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Computational Statistics & Data Analysis 51 (2007) 6123 – 6137 www.elsevier.com/locate/csda

A decomposition of Moran’s I for clustering detection Tonglin Zhanga,∗ , Ge Linb a Department of Statistics, Purdue University, 250 North University Street, West Lafayette, IN 47907-2066, USA b Department of Geology and Geography, West Virginia University, Morgantown, WV 26506-6800, USA

Received 31 January 2006; received in revised form 13 December 2006; accepted 14 December 2006 Available online 28 December 2006

Abstract The test statistics Ih , Ic , and In are derived by decomposing the numerator of the Moran’s I test for high-value clustering, lowvalue clustering, and negative autocorrelation, respectively. Formulae to compute the means and variances of these test statistics are derived under a random permutation test scheme, and the p-values of the test statistics are computed by asymptotic normality. A set of simulations shows that test statistic Ih is likely to be significant only for high-value clustering, test statistic Ic is likely to be significant only for low-value clustering, and test statistic In is likely to be significant only for negatively correlated spatial structures. These test statistics were used to reexamine spatial distributions of sudden infant death syndrome in North Carolina and the pH values of streams in the Great Smoky Mountains. In both analyses, low-value clustering and high-value clustering were shown to exit simultaneously. © 2007 Elsevier B.V. All rights reserved. Keywords: Clustering and clusters; Low-value clustering; High-value clustering; Negative autocorrelations; Positive autocorrelations; Random permutations

1. Introduction Spatial autocorrelation is a measure of spatial dependence between values of random variables over geographic locations. The most often used and cited one is Moran’s (1948) I, a single test statistic that indicates two types of spatial autocorrelation—positive autocorrelation and negative autocorrelation. These two autocorrelation indicators have been used widely to capture three types of spatial relations: a positive autocorrelation captures the existence of both high-value clustering and low-value clustering, while a negative autocorrelation captures the juxtaposition of high-values next to low-values (Anselin et al., 2000; Haining, 1990; Lawson and Denison, 2000). In other words, only one dominant type of autocorrelation can be detected. If two structures, such as high-value clustering and low-value clustering, coexist, Moran’s I cannot distinguish them. Since Moran’s I was first published in 1948, there have been hundreds of applications and extensions of the test for spatial clustering. A quick search of the Science Citation Index and Social Science Citation Index since 1975 yielded more than 300 English-language articles that either directly cited the original paper on Moran’s I or included the name of the test in the title or abstract. Most of methodological papers on the test have been focused on estimation methods (Cliff and Ord, 1972; Lee, 2004; Whittemore et al., 1987), spatial distribution properties (Bennett and Haining, 1985; ∗ Corresponding author. Tel.: +1 765 496 2097; fax: +1 765 494 0558.

E-mail addresses: [email protected] (T. Zhang), [email protected] (G. Lin). 0167-9473/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.12.032

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Getis and Aldstadt, 2004), and ways of incorporating population size when applying Moran’s I for count data (Assuncao and Reis, 1999; Oden, 1995; Tango, 1999; Waldhor, 1996). A concurrent theme is focused on local spatial statistics (Fotheringham, 1997). It was pointed out that spatial dependencies may differ from place to place (Fotheringham, 1999), and the global test of Moran’s I only captures the most dominant dependency. Some statisticians attempted to use spatial partition models, such as spatially weighted regression, to parameterize spatial variation (Brunsdon et al., 1999; Schlattmann and Bohning , 1993). Some have attempted to spatially decompose global autocorrelation measures such as Moran’s I into a local indicator of spatial association (LISA) (Anselin, 1995). LISA, when coupled with other exploratory spatial data-analysis tools such as Moran’s I scatter plot, is able to locate spatial associations, such as high-value clusters, low-value clusters, and negative autocorrelations. However, in the absence of the global test, LISA is often too sensitive to values outside a potential clustered area (Tiefelsdorf, 1997), because the traditional permutation test may not be appropriate for handful local regions. Ord and Getis (2001) noticed this problem, and they applied a common method of partitioning regions into nonoverlapping groups. In either cases, it is suggested that local indicators should be interpreted according to the degree of the corresponding global autocorrelation (Sokal et al., 1998; Tiefelsdorf, 2002), which calls for a set of global autocorrelation tests that would explicitly confirm or refute the existence of the three types of spatial relations simultaneously. There are other reasons to detect high-value and low-value clustering separately from a spatial autocorrelated structure. Even though established methods of spatial autocorrelation often emphasize the spatial concentration of elevated risks or the tendency toward high-value clustering (Lawson and Kulldorff, 2000), the importance of testing for low-value clustering cannot be ignored (Getis and Ord, 1992; Lin and Zhang, 2004). The detection of negative autocorrelations and low-value clustering is imperative, not only for more precise testing of high-value clustering, but also for identifying substantive spatial problems, such as spatial configuration of imageries, spatial process of employment, and ethnic settlement patterns (Griffith, 2003). For instance, proper detection of low-value clustering can reveal the existence of a healthy community either because of environmental or genetic endowment or because of effective prevention programs. In addition, even though we can include a parameter that accounts for spatial autocorrelation in a spatial regression (Anselin and Griffith, 1988; Arbia, 1989), we nonetheless would want to know the nature of the autocorrelation. Properly evaluating regression residuals in terms of low-value clustering and high-value clustering could also isolate clustered areas that might help researchers to narrow the list of putative causes that affect the response variable (Zhang and Lin, 2006). In this paper, we propose a method to decompose the traditional Moran’s I test into three components so that each component represents a global test statistic. We denote the three components as Ih , Ic , and In to test, respectively, for the existence of high-value clustering, low-value clustering, and negative autocorrelation. In the section that follows, we derive these statistics, as well as their means and variances, according to a random permutation routine. In Section 3, we evaluate these tests with a set of simulations. We not only show that these tests are effective, but also demonstrate that even when the result of Moran’s I is very close to 0, there is still a possibility that high-value clustering, lowvalue clustering, and negative autocorrelation coexist (Anselin, 1995). In Section 4, we present two applications by reexamining pH values from streams located in the Great Smoky Mountains (Schmoyer, 1994) and sudden infant death syndrome (SIDS) data from North Carolina (Cressie and Chan, 1989). In the final section, we offer some concluding remarks. 2. Deriving Ih , Ic and In from Moran’s I Let xi be the variable of interest in region i for a study area A that has m regions (i = 1, . . . , m). Then Moran’s (1950) I is defined as I=

