Geographical Analysis ISSN 0016-7363
A Supplemental Indicator of High-Value or Low-Value Spatial Clustering Tonglin Zhang,1 Ge Lin2 1
Department of Statistics, Purdue University, West Lafayette, IN, 2Department of Geology and Geography, West Virginia University, Morgantown, WV
Most test statistics for detecting spatial clustering cannot distinguish between low-value spatial clustering and high-value spatial clustering, and none is designed to explicitly detect high-value clustering, low-value clustering, or both. To fill this void in practice, we introduce an adjustment procedure that can supplement common twosided spatial clustering tests so that a one-sided conclusion can be reached. The procedure is applied to Moran’s I and Tango’s CG in both simulated and real-world spatial patterns. The results show that the adjustment procedure can account for the influence of low-value clusters on high-value clustering and vice versa. The procedure has little effect on the original global testing methods when there is no clustering. When there is a clustering tendency, the procedure can unambiguously distinguish the existence of high-value clusters or low-value clusters or both.
Introduction Spatial clusters, according to Marshall (1991), are foci of particularly high or low incidence rates that are unlikely to happen by chance. Explicitly detecting highvalue or low-value clustering has both statistical and practical utility. However, the detection of spatial clustering is typically approached as a hypothesis-testing problem that is analogous to positive spatial autocorrelation. When the alternative of clustering is tested against the null hypothesis, a spatially constant mean is always assumed. The rejection of the null hypothesis only suggests the existence of an overall tendency of either high-value or low-value clustering or both. If this tendency is toward low-value clustering, when the interest is in high-value clustering, the two-sided testing result is not unambiguous. Consequently, information is lost in the spatial autocorrelation type of hypothesis testing. Such information loss can also lead to a potentially higher false alarm rate when one-sided clustering was inferred from a two-sided test that includes low-value clustering. Correspondence: Ge Lin, Department of Geology and Geography, West Virginia University, P.O. Box 6300-406, White Hall, Morgantown, WV 26506-6800 e-mail:
[email protected]
Submitted: November 23, 2004. Revised version accepted: September 16, 2005. Geographical Analysis 38 (2006) 209–225 r 2006 The Ohio State University
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The problems of detecting high-value or low-value clusters separately have been recognized in the literature and in practice. Although not specifically commented on, the scan statistic of Kulldorff (1997) is able, in practice, to distinguish whether a detected cluster is higher or lower than the expected value. However, because the test is based on a Monte Carlo-simulated P value from the observed data, the presence of a low-value cluster may cause a greater difference of relative risks between those in a high-value clustered region and those in the reference surface, so the method is sensitive to the number of high-value and lowvalue clusters. Ord and Getis (2001) propose a solution to the detection problem by treating a high-value cluster separately from the rest of the reference surface, but the problem remains if a low-value cluster contributes too much to the remainder of the reference surface. Lin (2003) proposes a spatial logit model that accounts for both high-value and low-value clusters in a stepwise likelihood-ratio test. Because the test is model-based, it is sensitive to both sample size and estimated parameters. More recently, Lin and Zhang (2004) introduced a method for testing low-value clustering for rare diseases based on the general test of Tango’s CG (Tango 1995), but they did not report any statistical properties of their method. Because of its purpose, this low-value test method cannot be applied consistently to test for both high-value clustering and low-value clustering. In this article, we extend their research in three ways. First, we provide two consistent test indicators for an existing one-side test for spatial clustering, so that it can detect high-value or low-value clustering or both. Second, we provide statistical properties for the proposed procedure. Finally, we apply the procedure to both Poisson and continuous variables. In the next section, we introduce the procedure in the context of Poisson count data. We then implement the procedure for the general tests for Tango’s CG and Moran’s I by using Monte Carlo simulation in the third section and Minnesota leukemia data in the fourth section. We make some concluding remarks in the final section.
