy, respectively. Then, LQi = ( ei Ei) ( e E). (1) Equal ratios provide the reference value of LQ i = 1; if LQ i < 1, then the ith place of observation has a relatively low level of employment in the sector in question, and if LQ i > 1, then it has a relatively high level. While the magnitude of a place s departure from the implicit null expectation that LQ i = 1 is indicated, whether that departure is statistically significant or simply occurs by reasonable chance is also of interest in attempting to understand whether locational patterns are meaningful in a functional as well as visual way. Important work about significance testing has been done with respect to computing confidence intervals for LQs using the delta method (Moineddin, Beyene, and Boyle 2003; Beyene and Moineddin 2005). The confidence limits provide probabilities concerning an individual LQ s difference from one. More recent work has been done by Dira, Schaarschmidt, and Fayissa (2010), who accommodate the negative comparisons among LQs in a sample (due to offsetting values among observations with respect to one) in their inferential treatment. The great maority of LQ analyses, however, employ the statistic purely for descriptive purposes. The second limitation affecting LQs in spatial analysis is that they are calculated without regard to neighborhood or other distance-related interaction effects among the units of observation. Useful explicitly spatial alternatives to LQs have been developed in the family of local indicator of spatial association (LISA) statistics, including the G statistics developed by Getis and Ord (1992) and the local Moran s I statistic (Anselin 1995). LISA statistics, like LQs, are calculated for individual units of observation. However, unlike LQs, LISA statistics incorporate the spatial structure of the units of observation in their computation and typically are subected to significance testing as a component of their interpretation. Regionalization or spatial cluster identification based upon LISA statistics can be evaluated not only by a visual criterion, as with LQs, but also by an inferential criterion (Jacquez 2008). Carroll, Reid, and Smith (2008) argue that LQs and G statistics are useful as complements in their analysis of automotive clusters in the United States, with the latter statistic providing the spatial interaction measure that is especially useful in regionalization. The purpose of this article is to provide a description of a focal LQ (FLQ) a relative ratio measure that incorporates the spatial structure of the observations in the reference group directly into its computation and that can be subected to significance testing in its interpretation. Like the LQ and the G statistic, the FLQ is a measure of spatial concentration. Unlike the standard LQ, however, the FLQ incorporates spatial information in its calculation and is a geographically weighted statistic. Further, unlike the G statistic, the computation of the FLQ uses geographically weighted aggregation for spatially intensive data. Therefore, the FLQ extends the foundations for measuring spatial concentration across areal units of observation. 2

3 Robert G. Cromley and Dean M. Hanink Focal Location Quotients Methodological foundation An emphasis on developing local statistical functions has led to somewhat of a local versus global dichotomy in spatial analysis. Calculated local statistics vary by location, and the output value is a function of more than one input location. In contrast, a global statistic is the same for all locations, and the output value is a function of all input locations (Fotheringham, Brunsdon, and Charlton 2002). Global and local statistics are based on the notion of an entire map pattern versus local map patterns (Openshaw 1991). Under this rubric, the LQ should be considered a local statistic because it describes the relationship at a location (an areal unit) within the context of a larger domain and can be mapped. However, a wider range of spatial definitions than this local/global dichotomy is necessary to distinguish between using information only within the areal unit itself versus also using information from areal units within a neighborhood. In defining cell-based, spatial functions for GIS operations used in cartographic modeling in general (Tomlin 1991) and ARCGRID in particular (ESRI 1994), a richer classification is used that encompasses local, focal, zonal, and global functions. Local functions compute an output value for each location based on only the input value(s) at that location, whereas global functions compute the output value at each location based on values at all other locations. Although it could, a global function is not required to produce the same value for all locations, so it differs from the concept of a whole-map pattern. The LQ should be a local function under this definition. The concept of the ARCGRID local function, however, differs from the local statistics used as indicators of spatial association (Getis and Ord 1992; Anselin 1995; Ord and Getis 1995) or for exploratory data analysis (Brunsdon, Fotheringham, and Charlton 2002) because no value for another location is used in their computations. The focal and zonal functions best match the LISA definitions of localized statistics. Focal functions calculate an output value for a location based on values from other locations within a specified neighborhood, and zonal functions compute a location s output value based on other locations within the same zone (or region). The former allows for a more continuous surface of output values, whereas the latter produces the same output value for every location within the same zone, producing a discrete, stepped surface of output values. Focal functions are most similar in nature to geographically weighted statistics as defined by Brunsdon, Fotheringham, and Charlton (2002). These scholars consider the FocalMedian function used in cartographic modeling to be a localized statistic. We prefer to use the nomenclature of cartographic modeling that is, focal LQ to distinguish the FLQ from the traditional LQ that is a local statistic. Because the denominator in the traditional LQ is a scalar, the output value is dependent only on input values from the location under investigation, and hence the LQ itself is a local operator. Furthermore, the numerator is a ratio that also is a local operator. In addition to being a ratio of ratios, the LQ also can be written as LQi = ei [ Ei( e E)]. (2) Here, the numerator is the observed number of events, and the denominator is the expected number of events, for the ith observational unit. As a measure of concentration, values greater than one signify that more of an event is occurring, and for values less than one, less than expected is occurring. Geographically weighted statistics commonly are a linear combination of variate values: LS = w u, (3) i i 3