m m 1  wij (xi − x)(x ¯ j − x), ¯ S 0 b2

(1)

i=1 j =1

m m

 m where S0 = i=1 j =1 wij , x¯ = m ¯ k /m and wij with wii = 0 is an element of the spatial i=1 xi /m bk = i=1 (xi − x) weight matrix that measures spatial distance or connectivity between regions i and j. Usually wij is assigned according to a spatial connectivity that is wij = 1 if regions i and j are adjacent and wij = 0 otherwise. In most situations, a spatial weight matrix is symmetric, that is, wij = wj i for all values of i and j, but we do not need to use this constraint in our study.

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Moran’s I coefficients usually range between −1 and 1, with a coefficient close to −1 indicating neighborhood dissimilarity and a coefficient close to 1 indicating neighborhood similarity. If the values in neighboring regions for a given location are correlated positively, then the regions around that location are clustered either in high-value or low-value clusters. If the values in neighboring regions for a given location are negatively correlated, then the regions around that location are juxtaposed between high values and low values as a chessboard pattern. When the coefficient of Moran’s I is close to 0, it indicates spatial randomness or independence. Typically, the null hypothesis for Moran’s I is spatial independence, and the alternative hypothesis is the existence of either a positive or negative autocorrelation. In the absence of other spatial associations and the first order trend, Moran’s I in general is sensitive to a local association or structure. For example, a high-value cluster in an otherwise spatially random pattern is usually indicated a clustering tendency by Moran’s I, and applied to any single spatial structure. When multiple spatial associations exist, a statistically significant and positive coefficient indicates the dominance of similarity in spatial values (i.e., high-value clustering or low-value clustering), and a statistically significant negative coefficient indicates the dominance of dissimilarity between neighboring regions. In order to distinguish a high-value clustering tendency from a low-value clustering tendency, we need to have alternative hypotheses that correspond to the existence of high-value clusters, low-value clusters, or negative correlations. The three distinct null hypotheses are: the nonexistence of high-value clustering (Ih ), the nonexistence of low-value clustering (Ic ) or the nonexistence of negative autocorrelation (In ). We note that spatial statistics can often be decomposed via their numerators. For examples, Tango’s CG index has been decomposed into spatial and aspatial components of the 2 statistic (Rogerson, 1999). Ord and Getis (2001) used an approach of partitioning means into groups, and compared the mean within a distance range against the mean in the rest of study area. Follow these lines of reasoning, we can also decompose the numerator of Moran’s I to be correspondent to the three null hypotheses above. Since high-value clustering in general is above the mean value and low-value clustering below the mean, and the denominator of Moran’s I in (1) does not vary under a random permutation scheme, we can partition the numerator of Moran’s I into the following three components: I = Ih + Ic + In ,

(2)

where Ih = I˜h /(S0 b2 ), Ic = I˜c /(S0 b2 ) and In = I˜n /(S0 b2 ), with I˜h =

m m  

wij (xi − x) ¯ + (xj − x) ¯ +=

i=1 j =1

I˜c =

m  m 

 

wij (xi − x)(x ¯ j − x), ¯

i∈Ah j ∈Ah

wij (xi − x) ¯ − (xj − x) ¯ −=

i=1 j =1

 

wij (xi − x)(x ¯ j − x), ¯

(3)

i∈Ac j ∈Ac

and I˜n = −

m  m  i=1 j =1

(wij + wj i )(xi − x) ¯ + (xj − x) ¯ −=

 

(wij + wj i )(xi − x)(x ¯ j − x), ¯

i∈Ah j ∈Ac

¯ where a + and a − , respectively, denote the positive and negative components of the real number a, and Ah = {i : xi  x} and Ac = {i : xi < x}. ¯ Notice that in our partition, the terms in In are all negative and the terms in Ih or Ic are positive, with Ih being the sum of the product of positive terms and Ic being the sum of the product of negative terms. We compute the mean and variance of the Ih , Ic , and In under random permutations. As is the case with the original Moran’s I, a positive and statistically significant Ih and Ic coefficients indicate high-value clustering and low-value clustering, respectively. A negative and statistically significant In coefficient indicates value dissimilarity between neighboring regions. Unlike the original Moran’s I test, the effect of Ih , Ic or In alone is not to be dominated by the other two effects. √ For permutation distributions of Ih,std = [Ih − E(Ih )]/ Var(Ih ), Ic,std = √ tests, we are interested in the asymptotic √ [Ic − E(Ic )]/ Var(Ic ) and In,std = [In − E(In )]/ Var(In ) when m → ∞. Usually the means and variances of the