The adjustment procedure Following Marshall’s definition, geographical clusters can be viewed as abnormally high values or abnormally low values clustered somewhere in the study area. If there is no clustering of abnormally low values, then a general clustering test, such as Moran’s I or Tango’s CG, is able to conclude the existence of high-value clustering when it rejects the null hypothesis of spatially constant mean. In the real world, however, high-value and low-value clusters can coexist in any single study area. If the data are not location-specific, abnormally high or low values are part of the normal distribution, in which clustering cannot be defined. Only when data are location-specific do spatially clustered values become abnormal in the spatial sense. If high-value and low-value clusters are unbalanced in terms of mean, the relative risk of the reference surface can be either upward or downward from the null hypothesis. To turn a two-sided test result into a one-sided conclusion, it is 210
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necessary to reduce any potentially unwanted influences caused by either highvalue or low-value clusters. Extending the approach of Ord and Getis (2001), who contrast regions within a potential cluster with the remainder regions in the study area, we suggest the following scheme of spatial clustering structures: (i) (ii) (iii) (iv)
There is no clustering, which is equivalent to a spatially constant mean or risk. There is at least a low-value cluster, but no high-value cluster. There is at least a high-value cluster, but no low-value cluster. Both high-value and low-value clusters exist.
Many cluster testing methods, such as Moran’s I and Tango’s CG, would simply group cases (ii)–(iv) into a single test statistic against (i). Case (ii), however, implies the presence of low-value clustering, and when a two-sided clustering test, such as Tango’s CG or Rogerson’s R (Rogerson 1999) is significant, it could be misinterpreted as having high-value clusters, thus causing a false alarm. The partition method of Ord and Getis could be feasible for (iii) but not for (iv). Lin’s logit model likely works for any of the four structures, provided that the shifting up and down of the grand mean in the logit model depends on the number of high-value or lowvalue clusters. Our adjustment procedure is designed to overcome these information deficiencies in cluster detection. Although we demonstrate the method for Poisson data, it can be easily extended to Gaussian data. Let us consider a study area that has m regions indexed by i. The total popP Pm ulation size is x ¼ m i¼1 xi and the total disease count is N ¼ i¼1 Ni , wherein xi and Ni are the i-th region’s population size and disease occurrence, respectively. Let yi, usually unknown in practice, be the relative risk for region i and assume that Ni are independent Poisson distributions with observations ni (Rogerson 1999; Diggle 2000). That is Ni Poission(yilxi), wherein the regular disease rate l is a constant. Based on these notations, when regions with a yi value significantly less than 1 are close to each other, they form a low-value cluster; when regions with a yi value significantly greater than 1 are close to each other, they form a high-value cluster. If we can make sure in a two-sided test that there is no low-value cluster, then the significant test result for clustering would indicate high-value clustering. Having a method to deal with the potential existence of low-value clusters, or vice versa, in a two-sided test is the key to the one-sided indictor. Because of symmetry, it is sufficient to introduce our adjustment procedure to test for the existence of high-value clustering. We call this procedure the adjustment for a high-value clustering test (AHVCT) to contrast with the adjustment for a low-value clustering test (ALVCT). We introduce the procedure by assuming that the l value is known. When the l value is unknown, an estimated one can be used instead. As mentioned above, the null hypothesis of most test statistics for spatial clustering ignore the influence of low-value clusters on the testing procedure for high-value clusters, or vice versa. Following Marshall’s definition, we treat both 211
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low-value and high-value clusters as unusual. Hence, we have to be able to reduce the potential influence of low-value and high-value spatial clusters if we want to use a two-sided test to draw a one-sided conclusion (Gangnon and Clayton 2000). In the real world, a region with a relative risk (yi) significantly less than 1 could either be part of a low-value cluster or be spatially random. Even though the objective is to reduce the influence of a low-value cluster, without prior knowledge we cannot directly determine whether a region belongs to a low-value cluster or not. If we could determine that a region belongs to such a cluster, we would then be able to determine if a region belongs to a high-value cluster and it would not be necessary to use the procedure. One characteristic of a low-value cluster is that memberships or regions within the cluster all tend to have fairly low values. So when the l value is known, we can indirectly reduce the influence of low-value clustering by replacing observed low values with randomly generated low values based on a set of threshold values, such that the clustering effect is randomized. Theoretically, if there is no low-value cluster, this random replacement is equivalent to a random permutation of low values because of the known l (Mihram 1972). If there are lowvalue clusters, their clustered effects are destroyed by randomly regenerated low values that reflect the real distribution according to the known l. In the AHVCT, we choose lxi to be the threshold for region i, because E(Ni) 5 lxi if yi 5 1. A Poisson random variable Ni0 with parameter lxi is repeatedly generated until Ni0 < lxi ; it then becomes ni0 , which is to be used to replace ni if niolxi. Clearly, Ni0 can only take values at 0, 1, . . ., blxi c, where function bxc equals the greatest integer less than x. Let fm and Fm be the probability mass function (PMF) and cumulative distribution function (CDF) of the Poisson distribution with mean m for a given m, the PMF of Ni is expressed as P ðNi0 ¼ ni0 Þ ¼
flxi ðni0 Þ : Flxi ðblxi cÞ
ð1Þ
Equation (1) is the PMF of the truncated Poisson distribution. The equation suggests that the truncated lower part of the original Poisson distribution will change little when there is no low-value cluster; when there is a low-value cluster, the PMF of Ni0 is upwardly adjusted, so that its influence on the test of low-value clustering will be reduced. Consequently, to reduce the influence of potential lowvalue clusters, we can utilize the AHVCT as follows (the steps for the ALVCT can be symmetrically derived by changing the conditional statements): 1. 2. 3.