4 Geographical Analysis where w i is a standardized geographic weight, and u is the th value of the variate of interest. Weights may be based on some type of kernel density functions, such as a box kernel, a Gaussian kernel, or a bisquare kernel (see Fotheringham, Brunsdon, and Charlton 2002, pp , for specific definitions). When observations correspond to areal units, the variate values can be either spatially extensive (e.g., counts) or spatially intensive (e.g., rates, proportions, and ratios; Goodchild and Lam 1980). A geographically weighted statistic for count data uses the form given by equation (3). However, for spatially intensive data in which z = x /y, one can either geographically weight the z values, as in equation (3), or geographically weight the numerators (x ) separately from the denominators (y ) of the intensive values before their ratio is taken, such as. (4) LS wx wy i = ( i ) ( i ) The latter approach treats the statistic as a geographically weighted aggregation of the individual obects within each original areal unit that gives rise to the spatially intensive value rather than a direct measurement of the areal units themselves such as perimeter. Because the LQ is defined as a ratio of ratios, it is a ratio of two spatially intensive values, and this alternative weighting approach is used here to calculate FLQs as ( ) (5) FLQ w e w E e E i = ( i i ) The attributes of any areal aggregation unit are themselves based on geographically weighted summaries of individual data contained within an areal unit. The weight is equal to one for each individual located within the boundary of that areal unit and is equal to zero for each individual located outside the boundary of that areal unit. The focal approach extends the aggregation to locations outside an initial areal unit. The LQ is normally calculated for discrete areal units because most economic data collected by governments are reported only at higher levels of aggregation due to confidentiality restrictions. However, data for individual firm locations are available through private vendors, and no theoretical reason exists for precluding an FLQ at any location. This conceptualization mirrors Rushton and Lolonis (1996) conception of health rates as a continuous spatial distribution rather than as one based on administrative units, as well as the generation of regression parameters and summary statistics (Brunsdon, Fotheringham, and Charlton 2002) as a continuous surface. The FLQ is somewhat similar to the G i * statistic (Getis and Ord 1992) when using spatially intensive data but differs from it in certain important respects. The G i * statistic, unlike the FLQ, does not use geographically weighted aggregation and is defined as * = (6) G w z z i i where z is the variate value of interest. For spatially intensive data, i = ( i( ) ) ( ) G * w x y x y. (7) The G i * statistic is a ratio of the sum of weighted spatially intensive values to the sum of the respective unweighted values, which Ord and Getis (2001) note is simply a weighted moving average. A geographically weighted aggregation of spatially intensive values sums the weighted 4