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test statistics are unknown, and we compute them based on the permutation distributions of the test statistics. Since xi − x¯ is always nonnegative when i ∈ Ah and always negative when i ∈ Ac , we can compute the permutation mean and variance of Ih under a random permutation scheme that permutes the locations in Ah but not in Ac . In the same way, we can compute the permutation mean and variance of Ic by a random permutation of Ac locations only. Finally, since In is an interaction term, we can derive its mean and variance from the product of two random permutations: (1) a random permutation of Ah for xi − x¯ given i ∈ Ah and (2) a random permutation of Ac for xj − x¯ given j ∈ Ac . The first step permutes the locations in Ah but not in Ac ; the second step permutes the locations in Ac but not in Ah . In other words, changes in locations for Ah are determined in the first step, and changes in locations for Ac are determined in the second step. We derived the permutation means and variances of Ih , Ic and In in Appendix A, and provide the main results below. We put the derivations of the permutation means and variances of Ih , Ic and In in Appendix A, and provide the main results below. It is worth pointing out that even though the derivation is based on the assumption that m → ∞, it is sufficient if m50. Walter (1992) as well asour own simulations.  m  This number is based on Denote mh = m ¯ k /mh and ck = i∈Ac (xi − x) ¯ k /mc , I =#(A ), m = ¯ h c i=1 Xi  X i=1 IXi <X¯ =#(Ac ), hk = i∈Ah (xi − x) where IXi  X¯ denotes the indicator function and #(S) denotes the number of elements within a set S. Write wi·h =      h  c  c  j ∈Ah wij , S1h = i∈Ah j ∈Ah (wij + j ∈Ah wij , w·i = j ∈Ah wj i , wi· = j ∈Ac wij , w·i = j ∈Ac wj i . Let S0h = i∈Ah      2 2 2 h h wj i ) /2 and S2h = i∈Ah (wi· + w·i ) ; let S0c = i∈Ac j ∈Ac wij , S1c = i∈Ac j ∈Ac (wij + wj i ) /2 and S2c =       2 c c 2 c i∈Ac (wi· + w·i ) ; and let S0n = i∈Ah j ∈Ac (wij + wj i ) and S1n = i∈Ah j ∈Ac (wij + wj i ) , S2n = i∈Ah (wi· +  2 2 h ) . The permutation means and mean squares of I , I and I are w·ic ) , and S3n = j ∈Ac (wjh· + w·j n c n Eh (Ih ) =

S0h (mh h21 − h2 ) , S0 b2 (mh − 1)

S0c (mc c12 − c2 ) , S0 b2 (mc − 1) S0h h1 c1 En (In ) = , S 0 b2

Ec (Ic ) =

(4)

and Eh (Ih2 ) =

S1h (mh h22 − h4 ) (S0 b2 )2 (mh − 1) +

Ec (Ic2 ) =

En (In2 ) =

(S0 b2 )2 (mh − 1)(mh − 2)(mh − 3)

(S0 b2 )2 (mc − 1)

+

(S0 b2 )2 (mc − 1)(mc − 2)

(S0 b2 )2 (mc − 1)(mc − 2)(mc − 3)

(S0 b2 )2

+

,

(S2c − 2S1c )(m2c c12 c2 − 2mc c1 c3 − mc c22 + 2c4 )

2 + S − S )(m3 c4 − 6m2 c2 c + 8m c c + 3m c2 − 6c ) (S0c 1c 2c c 1 3 c 2 4 c 1 c 1 2

S1n h2 c2

+

(S0 b2 )2 (mh − 1)(mh − 2)

2 + S − S )(m3 h4 − 6m2 h2 h + 8m h h + 3m h2 − 6h ) (S0h 1h 2h h 1 3 h 2 4 h 1 h 1 2

S1c (mc c22 − c4 ) +

+

(S2h − 2S1h )(m2h h21 h2 − 2mh h1 h3 − mh h22 + 2h4 )

(S2n − S1n )[h2 (mc c12 − c2 )] (S0 b2 )2 (mc − 1)

+

(S3n − S1n )[c2 (mh h21 − h2 )] (S0 b2 )2 (mh − 1)

2 + S − S − S )(m h2 − h )(m c2 − c ) (S0n 1n 2n 3n h 1 2 c 1 2

(S0 b2 )2 (mh − 1)(mc − 1)

,

,

(5)

where Eh (·), Ec (·) represents the expected value under the random permutation over Ah , over Ac , respectively, and En (·) denotes the expected value under the product of a random permutation of Ah and a random permutation of Ac . The permutation variances of Ih , Ic and In can be easily obtained by Varh (Ic )=Ec (Ic2 )−[Ec (Ic )]2 , Varc (Ic )=Ec (Ic2 )− [Ec (Ic )]2 and Varn (In ) = En (In2 ) − [En (In )]2 based on these calculations, and the corresponding p-values are derived

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under the asymptotic normality of Ih , Ic or In as m → ∞. To prove the asymptotic normality, we need to impose the following regularity conditions used by Sen (1976) in his proof of the asymptotic normality of Moran’s I: (C1) wij = wj i 0.  (C2) There exists a constant C > 0 such that wi· = m j =1 wij C. (C3) The limits limm→∞ S0m /m = 0 , limm→∞ S1m /m = 21 and limm→∞ S2m /m = 22 exist and are positive,  m  m m 2 2 where S0m = m i=1 j =1 (wij + wj i ) /2 and S2m = i=1 j =1 wij , S1m = i=1 (wi· + w·i ) . We state the asymptotic normality theorem for Ih , Ic and In below, and provide the details of the proof in Appendix B. Theorem 1. Suppose X1 , . . . , Xm are iid nondegenerate random variables with finite fourth moments. Let Ih,std =

Ih − Eh (Ih ) , √ Vh (Ih )

Ic − Ec (Ic ) Ic,std = √ Vh (Ic )

and

In,std =

In − En (In ) . √ Vn (In )