Based on the observed count ni at region i for i 5 1, . . ., m, if ni blxi c, then generate a count ni0 by a Poisson random variable with mean lxi. If ni0 > blxi c, then regenerate the count ni0 until ni0 blxi c. Replace ni by ni0 for region i and perform steps 1 and 2 for a new region until all regions in the area have been checked.
To illustrate the above procedure graphically, we use Poisson distributions with lxi 5 10 for different yi values (Fig. 1). The threshold value for replacements is 212
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θ = 0.5 θ = 1.0 Threshold line
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marked at 10 in each graph, and the counts on the left side of the threshold will be replaced by a generated count in AHVCT, and the counts on the right side will be retained. When a yi value is small (solid line on the left plot), the cumulative probability on the left-hand side of the threshold is very large, and the probability for value replacement is very high. In particular, when yi values less than the threshold are spatially clustered, their effects will be destroyed by random replacements in AHVCT. When yi is large (solid line on the right plot), however, the cumulative probability on the left-hand side of the threshold is very small, and the probability for a large yi value to be replaced is very low. Finally, when yi is close to 1, this procedure will not have much effect. To evaluate the above procedure analytically, the newly generated random variable in AHVCT can be denoted by NiH for region i, where the superscript indicates the type of test for adjustment, and in this case, H corresponds to AHVCT. In AHVCT, NiH is the observed count Ni0 if Ni is greater than or equal to the threshold value but it becomes the generated count Ni0 if Ni is less than the threshold value. In other words, NiH is a mixture of two independent truncated Poisson distributions. One is the truncated Poisson distribution with parameter lxi and truncation less than or equal to blxi c. The other is the truncated Poisson distribution with parameter yilxi and truncation greater than blxi c. The random variable NiH can then be expressed as NiH
¼
Ni0 ; Ni ;
when Ni blxi c when Ni > blxi c
ð2Þ
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wherein the PMF of Ni0 is given in (1). As one can see from (2), the PMF of NiH is identical to the PMF of Ni when yi 5 1. When yi is much larger than 1, the lower term on the right side of the equation becomes dominant. Accordingly, the probability that Ni will be replaced in AHVCT is expressed as q H ðyi ; l; xÞ ¼ P ðNi blxi cÞ ¼ Fyi lxi ðblxi cÞ ¼
blx Xi c
fyi lxi ðkÞ
k¼0
¼
blx Xi c
ðyi lxi Þk yi lxi e k! k¼0
and the PMF of NiH is p H ðk; yi ; l; xi Þ ¼ P ðNiH ¼ kÞ ¼
ð3Þ
8 < fyi lxi ðkÞ;
when k > blxi c
F ðblxi cÞ : Fyi lxiðblx cÞ flxi ðkÞ; lxi i
when k blxi c
ð4Þ
After the AHVCT is implemented, the new relative risk at region i becomes yH i ðyi ; l; xi Þ ¼
1 EðNiH Þ 1 X ¼ kp H ðk; yi ; l; xi Þ lxi lxi k¼0
ð5Þ
Because NiH in Equation (5) includes randomly generated values when Ni blxi c, AHVCT tends to upwardly adjust the relative risks for regions that have substantially low relative risks while retaining the relative risks for regions whose relative risks are 1. When a set of randomly low values are randomly replaced, the overall effect after the adjustment is nearly the same as the preadjustment one. When a set of values within a low-value cluster is replaced by the randomly low values, the cluster is randomized and its original effect is