5 Robert G. Cromley and Dean M. Hanink Focal Location Quotients numerators and sums the weighted denominators that form the spatially intensive values separately before taking the ratios, as follows: (( wx i wy x y ) ( i )) (( ) ( )), (8) which is the same as the FLQ. Because spatially intensive data for an areal unit are aggregated separately by numerator and by denominator before a ratio is calculated, aggregating geographically weighted numerators and denominators separately before ratios are calculated in computing statistics that encompass data beyond areal unit boundaries is more natural. This order of aggregation also will have an effect on significance testing. Although FLQs are easy to define as descriptive statistics, determining if any value is significantly different from one in order to identify meaningful hot (high FLQ) or cold (low FLQ) spots also is important. Recently, confidence intervals for traditional LQs have been established for this purpose (see Moineddin, Beyene, and Boyle 2003; Dira, Schaarschmidt, and Fayissa 2010). Significance tests for FLQs are produced empirically in this study using a technique developed by Rushton and Lolonis (1996) to determine the probability that an observed rate is significant. In the Rushton Lolonis method, a distribution of test statistics is created by 1,000 simulations in which each individual event is given a chance of acquiring the state under investigation. A random number is generated from a uniform distribution in the range 1 1,000. The rate observed for the whole region (the denominator of the FLQ), expressed as N per 1,000, is used to determine whether the state occurs for an individual event. The number of events is equal to the total number of employees in the case of sectoral employment or to the total number of births in the case of sudden infant death syndrome (SIDS). Each event has a geocoded areal address for aggregated data, such as county, or a geocoded (x,y) address for disaggregated data. If the random number is in the range 1 N, the individual event is given the simulated state; otherwise, the event does not have that state. This step is repeated for every individual event in the study region to produce a simulated pattern of FLQ. One thousand such simulations were executed, and 1,000 different FLQs for each location were rank ordered in a distribution. The observed FLQ for each location was compared with the simulated distribution of FLQs for that location to determine its significance level. Like G i *, the FLQ is scale invariant in the calculation of the statistic; however, its significance level changes as the size of the population and/or number of cases changes. Also, for spatially intensive data in which both the numerator and denominator are scaled by the same constant, the z variate values used to calculate the G i * statistic do not change even when the numerators and denominators change. Unlike the G i * statistic (whose significance levels are based on a classical distribution), the significance level of the FLQ based on simulations should be tighter for the data sets with much larger underlying populations (denominators) and/or cases (numerators). Empirical examples A shapefile for North Carolina counties was downloaded from the U.S. Bureau of the Census website for mapping and distance calculations. The 100 counties of North Carolina also are regionalized into the Mountains, the Piedmont, and the Coastal Plain regions of the state (Fig. 1). The original geographic coordinates were proected into North Carolina state plane coordinates using the WGS84 datum. For mapping purposes, the following statistical significance classes are used: less than 2.5%, 2.5% to 12.5%, 12.5% to 87.5%, 87.5% to 97.5, and greater than 97.5%. 5

6 Geographical Analysis Figure 1. North Carolina counties and regions. At the 95% significance level, those less than 2.5% are significantly below expectation (ª1.0), and those greater than 2.5% were significantly above expectation. The Cressie and Chan (1989) SIDS data set for North Carolina and North Carolina employment data for December 2007 constitute the illustrative data sets. All data are aggregated to the county level for the 100 counties of North Carolina. The SIDS rates for the period are examined first. Overall, 836 SIDS cases occurred for 422,392 births, a rate of 2 per 1,000. A single Monte Carlo simulation requires 422,392 events one for every birth. For each birth, a random number is generated in the range 1 1,000, and the birth is given the simulated state (a SIDS death occurs) whenever the number is less than or equal to two; otherwise, the birth does not have that state. Because births are spatially post-stratified by county, the simulated deaths among these births also are post-stratified by county. Each simulation generates a distribution of SIDS cases by county that can be used to calculate an LQ and a FLQ statistic for each county. A distribution of LQs and FLQs for each county can be created by executing 1,000 simulations. Fig. 2a presents the actual county-based LQs for SIDS, and Fig. 2b presents the associated significance level based on Monte Carlo simulation. Six counties, located mainly in the southern tier, are significantly higher, and 13 counties dispersed across the state are significantly lower than expected. The FLQs can be calculated using a box kernel with a 53-km (33-mile) bandwidth. This kernel is the same as originally used by Getis and Ord (1992) to compute their G i * statistic for the same data set. Distances used are calculated between county centroids. Fig. 3a presents the actual FLQs for SIDS, and Fig. 3b presents the associated significance levels based on Monte Carlo simulation. Now 12 counties have significantly high FLQs, and 11 counties, clustered mainly in a north-to-south band in the Piedmont region, have significantly low FLQs. The G i * map (Fig. 3c) shows a similar pattern of significance, although it contains fewer highly significant locations than the FLQs map. Because the significance tests for the FLQ are based on many more obects than for the G i * statistic (the number of events within the neighborhood of a given areal unit rather than ust the number of areal units within its neighborhood), deviations from the expected value are more likely to be significant. Next, the spatially continuous version of the FLQ (its value is not constant within an areal unit) is illustrated using the aggregated SIDS data because no spatial information about individual births and deaths is available. To compute the continuous FLQ surface, a Gaussian kernel with a 53-km (33-mile) bandwidth was applied to a regular square grid with 1-mile spacing between points (approximately 49,000 locations). Distances were calculated between each point 6