Then, under conditions (C1), (C2) and (C3), Ih,std , Ic,std and In,std are asymptotically N(0, 1) as m → ∞. Remark. Theorem 1 can be easily generalized to a model based Ih , Ic and In . Since the error terms of a general linear model are usually assumed independently normally distributed, the approximate normality for Ih , Ic and In are valid if the number of observations is large and the number of covariates is small. The p-value of this test statistic is 1 − (Ih,std ), where  is the standard normal CDF. Since the Ih test statistic is based on observations from Ah , we can select a nominal p-value of  for a one-sided test. If Ih is larger than z or the p-value for Ih is less than , we conclude that high-value clustering exist in the study area, where z is the upper  quantile of the standard normal distribution. As the case of Ih above, the p-value of the test statistic Ic is 1 − (Ic,std ). If Ic is larger than z or if the p-value for Ic is less than a nominal value , we conclude that low-value clustering exist based on a one-sided test. The p-value for In can be computed by using (In,std ) since we reject the null hypothesis of no negative autocorrelation for a small value of In . If the p-value for In is less than the nominal value , we conclude the existence of negative autocorrelations in the study area. In summary, the new test statistics Ih , Ic , and In are designed to detect, respectively, the existence of high-value clustering, low-value clustering, and negative autocorrelations for a single study area. We expect the test statistic Ih to be statistically significant if there is a high-value cluster regardless of the presence or absence of a low-value cluster or a negative autocorrelation; Ih is not to be significant in the absence of a high-value cluster regardless of the presence of other patterns. In contrast to Moran’s I test, the p-values of Ih , Ic , and In are computed under a one-sided normal approximation. A significantly large Ih value indicates the existence of a high-value cluster, a significantly large Ic value indicates the existence of a low-value cluster, and a significantly small In value indicates the existence of negative autocorrelation. 3. Simulation We now evaluate the performances of test statistics Ih , Ic and In for normal data. As a comparison, we also include results from the conventional Moran’s I test. Since the conventional Moran’s I concludes the existence of either a positive or a negative autocorrelation, we use I (p)√for positive Moran’s I results if Istd > z/2 and I (n) for negative results if Istd < − z/2 , where Istd = [I − ER (I )]/ VarR (I ), and where ER (·) and VarR (·) represent the permutation mean and variance under a full random permutation of all locations in the study area. As mentioned earlier, Moran’s I is sensitive to a local structure in the absence of other spatial associations, and it is sufficient to insert one or two spatial structures in an otherwise spatially independent pattern. We performed simulations on six spatial patterns on a regular 20 × 20 lattice: (a) a high-value cluster centered at (5, 5); (b) a low-value cluster centered at (15, 15); (c) a negative autocorrelation centered at (5, 15); (d) a high-value cluster centered at (5, 5) and a low-value cluster centered at (15, 15); (e) a low-value cluster centered at (15, 15) and a chessboard pattern centered at (5, 15); and (f) a high-value cluster centered at (5, 5) and a negative autocorrelation centered at (5, 15). Both clustered and autocorrelated areas cover a distance within 3 units from each center location. In our assessments, we fixed the

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nominal value of  at 0.05, and preliminary simulations showed that the results from both nominal values 0.05 and 0.01 were very similar. We denoted i = 20(ai − 1) + bi for ai = 1, . . . , 20 and bi = 1, . . . , 20. We generated xi independently from N (i , 1) at  location (ai , bi ). With  varying from 0 to 2 with stepped increments of0.1 in each pattern, we chose i =  if 2 2 (ai − 5)2 + (bi − 5)2 3 in patterns (a), (d), and  (f); we chose i = − if (ai − 15) + (bi − 15) 3 in patterns (b), (d), and (e); we chose i = −(−1)ai +bi  if (ai − 5)2 + (bi − 15)2 3 in patterns (c), (e) and (f); and we chose i = 0 for all the remaining locations at (ai , bi ). In each simulation, we chose wij = wj i = 1 if locations (ai , bi ) and (aj , bj ) were adjacent to each other and wij = wj i = 0 if they were not:  1 if |ai − aj | + |bi − bj | = 1, wij = wj i = 0 otherwise. Based on 10, 000 runs of every  value from patterns (a)–(f), the rejection rates of the tests based on Ih , Ic , In and Moran’s I are displayed in Fig. 1. For the single structured cases, the upper panel of the figure showed that Ih , Ic , and In were only likely to be significant when high-value clustering, low-value clustering, or negative autocorrelations, respectively, existed. The figure also showed that I (p) was likely to be significant for both high-value clustering and low-value clustering but not for negative autocorrelations and that I (n) was likely to be significant for negative autocorrelations only. For two-structured cases, the lower panel of Fig. 1 showed that Ih , Ic , and In were still likely to be significant for the same types of spatial associations in the upper panel. Comparing plots (b), (c), (e), and (f) in Fig. 1, we found that the rejection rates for Ih and Ic were reduced by a negatively autocorrelated structure and vice versa. However, plots (e) and (f) showed that neither I (p) nor I (n) was

a

b

c

One Low

One Negative

1.0

1.0

0.8

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0.6

0.4

Rejection Rate

1.0

Rejection Rate

Rejection Rate

One High

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0.0 0.5

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0.0 0.0

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0.0 0.0

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δ

d

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One Low, One Negative

One High, One Negative

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δ

Fig. 1. Rejection rates for normal data on a 20 × 20 lattice when  = 0.05.