reduced. There will be no low-value clusters after this procedure. If a two-sided test is significant on the basis of adjusted data, it can unambiguously conclude the existence of high-value clustering. On the other hand, when testing for low-value clustering, the corresponding ALVCT should be used to reduce the potential influence of a high-value cluster. Symmetric to Equation (2), the random variable obtained at region i in ALVCT, denoted by NiL , can be expressed as 00 Ni ; when Ni > ½lxi NiL ¼ ð6Þ Ni ; when Ni ½lxi wherein [x] equals a greater integer that is not greater than x. The PMF of Ni00 is P ðNi00 ¼ ni00 Þ ¼
flxi ðni00 Þ 1 Flxi ð½lxi Þ
ð7Þ
for integers ni00 > ½lxi . If yi is much less than 1, the lower term of Equation (6) becomes dominant. 214
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Likewise, the probability that Ni will be replaced in ALVCT is 1 X
q L ðyi ; l; xi Þ ¼ P ðNi > ½lxi Þ ¼
fyi lxi ðkÞ ¼
k¼½lxi þ1
The PMF of NiL is p L ðk; yi ; l; xi Þ ¼ P ðNiL ¼ kÞ ¼
1 X
ðyi lxi Þk yi lxi e k! k¼½lx þ1
ð8Þ
i
8 < fyi lxi ðkÞ;
when k ½lxi
1F ð½lx Þ : 1Fyi lxið½lx Þi flxi ðkÞ; lxi i
when k > ½lxi
ð9Þ
and the new relative risk is yL ðyi ; l; xi Þ ¼
1 1 X kp L ðyi ; l; xi Þ: lxi k¼0
ð10Þ
In summary, AHVCT is more likely to replace a low value (count) than a high value. The adjusted relative risk is much larger than the observed one when the observed one is much less than 1; the adjusted risk is nearly identical to the observed one when the observed one is 1. The reverse is true for ALVCT. Both AHVCT and ALVCT are theoretically very effective. In the following, we evaluate the adjustments by using simulations and a case study. Simulation To evaluate the proposed procedures, we constructed a regular 20 20 lattice (m 5 400), in which each lattice point represents a region. Simulations were performed on six spatial patterns on the lattice (Fig. 2) in the context of count data: (i) one low-value and (ii) one high-value cluster centered at (5, 5), (iii) two low-value and (iv) two high-value clusters centered respectively at (5, 5) and (15, 15), (v) one low-value and one high-value clusters centered respectively at (5, 5) and (15, 15), and (vi) two low-value and two high-value clusters centered respectively at (5, 5), (15, 15), (5, 15), and (15, 5). In all the simulations, regional populations were independently generated from the closest integer of the G(100, 0.001) distribution. Consequently, the mean of a regional population was 100,000, and the standard error was 10,000. In addition, the l value was fixed at 10 4, so that the replacement procedures could be implemented according to the theoretical derivations. The relative risk was 1 in the entire study area, except in the regions that were covered by a circular cluster with the radius of 3. A high-value cluster was designed to have the greatest value at the center and to have its value gradually decline toward its border. A low-value cluster, in contrast, was designed to gradually reduce its value toward its center. Specifically, if a region at (x, y) belonged to a cluster centered at (x0, y0), the relative risk was expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðd=3Þ ðx x0 Þ2 þ ðy y0 Þ2 ; the ‘‘1’’ was selected if the cluster was designed to be a high-value cluster and the ‘‘ ’’ was selected if the cluster was designed to 215
Geographical Analysis ii: One High Value Cluster
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Figure 2. The spatial layouts of low-value clusters (L) and high-value clusters (H) used in the simulations.