7 Robert G. Cromley and Dean M. Hanink Focal Location Quotients Figure 2. LQs and significance levels for North Carolina SIDS cases, (a) LQs for SIDS cases. (b) Significance level for LQs. and the county centroids. Fig. 4a presents a map of the continuous FLQ surface, and Fig. 4b portrays the associated significance levels (the significance was not adusted for multiple testing). The pattern of significance for the continuous surface conforms to that of the discrete rendition. LQs are most commonly used in analyzing economic data, and sectoral employment frequently is the variable of interest. The following analysis furnishes two such applications with employment data investigated for the finance and real estate sector and for the manufacturing sector (Bureau of Labor Statistics 2011). Only the discrete version of the FLQ is calculated and mapped because no individual firm data were available and the mechanics of the continuous version is outlined in the previous SIDS example. From a total employed labor force of 5,460,841, 209,117 individuals are employed in the finance and real estate sector, a rate of 38 per 1,000. A single Monte Carlo simulation in this instance has 5,460,841 events, and each event is assigned an employment case in the finance and real estate sector whenever the random number is less than or equal to 38. Financial and real estate services in North Carolina are highly concentrated in Mecklenburg County, the location of Charlotte, a maor regional banking and insurance center. That county is the only one in the state with an LQ greater than two (Fig. 5a). Besides Mecklenburg, other counties with LQs greater than one include those containing the cities of Winston-Salem, Greensboro, Durham, and Raleigh. The discrete FLQs are greater than one for counties in a band running through the Piedmont region from Winston-Salem and Greensboro south to Charlotte. The pattern of significance (Fig. 5b) exactly matches the pattern of FLQs: those counties with FLQs less than one are significantly below and those with FLQs 7

8 Geographical Analysis Figure 3. A comparison of the FLQ for SIDS cases at a 53-km (33-mile) distance and its significance level, and the significance level of the G i * Statistic at the same distance for North Carolina SIDS cases, (a) FLQs. (b) Significance level for FLQs. (c) Significance level for the G i * statistic. greater than one are significantly above the expected value. The pattern for the G i * statistic is somewhat different. No county is significantly below, and only Mecklenburg and four of its neighboring counties and the counties containing Winston-Salem and Greensboro are significantly above at the 5% level (Fig. 5c). From the total employed labor force, 559,913 individuals are employed in the manufacturing sector, a rate of 103 per 1,000. The Monte Carlo simulation for manufacturing uses the same number of events as the finance and real estate sector simulation, but a manufacturing case occurs 8

9 Robert G. Cromley and Dean M. Hanink Focal Location Quotients Figure 4. The pattern of the continuous FLQs and significance levels for SIDS cases at a 53-km (33-mile) bandwidth, for North Carolina SIDS cases, (a) Continuous FLQs. (b) Continuous FLQ significance levels. whenever the generated random number is less than or equal to 103. Manufacturing employment is more widely distributed in North Carolina than finance and real estate employment, with the lowest concentrations in the western mountain counties and the eastern Coastal Plain counties (Fig. 6a). The pattern of FLQs again effectively corresponds to their 5% significance level; with one exception, all counties with a FLQ value below one are significantly lower and all counties above one are significantly higher than expected (Fig. 6b). The G i * pattern is significantly higher in the Piedmont region (Fig. 6c), while some areas in the west and coastal tidewater region are significantly lower. To examine the impact of differences in the number of cases and population size, the number of SIDS cases and the number of live births for each county were first multiplied by a factor of 6 and then by a factor of 12; these multiplicative factors increase the population size without changing the SIDS mortality rate for the entire region. As the size proportionally increases, the number of extreme FLQs increases. For the original values, only 22 counties have extreme values. The number of extreme values increases to 58 with the sixfold increase in SIDS cases and live births, and to 71 with the 12-fold increase. All FLQs will be extreme (in either the upper or the lower tail of the simulated distribution) when the population reaches a certain size, unless a calculated value approximates the expected value of one. Finally, the overall SIDS mortality rate increases from 2 per 1,000 to 20 per 1,000 when the numerator increases 10-fold for every county; this change does not alter any FLQ values. With this latter 9

10 Geographical Analysis Figure 5. Comparison of the pattern of LQs and significance levels for the FLQ and G i * statistic for the financial and real estate sector employment in North Carolina, December (a) LQs. (b) Significance level for the FLQs. (c) Significance level for the G i * statistic. increase, the number of extreme FLQs increases from 22 to 65. Again, increasing the size of the number of cases in the simulated distribution increases the likelihood that any variation is statistically significant. Therefore, the number of extreme values is a function of the size of the population as well as of the rate of the event within the population for FLQs, which is not the case for the G i * statistic. Summary and conclusions The LQ is a descriptive statistic of concentration widely used in economic geography but easily applied to a variety of spatial data. Its utility as a concentration measure can be further enhanced 10