0.5

1.0 δ

1.5

2.0

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likely to be significant when a negative autocorrelation coexisted with a low-value or a high-value cluster. This finding suggests that negative and positive autocorrelations tend to cancel out each other in the conventional Moran’s I test. The simulations also showed that when high-value clustering and low-value clustering coexisted (plot (d)), the rejection rate for Moran’s I was higher than the rejection rates of Ih and Ic , because Moran’s I is supposed to be positively significant for either high-value clustering or low-value clustering. In summary, Ih , Ic and In are able to detect the existence of high-value clustering, low-value clustering and negative autocorrelation separately for a single study area. In the presence of other spatial structures, the false alarm rates of Ih and Ic are unlikely to increase substantially. The conventional Moran’s I test cannot separately detect the coexistence of high-value clustering and low-value clustering. Tests based on Ih , Ic , and In provide a stronger indication of the types of spatial structures, although sometimes they could have a weaker statistical power than the conventional Moran’s I. 4. Application In this section, we apply our test statistics to published normal and Poisson data examples that were deemed to have a high-value cluster. The normal data example was first examined by Schmoyer (1994) on the spatial distribution of pH values of streams in the Great Smoky Mountains. The original study found positive spatial autocorrelation while controlling for elevation. The second example is based on Cressie and Chan’s (1989) study of county-level SIDS in North Carolina for two periods: 1974–1978 (period 1) and 1979–1984 (period 2). In the original article, a significant autocorrelation was found in period 1, and the data for this period was used extensively for spatial clustering analyses, and was included in many spatial statistical packages. For this reason, we restrict our analysis to period 1 only. In both examples, we attempt to explicitly detect high-value clustering and low-value clustering. 4.1. Stream pH value study Schmoyer’s study is based on stream water samples collected from 75 locations in the Great Smoky Mountains in 1988. The data set included sample pH values of water samples and elevations of the locations in a Cartesian coordinate. Based on the sample locations, Schmoyer designed a weight, wij , based on Denuly triangulation, where wij = 1 if locations i and j are connected by an edge and are 0 otherwise. We adopted Schmoyer’s approach and used the residual Moran’s I in an OLS regression. Specifically, we took xi in Eq. (1) to be the residuals of the regression model with pH value as the response variable. Following Schmoyer’s practice, we added elevation and east–west coordinate variables to remove global trends, i.e., the first order trends. Our results were consistent with the original study. The coefficient for elevation was −1.1068 with a p-value of 2.69 × 10−7 , and the coefficient of east–west coordinate was 0.00819 with a p-value of 1.23 × 10−6 . The R 2 value was 0.4047. Using xi as the ith residual of the regression model, Table 1 lists values of conventional Moran’s I, Ih , Ic , and In together with their p-values. The results from the conventional Moran’s I test suggested the existence of positive autocorrelation, and the results from the Ih and Ic tests further confirmed, respectively, the existence of high-value clustering and low-value clustering. The p-value for Ih was 0.044 indicating that the effect of the high-value clustering was weak. The p-value of Ic was 0.0191 indicating the existence of a low-value cluster. Judging by these two p-values, the effect of low-value clustering was stronger than that of high-value clustering. We found no evidence of negative spatial autocorrelation. It is suggested from the analysis of residuals that a high-value cluster was likely to be situated in the northeastern part of the Great Smoky Mountains, and a low-value cluster was likely to be located in the southeastern part. Both results were useful; a low-value cluster indicates a cluster of acidic water and a high-value cluster indicates a cluster of alkaline water. Table 1 Values and p-values of test statistics for the Great Smoky Mountain data

Values p-values

Istd

Ih,std

Ic,std

In,std

3.570 0.0004

1.705 0.044

2.073 0.0191

0.717 0.733

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4.2. SIDS in North Caroline SIDS is defined as the sudden and unexpected death of any infant 12 months old or younger that remains inexplicable after performing an adequate postmortem examination. The number of SIDS cases and the number of live births for each North Carolina county together with the locations of county-seats and spatial weights are provided in Cressie and Chan (1989). These data have also been analyzed by other authors, such as Getis and Ord (1992) and Kulldorff (1997). None of them, however, has studied the problems of the existence of significant low-value clustering. Since Kulldorff’s SatScan test has an option of detect either high-value or low-value clusters Kulldorff (2006), we also used it to check the performance of Ih , Ic .  Let ni , i be the number of cases and at risk population in county i, n = m i=1 ni be the total number of SIDS cases,  be the total newborn population. Since spatial variation in population sizes may significantly influence and  = m i=1 i the underlying distribution of Moran’s I if one takes xi = ni /i to be the ith county observed rate (Assuncao and Reis, 1999; Oden, 1995; Waldhor, 1996), we followed approaches in Burr (2001), Walter (1992) and Waller and Gotway (2004, p. 229),  who introduced a variance stabilization transformation of the rates by using the Pearson residual, i.e., xi = (ni − nˆ i )/ nˆ i , where nˆ i = i (n/).  Under the assumption that ni is independently observed from Poisson(i ), the random variable Zi = (ni − i )/ i is independently distributed with mean 0 and variance 1. Note that Zi is approximately N(0, 1) when i is large, the Pearson residual can be used as the variable of interest in the formulation of Moran’s I by simply replacing the unknown parameter with its estimated value ˆ = n/. As did Cressie and Chan, we excluded Anson County as an outlier. Using the same spatial weight and taking xi to be the Pearson residual of the SIDS rate, we derived the values and the corresponding p-values of Istd , Ih,std , Ic,std , and In,std (Table 2). The results from Moran’s I test indicated a significant positive autocorrelation or spatial clustering. Our test-statistics Ih and Ic showed the existence of both high-value and low-value clustering but not of negative autocorrelation. Cressie and Chan suggested that the significant high-value clustering was likely to be located in the northeastern part of North Carolina but did not elaborate on potential low-value clustering. Based on the Ic test and on an inspection of the spatial distribution of the SIDS rates on a map (Fig. 2), we suspected the existence of two low-value clusters, one centered around Ashe county in northwestern corner, and the other centered around Rowan county. Using the Scan, we identified a circular cluster around Davie county, which is consistent with one of our

Table 2 Values and p-values for test statistics in the North Carolina SIDS data between 1974 and 1978 Istd

Ih,std

Ic,std

In,std

6.310 1.3 × 10−10

3.545 0.0002

2.561 0.0052

2.638 0.9958

Values p-values

Ashe Davie

0.00 - 0.84 0.84 - 1.50 1.50 - 2.04 2.04 - 2.78 2.78 - 9.55

Fig. 2. Crude rate for sudden infant death syndrome in North Carolina.