be a low-value cluster. The quantity d signifies the magnitude of a cluster ranging from 0 to 1. When d 5 0, no cluster exists; when d 5 1, the value at the center of a cluster is twice as large as the relative risk for a high-value cluster. We ran 5000 simulations for each d value from 0 to 1 in 0.05 increments. We used Moran’s I and Tango’s CG tests in our simulations, because both are two-sided general tests. Tango’s CG is based on Poisson count data, while Moran’s I is based on continuous data. The latter may not be appropriate when the sample size within each region is too small for the normal approximation (Waldho¨r 1996). Because we used a fairly large sample in the simulations, the normal assumption for Moran’s I is likely to be held. Given m ( 5 400) regions, Tango’s general CG is defined as CG ¼ ðr pÞt W ðr pÞ
ð11Þ
P wherein r 5 (r1, . . ., rm) with ri ¼ ni = m i¼1 ni ; p ¼ ðp1 ; . . . ; pm Þ, with pi ¼ Pm xi = i¼1 xi , and W 5 (wij)m m with wij ¼ e dij =t , which is a decreasing function according to the Euclidean distance dij between regions i and j with t as a scale parameter. We fixed t at 1. Like the chi-squared statistic, Tango’s CG is always positive. A large CG value indicates the existence of either a high-value cluster or a low-value cluster, but it may also indicate nonclustered spatial variation (Lin 2004), 216
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because CG can be partitioned into spatial and aspatial components (Rogerson 1999). Moran’s I is defined as Pm P Þðxj x Þ i¼1 j6¼i wij ðxi x i I ¼ hP Pm ihPm P m 2 Þ =m i¼1 i¼1 ðxi x i¼1 j6¼i wij wherein xi 5 ni/xi is the disease rate in the i-th region and wij are elements of the weight matrix. The weight can be either a 0–1 adjacency indicator or a decreasing function of the distance between regions i and j. For the ease of comparison with Tango’s CG, we opted to use wij ¼ e dij =t as it is defined in Tango’s CG. Moran’s I measures the spatial dependence. A positive and significant I value indicates the existence of either high-value or low-value clustering. A negative and significant I value indicates a negative spatial autocorrelation or a tendency of high values to be juxtaposed with low values. If there is no spatial dependence, the value of I is close to 0 when m is large. Using the original results from the I and CG tests as a reference, we first evaluated the performance of the adjustment procedures for type I errors. In other words, we evaluated the rejection rate of the null hypothesis of no spatial clustering for I and CG in Fig. 3 according to four simple clustered patterns (Fig. 2a–d). Based on the null hypothesis of no high-value clustering for the AHVCT, we evaluated One Low Value Cluster
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Figure 3. Rejection rates of Tango’s CG and Moran’s I. 217
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adjusted Moran’s I and Tango CG, denoted respectively by I(h) and CG(h) tests when there are low-value clusters but no high-value clusters (Fig. 3a). Based on the null hypothesis of no low-value clustering for the ALVCT, we evaluated the adjusted Moran’s I and CG tests, denoted respectively by I(‘) and CG(‘) when there are highvalue clusters but no low-value cluster. As expected, when a high-value cluster and a low-value cluster were simulated in the same way, the original Moran’s I and Tango’s CG were unable to distinguish between them. For example, the simulations for the original Moran’s I test produced similar rejection curves for one low-value cluster (Fig. 3a) and one highvalue cluster (Fig. 3b), but we would not be able to tell if they indicate a high-value clustering or low-value clustering. When there was no cluster (d 5 0) or there were weak high-value or low-value clusters (small d values), the rejection rates of the null hypothesis of no clustering for both the original Moran’s I and Tango’s CG were close to 5%. As we made the high-value or low-value clusters stronger by increasing the d value toward 1, the rejection rates increased rapidly in a similar fashion. Both results suggest that these original tests have performed fairly well if we do not want to indicate a one-sided clustering tendency, that is, the existence of highvalue clustering, low-value clustering, or both. When there is a possible low-value cluster (Fig. 3a and c), the results from Moran’s I(h), on the other hand, consistently rejected the null hypothesis of no high-value clustering at the 5% level until d was 40.8. At the point of a strong cool-spot (d 5 1), the rejection rate for Moran’s I(h) was about 15%, much less than the 100% rejection rate from the unadjusted Moran’s I test. These general patterns were also observed in two low-value (Fig. 3c) and two high-value clusters (Fig. 3d). Combining the simulation results of one and two low-value clusters, we found that the original test results from Tango’s CG and Moran’s I could cause high falsealarm rates if the existence of a low-value cluster were interpreted as having a highvalue clustering tendency. The AHVCT can effectively reduce the influence of the low-value clusters on the CG(h) and I(h) tests, thereby lowering the potential falsealarm rates. Similarly, in the cases of one and two high-value clusters, I(‘) and CG(‘) would reduce the potential false-alarm rates by 85% (Fig. 3b and d). To assess the powers of the adjusted tests against the original I and CG tests, which is reflected by the rejection rates under the alternative hypothesis (Fotheringham and Brunsdon 2004) we used I(h) and CG(h) to test for the existence of high-value clustering and I(‘) and CG(‘) for low-value clustering (Figs. 4 and 5). The powers for the CG, CG(h), and CG(‘) tests were broadly similar to those for the corresponding Moran’s I, I(h), and I(‘) tests. Apparently, the rejection rates from the CG tests were slightly higher than those from the corresponding Moran’s I tests. Because the two sets of test results were similar, we primarily discuss the results from Moran’s I. When there was only a single high-value cluster (Fig. 4a), AHVCT had little effect because the curves of the rejection rates for I(h) and I were almost identical. Specifically, the rejection rates for I(h) were slightly above 95% when d 5 1 and 218
One High Value Cluster
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Supplemental Indicator of Spatial Clustering
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Tonglin Zhang and Ge Lin
I(h) I I(l)
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Figure 4. Powers of Moran’s I under the alternative of high-value and low-value clustering tests.