11 Robert G. Cromley and Dean M. Hanink Focal Location Quotients Figure 6. Comparison of the pattern of LQs and significance levels for the FLQs and G i * statistic for manufacturing sector employment in North Carolina, December (a) LQs. (b) Significance level for the FLQs. (c) Significance level for the G i * statistic. if its geographic domain is made variable. The concept of FLQs developed here is an attempt to implement a more spatially versatile measure that can detect levels of concentration in event outcomes across a range of geographic scales. The FLQ is an alternative to other local concentration measures, such as those derived from Moran s I and the G statistics, when using spatially intensive variate values associated with aggregated areal data. In this situation, the FLQ is shown to be a geographically weighted aggregation counterpart to the G i * statistic. In addition to calculating FLQs for discrete areal units, this article shows how to generate a continuous surface of FLQs based on individual point data. The purpose of the continuous 11

12 Geographical Analysis measure is to reflect a distribution unencumbered by an arbitrary partitioning of space. In the absence of individual data, spatially aggregated data could be used, which is illustrated with Cressie and Chan s North Carolina SIDS example. The discrete and continuous versions of the FLQ can be used by health care policy makers to examine disparities in health status over a range of bandwidths to determine the scale at which the disparities occur. Significance tests were developed using a Monte Carlo simulation of alternative data distributions. In empirical tests involving SIDS data and economic employment sector data for the state of North Carolina, extreme values are shown to be a function of the overall size of the population and of the rate of the event within the entire population. When using spatially intensive data, an analyst should be aware of the overall population size and may want to investigate spatial clusters using FLQs as an alternative to existing measures. Spatially intensive variate values for areal obect data should be handled differently than variate values for field data, and the aggregation aspect of the former data should be recognized. Spatially intensive variate values based on larger denominators are more significant than ones based on smaller denominators; this size effect is lost when the data are viewed as directly measured variate values. FLQs should be found especially useful in providing an explicitly spatial component to those types of analyses that commonly use traditional LQs as building blocks for regionalization. In addition, FLQs also can eliminate the necessity of using traditional LQs in tandem with LISA statistics in assessing univariate spatial distributions. References Anselin, L. (1995). Local Indicators of Spatial Association-LISA. Geographical Analysis 27(2), Beyene, J., and R. Moineddin. (2005). Methods for Confidence Interval Estimation of a Ratio Parameter with Application to Location Quotients. BMC Medical Research Methodology 5, 32. Brown, L., and S.-Y. Chung. (2006). Spatial Segregation, Segregation Indices and the Geographical Perspective. Population, Space and Place 12(2), Brunsdon, C., A. S. Fotheringham, and M. Charlton. (2002). Geographically Weighted Summary Statistics A Framework for Localised Exploratory Data Analysis. Computers, Environment and Urban Systems 26(6), Bureau of Labor Statistics. (2011). Local Area Unemployment Statistics. Available at (accessed on 15 January 2011). Carroll, M., N. Reid, and B. Smith. (2008). Location Quotients Versus Spatial Autocorrelation in Identifying Potential Cluster Regions. Annals of Regional Science 42(2), Cressie, N., and N. Chan. (1989). Spatial Modelling of Regional Variables. Journal of the American Statistical Association 84(406), Dira, G., F. Schaarschmidt, and B. Fayissa. (2010). Inferences for Selected Location Quotients with Applications to Health Outcomes. Geographical Analysis 42(3), ESRI. (1994). Cell-Based Modeling with GRID. Redlands, CA: Environmental Systems Research Institute. Fernhaber, S., B. Gilbert, and P. McDougall. (2008). International Entrepreneurship and Geographic Location: An Empirical Examination of New Venture Internationalization. Journal of International Business Studies 39(2), Fotheringham, A. S. (1997). Trends in Quantitative Methods I: Stressing the Local. Progress in Human Geography 21(1), Fotheringham, A. S., C. Brunsdon, and M. Charlton. (2002). Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. West Sussex, U.K.: Wiley. Getis, A., and J. Ord. (1992). The Analysis of Spatial Association by Use of Distance Statistics. Geographical Analysis 24(3),

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