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suspected clusters. Since the Scan test takes account of multiple testing and boundary conditions, the cluster around Davie is more trustworthy. 5. Conclusion Herein, we proposed three decomposed statistics, Ih , Ic and In , and derived their means and variances. Like many other decomposed test indicators, the sum of Ih , Ic , and In is the conventional Moran’s I, and each test statistic is associated with its null hypothesis. We proved that Ih , Ic , and In are asymptotically normal as is the conventional Moran’s I (Sen, 1976), and computed one-sided p-values for these test statistics based on the asymptotic normality of permutation distributions under the null hypothesis. In the future, we would like to estimate the strength of Ih , Ic and In under the alternative hypothesis. Our simulations showed that the test statistics were able to detect the existence of high-value clustering, low-value clustering and negative autocorrelation. In the case of a single high- or low-value clustering, our simulations showed that Ih or Ic had a statistical power greater than or equivalent to Moran’s I. But when both high-value clustering and low-value clustering coexisted, the statistical powers of Ih or Ic were understandably weaker than that of Moran’s I. The weaker power of Ih and Ic is compensated for, however, by explicitly indicating the existence of high-value clustering or low-value clustering, which cannot be achieved with the conventional Moran’s I test. In our analyses of published data, pH values of water in Great Smoky Mountain streams and SIDS cases in North Carolina, we extended the findings of early studies by demonstrating the existence of both high- and low-value clustering. Our finding of low-value clustering in the Great Smoky Mountain is particularly interesting, for high-value clustering and low-value clustering are both equally important in monitoring water quality. When spatial dependence is concluded, finding ways to distinguish high-value clustering from general spatial dependence then becomes a subject for further investigation. Since Moran’s I is unable to provide a signed indicator for the existence of a high-value cluster or a low-value cluster, a positive and statistically significant Moran’s I test is often followed by a visual search of observed values for potential high-value clustering. However, since a single low-value clustering can cause a positive autocorrelation, visual inspection without a signed global test should proceed with caution. In this regard, our proposed test statistics can provide auxiliary indicators for both global Moran’s I and local Moran’s Ii . If Moran’s I is positive and significant, Ih or Ic or both will likely also be significant. If Moran’s I is negative and significant, In will likely also be significant. Based on existing relationships between Moran’s I and Ii (Sokal et al., 1998), our simulations suggest that if a local Moran’s Ii identifies a high-value cluster and if Ih is significant, then the results from the local Ii are supported by the global Ih . Likewise, if a local Moran’s Ii identifies a low-value cluster, and Ic is significant, then the results from the local Ii are supported by the global Ic . It is necessary, therefore, to check Ih and Ic when undertaking local and global Moran’s I tests. Appendix A. Derivation of the permutation mean and variance We derive the first and second permutation moments of Ih , Ic and In in this part of the appendix. Since the denominator S0 b2 in (2) is permutation invariant, we have Eh (Ih )=Eh (I˜h )/(S0 b2 ), Ec (Ic )=Ec (I˜c )/(S0 b2 ), En (In )=En (I˜n )/(S0 b2 ), Eh (Ih2 ) = Eh (I˜h2 )/(S0 b2 )2 , Ec (Ic2 ) = Ec (I˜c2 )/(S0 b2 )2 and En (In2 ) = En (I˜n2 )/(S0 b2 )2 . To compute the first and second moments of I˜h and I˜c , we rely on the results for the moments of the statistic  M Is−t = M i=1 j =1 dij Yij (dii = 0) under a general random permutation scheme over a set S that has M elements (Cliff and Ord, 1981, p. 36): ER (Is−t ) = S0 ER (Yij )

(6)

2 ER (Is−t ) = S1 ER (Yij2 ) + (S2 − 2S1 )ER (Yij Yik ) + (S02 + S1 − S2 )ER (Yij Ykl ),

(7)

and

where i, j, k, and l index spatial weights between original and permutation R (·) denotes  the M Mregions, and E Mthe mean of a2 M 2 random permutation over S, and where S0d = M d , S = (d + d ) /2, S = ji 2d i=1 j =1 ij 1d i=1 j =1 ij i=1 (di· + d·i ) M M with di· = j =1 dij and d·i = j =1 dj i .

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To simplify the notations, zi =  xi − x¯ for i = 1, . . . , m. Let Eh (·) be the mean of a random permutation  let us denote ¯ k /mh = i∈Ah zik /mh . Straightforwardly, we have over Ah , and let hk = i∈Ah (xi − x) Eh (zi zj ) =

 1 mh (mh − 1)



z i zj

i∈Ah j ∈Ah ,j  =i

= = Eh (zi2 zj2 ) =

1 m(m − 1)



zi (mh h1 − zi )

i∈Ah

mh h21 − h2 , mh − 1

(8)

 1 mh (mh − 1)



i∈Ah j ∈Ah ,j  =i

zi2 zj2 =

mh h22 − h4 , mh − 1

Eh (zi2 zj zk ) =

 1 mh (mh − 1)(mh − 2)



=

 1 mh (mh − 1)(mh − 2)





i∈Ah j ∈Ah ,j  =i k∈Ah ,k =i,j

i∈Ah j ∈Ah ,j  =i

=

(9)

zi2 zj zk

zi2 zj (mh h1 − zi − zj )

 1 zi2 [(mh h1 − zi )2 − mh h2 + zi2 ] mh (mh − 1)(mh − 2) i∈Ah

=

m2h h21 h2

− 2mh h1 h3 − mh h22 + 2h4 , (mh − 1)(mh − 2)

(10)

and Eh (zi zj zk ) =

m2h h31 − 3mh h1 h2 + 2h3 . (mh − 1)(mh − 2)

(11)

Entering Eh (zi2 zj zk ) by expression (10) and Eh (zi zj zk ) by expression (11) in the third equation below, we have Eh (zi zj zk zl ) =

 1 mh (mh − 1)(mh − 2)(mh − 3)



 1 mh (mh − 1)(mh − 2)(mh − 3)







z i zj z k z l

i∈Ah j ∈Ah ,j  =i k∈Ah ,k =i,j l∈Ah ,l =i,j,k

=



zi zj zk (mh h1 − 3zi )

i∈Ah j ∈Ah ,j  =i k∈Ah ,k =i,j

=

3 mh h1 Eh (zi zj zk ) − Eh (zi2 zj zk ) mh − 3 mh − 3

=

m3h h41 − 6m2h h21 h2 + 8mh h1 h3 + 3mh h22 − 6h4 . (mh − 1)(mh − 2)(mh − 3)