were almost identical to those from the original Moran’s I. On the other hand, when there was no high-value cluster, the rejection rates of the corresponding null hypotheses for both I and I(h) were all close to 5%. The statistical powers increased slightly in the simulation of two high-value clusters. Similar results were achieved in the simulations of one or two low-value clusters. When both low-value and high-value clusters existed (Fig. 4e and f), the powers of Moran’s I(h) or I(‘) were slightly lower than those of Moran’s I. The greater power of the original Moran’s I test is expected, because it considers the existence of either low-value or high-value clusters together, while the I(h) and I(‘) tests consider separately the existence of either high-value or low-value clusters. Because there are two test results for the single pattern of coexistence of high-value and lowvalue clusters, together I(h) and I(‘) provide richer information than the original Moran’s I statistic. To briefly summarize, both AHVCT and ALVCT worked as expected in the simulations. When there was no cluster, both had little effect on the existing test statistics. When there was only a high-value cluster, AHVCT had little effect on the original test statistics, except that it helped to unambiguously confirm the existence of high-value clustering. Likewise, when there was only one or two low-value clusters, ALVCT had little effect on the original test statistics, except that it helped to unambiguously confirm the existence of low-value clustering. When high-value and low-value clusters coexisted, the powers of the I(h) or I(‘) and CG(h) or CG(‘) 219
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0.6 0.4 0.2 0.0
0.0 0.2 0.4 0.6 0.8 1.0 δ
0.0 0.2 0.4 0.6 0.8 1.0 δ
Figure 5. Powers of Tango’s CG under the alternative of high-value and low-value clustering tests.
were slightly lower than those from their original tests. Having two test indicators for a single spatial pattern, whether they were I(h) and I(‘) or CG(h) and CG(‘), allowed us to confirm separately the existence of low-value and high-value clustering, which supplement the original Moran’s I and Tango’s CG tests.
A case study This case study employed the 5-year (1992–1996) county-level leukemia mortality data from the Minnesota Cancer Surveillance System, which records county-specific leukemia deaths for males and females. According to the U.S. National Cancer Institute, the 5-year (1990–1994) leukemia mortality rate for white males in Minnesota is the second highest in the United States, or about 11% higher than the national rate. The mortality rate for white females in Minnesota was 23rd, just slightly higher than the national average. We compared the U.S. 1990 Census populations with the 1994 county estimates (the mid-year of 1992–1996) and found a slight change in population for most of the Minnesota counties. For this reason, we decided to use the 1990 U.S. Census data rather than the county estimates. We used Euclidean distance to calculate wij and set t 5 35 miles, or an average distance between any two adjacent counties. Preliminary assessments suggest that the choice of t values ranging from 20 to 50 were mostly consistent. 220
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Table 1 P-values of Tango’s CG and Moran I in Minnesota Leukemia Data Set Null hypothesis
Two-sided clustering
Tests
CG
Males Females
High-value clustering I
12
6 10 2 10 6
3 10 0.030
CG(h) 6
2 10 0.001
Low-value clustering I(h)
6
0.005 0.240
CG(‘) 1 10 0.010
I(‘) 7
1 10 4 0.656
Across Minnesota, there were 1226 cases of leukemia in females and 1718 in males. The total female population in the state was 2,229,916 and the mortality rate was 5.6 per 10,000. The total male population was 2,145,183 and the mortality rate was 8 per 10,000. Because the true values of the disease rates for females and males ^F ¼ 5:5 104 and l ^M ¼ 8:0 104 , were unknown, we used the estimated values l respectively, for females and males in the AHVCT and ALVCT. The procedures were applied to Tango’s CG test and Moran’s I test for females and males separately. Because each adjustment is random, the corresponding P value also comes with random noise, as such one random adjustment may fall into an extreme case. To reduce the influence of the randomness of the adjustment procedures, it is sufficient to run the procedures hundreds of times, but it requires only a few minutes for 10,000 runs. The final P values were the median P values of those tests in the 10,000 repeated runs (Table 1). The results show that all the P values were o0.05 for males but not for females from I(‘) and I(h) results. In general, the P values derived from Tango’s CG test were much smaller than those derived from Moran’s I test. This finding is