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Let Yij = zi zj , Yik = zi zk and Ykl = zk zl for different i, j, k, l in Eqs. (6) and (7). Enter Eh (zi zj ) by expression (8) in Eq. (6) and enter Eh (zi2 zj2 ) by expression (9), Eh (zi2 zj zk ) by expression (10) and Eh (zi zj zk zl ) by expression (12) in Eq. (7). Let S = Ah , M = #(Ah ) = mh , dij = wij , and Yij = zi zj . We have the expressions of Eh (I˜h ) and Vh (I˜h ), and the expressions of Eh (Ih ) and Vh (Ih ) in Eqs. (4) and (5). Similarly, by letting S = Ac , M = mc = #Ac , andYij = (xi − x)(x ¯ j − x), ¯ we have Ec (I˜c ) and Vc (I˜c ) as well as the expressions of Ec (Ic ) and Vc (Ic ) in Eqs. (4) and (5).

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The moments of I˜n are determined jointly by the product of the random permutations for xi − x¯ on Ah and for xj − x¯ on Ac in Eq. (3). The mean of I˜n is En (In ) =

 

(wij + wj i )Eh (xi − x)E ¯ c (xj − x) ¯ = S0n h1 c1 .

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i∈Ah j ∈Ac

Then, we have the expression for En (In ) in Eq. (4). To compute the second moment of I˜n , we use the following identities:    (wij + wj i )(wil + wli ) = S2n − S1n , i∈Ah j ∈Ac l∈Ac ,l=j

 



(wij + wj i )(wkj + wkj ) = S3n − S1n ,

j ∈Ac i∈Ah k∈Ah ,k =i

and  





2 − S2n − S3n + S1n . (wij + wj i )(wkl + wlk ) = S0n

i∈Ah j ∈Ac k∈Ah ,k =i l∈Ac ,l =j

Based on the following calculation, we have En (In2 ) = S1n Eh [(xi − x) ¯ 2 ]Ec [(xj − x) ¯ 2 ] + (S2n − S1n )Eh [(xi − x) ¯ 2 ]Ec [(xj − x)(x ¯ l − x)] ¯ + (S3n − S1n )Eh [(xi − x)(x ¯ k − x)]E ¯ ¯ 2] c [(xj − x) 2 + (S0n − S2n − S3n + S1n )Eh [(xi − x)(x ¯ k − x)]E ¯ ¯ l − x)]. ¯ c [(xj − x)(x

We derive the expression for Vn (In ) in Eq. (5). Appendix B. Asymptotic normality To prove the asymptotic normality of Ih , Ic and In in Theorem 1, we rely on the Martingale Central Limit Theorem (Billingsley, 1995, p. 478). Since the proofs for Ih , Ic and In are almost identical, we opt to show the proof for Ih by assuming the regularity conditions (C1)–(C3). Suppose that X1 , . . . , Xm are iid copy of a nondegenerate random variable X with the finite fourth-moment. Denote = E(X), 2 = V (X), p = P (X  ), h = E[(X − )+ ], 2h = V [(X − )+ ] and kh = E{[(X − )+ − h ]k }. Let  m m m m m 2 2 wi· = m j =1 wij , S0 = S0m = i=1 j =1 wij , S1 = S1m = i=1 j =1 (wij + wj i ) /2 and S2 = S2m = i=1 (wi· + w·i ) . + We write a ∧ b = min(a, b) and a ∨ b = max(a, b) for any real numbers a and b. We always denote Ui = (Xi − ) − h P

and Vi = IXi  − p. The asymptotic is evaluated under m → ∞, and we write → as convergence in probability, L

and →as convergence in distribution. P

P

P

Lemma 1. (S0h − p 2 S0 )/m → 0, (S1h − p 2 S1 )/m → 0 and [S2h − p 3 S2 − 2p 2 (1 − p)S1 ]/m → 0. Proof. A proof of this can be obtained by simply using the Chebyshev inequality and we omit it. P

P

P



P

Lemma 2. As m → ∞, h1 → h /p, h2 →( 2h + 2h )/p, h3 → 3h /p and h4 → 4h /p. P ¯ + (Xi − )+ ||X¯ − |. We have Proof. Note that mh is approximate Bin(m, p) so mh /m → ∞ and |(Xi − X) P

P

mh h1 /m → E[(Xi − )+ ] so h1 → h /p as m → ∞. Similarly, we can prove the rest of the statements.



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In the following, we only provide the main conclusions and a couple of important proofs. √ Lemma 3. I˜h / m = M1m + M˜ 1m + op (1), where ⎫ ⎧ m m  m ⎬ ⎨   1 ¯ wij Ui Uj + 2 h wi· Ui + 2p h S0 ( − X) , M1m = √ ⎭ m⎩ i=1 j =1

i=1

1 M˜ 1m = √ S0 2h . m

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Proof. Straightforwardly, we have I˜h √ = M1m + M˜ 1m + r1m + r2m + r3m + r4m + r5m + r6m , m

(15)

where 1  ¯ + − (Xi − )+ ][(Xj − X) ¯ + − (Xj − )+ ], wij [(Xi − X) r1m = √ m m

m

i=1 j =1

2  ¯ + − (Xj − )+ + (X¯ − )I ¯ (Xj )], r2m = √ wij Ui [(Xj − X) [ ∨X,∞) m m

m

i=1 j =1

¯  2( − X) wij Ui [I[ ∨X,∞) (Xj ) − p], √ ¯ m m

m

r3m =

i=1 j =1

m ¯  2p( − X) r4m = wi· Ui , √ m i=1

2  ¯ + − (Xi − )+ + (X¯ − )I ¯ (Xi )], r5m = √ h wi· [(Xi − X) [ ∨X,∞) m m

i=1

¯  2 h ( − X) wi· [I[ ∨X,∞) (Xi ) − p]. √ ¯ m m

r6m =

(16)

i=1

It is sufficient to show that r1m , r2m , r3m , r4m , r5m and r6m approach 0 in probability. The idea of the proof is to use the √ L ¯ + − (Xi − )+ | | − X| ¯ Chebyshev inequality based on the following three facts: m(X¯ − ) → N(0, 2 ), |(Xi − X) and ¯ + − (Xi − )+ + (X¯ − )I ¯ (Xi )] E|( − X)I ¯ ( ∧ X¯ Xi < ∨ X)| ¯ E[(Xi − X) [ ∨X,∞) → 0, as m → ∞.