expected, because Tango’s CG also absorbs significant nonclustered spatial variation (Lin 2004), while Moran’s I only picks up the most dominant spatial pattern (Assuncao and Reis 1999). The results for males derived from both the CG and I tests suggest a tendency toward high-value and low-value clustering. The results for females were inconsistent, because only CG(h) and CG(‘) were significant for a one-sided test. Even though the use of Tango’s CG is more appropriate than that of Moran’s I due to the nature of regional count data, we cannot exclude the concern that Tango’s CG may pick up nonclustered variations between counties. It is especially true for the result of low-value clustering for females, where the P value for CG(‘) was 0.010 while the P value for I(‘) was 0.656. We then attempted to identify the locations of high-value and low-value clusters by using maps, local Tango’s CF (Tango 1995) test, and local Moran’s Ii test (Anselin 1995). We used the Bonferroni method to account for the multiple testing problem (see Neter et al. 1996, p. 153). The Bonferroni method adjusts the number of observations, in our case the number of counties for the multiple testing problem. If there are potentially 100 units to contribute to the problem, it divides the resultant P value by 100. Because Minnesota has 87 counties, the method provides a P value less than 0.00057 ( 5 0.05/87) for significance. 221
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Both the CF and Ii tests indicated a significant local cluster of high values around Otter Tail County in western Minnesota. The P values for both tests were o0.0001, and they were consistent with different t values ranging from 20 to 50. The local tests also suggested other clusters in western Minnesota, but the results were less consistent with different t values. We concluded, therefore, that a highvalue cluster existed and was centered around Otter Tail County. As for low-value clusters, we found that some of the counties surrounding Sibley County, which is located southwest of Minneapolis, formed a consistent cluster. This cluster was confirmed by both the CF and Ii tests, even though there were some discrepancies in the size of the cluster when different t values were used. As with the high-value cluster tests, when we shifted the cluster center to a county not adjacent to Sibley County, both the cluster sizes and the levels of significance changed, sometimes drastically. Consequently, we concluded that the low-value cluster for males centered around Sibley County within a 35-mile radius. As for females, only a high-value cluster was consistently registered from the results of both local tests; the P value for each test was o0.001. This potential cluster centered around Pennington County in northwestern Minnesota. We stress the potential cluster, because only CG(h) rejected the null hypothesis of no highvalue clustering, while the global I(h) accepted it. The results from the low-value local tests were inconclusive. Based on the CF test, there were about 13 counties around Minneapolis that could be a low-value cluster, but the corresponding Ii test was not significant. Because the counties surrounding Minneapolis tend to be smaller than the northern counties, we also used smaller t values, such as 20 or 25. In these cases, the results of the CF test became insignificant. We concluded that there was no low-value cluster for females, which is consistent with the results from the general tests (Figs. 6 and 7). In summary, we identified one high-value cluster and one low-value cluster for male leukemia mortality and one high-value cluster for female leukemia mortality. When we shifted the cluster centers, the results were mixed, depending on the cluster-size criteria and the type of local tests. For this reason, it is difficult to determine the shapes and boundaries of these clusters, except in the counties immediately adjacent to them for the high-value clusters and in the counties that are within a 35-mile radius of the low-value cluster center.
Discussion Unlike nonspatial statistical tests, the use of a quadratic form or its variant in spatial clustering tests often leads to a two-sided spatial inference. If the two-sided spatial autocorrelation inference needs to be made one-sided (e.g., Pearson correlation), a statistical method that either treats data or the underlying distribution has to be developed. Our data treatment procedure is the first step in this direction in the absence of a one-sided spatial clustering test. Following the classical definition of spatial clusters, we consider both high-value and low-value clusters to be highly 222
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Figure 6. Female leukemia mortality rates per 1000 in Minnesota.
unusual. By accounting for the potential influence of one type of cluster over the other, our data-adjustment procedure is able to supplement Moran’s I and Tango’s CG with one-sided test results. When testing for high-value clustering, the procedure accounts for the potential impact of low-value clusters. When testing for lowvalue clustering, the procedure accounts for the potential impact of high-value clusters. Overall, the procedure has little effect on the original global test statistics, when there is no cluster. When there is a tendency toward clustering, the procedure is not only able to reduce a one-sided cluster effect, but is also able to unambiguously distinguish the existence of high-value, low-value, or both types of clustering. Because the procedure treats data rather than a test statistic, it can be easily applied to other spatial statistics, such as the Getis–Ord G test (Getis and Ord 1992) and Cuzick and Edwards k-NN (Cuzick 1990) test that are based on existing distributions, or the scan test (Kulldorff 1997) that is based on simulated disease risks. Several issues warrant further investigations. First, we offered a one-sided data adjustment to randomize the one-sided spatial arrangement of the observed values. However, this method does not preclude other data adjustment procedures. We prefer to preserve a large number of observed values by considering the threshold value close but not equal to lxi in the procedure. On the other hand, if one does not believe that low-value clusters are unusual, one can simply randomize the 223
Geographical Analysis
Figure 7. Male leukemia mortality rates per 1000 in Minnesota.
observed low values by their locations to destroy potential low-value clusters while retaining all the observed values. Second, because the main concern about the procedure is the efficiency of reducing unwanted influence, it is necessary to identify a more efficient value replacement method that would probably use a Bayesian procedure. Finally, even though the proposed random data adjustment could be further improved, it is not a test statistic by itself. The ultimate goal is to develop a one-sided test statistic. Acknowledgements We appreciate valuable comments from three anonymous reviewers. We would also like to acknowledge the adoption of a graphical illustration provided by a reviewer. References Anselin, L. (1995). ‘‘The Local Indicators of Spatial Association—LISA.’’ Geographical Analysis 27, 93–115. Assuncao, R., and E. Reis. (1999). ‘‘A New Proposal to Adjust Moran’s I for Population Density.’’ Statistic in Medicine 18, 2147–62. 224
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Cuzick, J., and R. Edwards. (1990). ‘‘Spatial Clustering for Inhomogeneous Populations (with Discussion).’’ Journal of Royal Statistical Society B 52, 73–104. Diggle, P. J. (2000). ‘‘Overview of Statistical Methods for Disease Mapping and its Relationship to Cluster Detection.’’ In Spatial Epidemiology: Methods and Applications, 87–103 edited by P. Elliott, J. Wakefield, N. Best, and D. Briggs. Oxford, UK: Oxford University Press. Fotheringham, S. A., and S. Brunsdon. (2004). ‘‘Some Thoughts on Inference in the Analysis of Spatial Data.’’ International Journal of Geographical Information Science 18, 447–57. Gangnon, R., and M. Clayton. (2000). ‘‘A Weighted Average Likelihood Ratio Test for Spatial Clustering.’’ Statistics in Medicine 20, 2977–87. Getis, A., and J. Ord. (1992). ‘‘The Analysis of Spatial Association by use of Distance Statistics.’’ Geographical Analysis 24, 189–206. Kulldorff, M. (1997). ‘‘A Spatial Scan Statistic.’’ Communications in Statistics, Theory and Methods 26, 1481–96. Lin, G. (2003). ‘‘A Spatial Logit Association Model for Cluster Detection.’’ Geographical Analysis 35, 329–40. Lin, G. (2004). ‘‘Comparing Three Spatial Clusters Tests from Rare to Common Diseases.’’ Computers, Environment and Urban Systems 28, 691–99. Lin, G., and T. Zhang. (2004). ‘‘A Method for Testing Low-Value Spatial Clustering for Rare Disease.’’ ACTA Tropica 91, 279–89. Marshall, R. J. (1991). ‘‘A Review of Methods for the Statistical Analysis of Spatial Patterns of Disease.’’ Journal of the Royal Statistical Society Series A 154, 421–41. Mihram, G. A. (1972). Simulation: Statistical Foundations and Methodology. New York: Academic Press. Neter, J., M. H. Kutner, C. Nachtsheim, and W. Wasserman. (1996). Applied Linear Statistical Models, 4th ed. New York: McGraw Hill. Ord, K., and A. Getis. (2001). ‘‘Testing for Local Spatial Autocorrelation in the Presence of Global Autocorrelation.’’ Journal of Regional Science 41, 411–32. Rogerson, P. A. (1999). ‘‘The Detection of Clusters using a Spatial Version of the Chi-Square Goodness-of-Fit Statistics.’’ Geographical Analysis 31, 130–47. Tango, T. (1995). ‘‘A Class of Test for Detecting ‘General’ and ‘Focused’ Clustering of Rare Diseases.’’ Statistics in Medicine 14, 2323–34. Waldho¨r, T. (1996). ‘‘The Spatial Autocorrelation Coefficient Moran’s I under Heteroscedasticity.’’ Statistics in Medicine 15, 887–92.
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