We decided to state Lemmas 4 and 5 because their proofs are almost identical to the proof of Lemma 3.

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√ Lemma 4. Eh (I˜h )/ m = M2 + M˜ 2 + op (1), where ⎡ M2 = M˜ 2 =

S0 ⎣ m5/2

m m  

Ui Uj + 2 h m

i=1 j =1,j  =i

m 

⎤ ¯ ⎦, Ui + 2p h m2 ( − X)

i=1

m(m − 1) 2h S0h √ . mh (mh − 1) m

Lemma 5. M˜ 2 − M˜ 1 =

p

2h √ 2

m  m 

m

i=1 j =1

2 2  w˜ ij Vi Vj + √h w˜ i· Vi + op (1), p m m

i=1

where w˜ ij = wij − S0 /[m(m − 1)] if i  = j and w˜ ii = 0, and w˜ i· =

m

˜ ij . j =1 w

The rest of the work is to show the asymptotic normality of Tm = M1m − M2m + M˜ 1m − M˜ 2m     m m m 2h 2 h  1  h =√ w˜ ij Ui Uj − 2 Vi Vj + √ w˜ i· Ui − Vi . p p m m i=1 j =1

(17)

i=1

Note that E(Vi ) = 0, V (Vi ) = p(1 − p), E(Ui ) = 0, V (Ui ) = 2h and E(Ui Vi ) = h (1 − p). We have E(Tm ) = 0 and     4 (1 − p)2 4 (1 − p)2 2 (1 − p) 2 (1 − p) ˜ 2 2 S 2 2 S˜1 h +

4h + h 2

2h + h − h − h 2 V (Tm ) = m p p m p p      4 (1 − p)2 4 (1 − p) 4S 2 S1 S2 = + 2h 2h − h

4h − h 2 − 20 + o(1), m p m m p where S˜0 , S˜1 and S2 are defined as same as S0 , S1 and S2 by replacing wij with w˜ ij . √ P Lemma 6. [Vh (I˜h / m) − V (Tm )] → 0. Proof. Ignoring the small order terms based on Lemma 2, we have  Vh

I˜h √ m



S1h (h2 − h21 )2 + = m



S2h 4S0h − 2 m m

 (h21 h2 − h41 ) + op (1).

Also based on Lemma 1, we have  Vh

I˜h √ m



     4 (1 − p)2 4 (1 − p) 2 4S S S1 2 + 2h 2h − h

4h − h 2 − 20 + op (1). m m p m p

√ P Therefore, Vh (I˜h / m) − V (Tm ) → 0.



(18)

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Theorem 2. Ih − Eh (Ih ) L → N (0, 1). √ Vh (Ih ) √ L Proof. Condition (C3) ensures that limm→∞ V (Tm ) exists and is positive. To show Tm / V (Tm ) → N(0, 1) by the Martingale Central Limit Theorem, we define the martingale by     k k k 2 2h  2h 1  h w˜ ij Ui Uj − 2 Vi Vj + √ w˜ i· Ui − Vi Tmk = √ p p m p m i=1 j =1

i=1

for k = 1, 2, . . . , m and define Tm0 = 0. The limiting distribution can be implied by Lindeberg–Feller condition (Billingsley, 1995, p. 376), which can also be implied by Lyapnounov condition (Billingsley, 1995, p. 362). Let m Zmk = Tmk − Tm,k−1 be the martingale difference. If there is an > 0 such that limm→∞ i=1 E(|Zmk |2+ ) = 0, then 4 ) = O(1/m2 ). This implies Lyapnounov condition holds. By straightforward computation (omitted), we have E(Zmk the Lyapnounov condition and therefore the conclusion of the theorem.  References Anselin, L., 1995. Local indicators of spatial association-LISA. Geographical Anal. 27, 93–115. Anselin, L., Griffith, D.A., 1988. Do spatial effects really matter in regression analysis? Paper of Regional Sci. Assoc. 65, 11–34. Anselin, L., Cohen, J., Cook, D., Gorr, W., Tita, G., 2000. Spatial analyses of crime. In: Duffee, D. (Ed.), Criminal Justice 2000, Volume 4, Measurement and Analysis of Crime and Justice. National Institute of Justice, Washington, DC, pp. 213–262. Arbia, G., 1989. Spatial Data Configuration in Statistical Analysis of Regional Economic and Related Problems. Kluwer Academic Publishers, Dordrecht. Assuncao, R., Reis, E., 1999. A new proposal to adjust Moran’s I for population density. Statist. Med. 18, 2147–2162. Bennett, R.J., Haining, R.P., 1985. Spatial structure and spatial interaction modeling approaches to the statistical analysis of geographic data. J. Roy. Statist. Soc. A 48, 1–36. Billingsley, P., 1995. Probability and Measure. Wiley, New York. Brunsdon, C., Aitkin, M., Fortheringham, S., Charlton, M., 1999. A comparison of random coefficient modeling and geographically weighted regression for spatial non-stationary regression problems. Geographical Environ. Modeling 3, 47–62. Burr, T., 2001. Maximally selected measure of evidence of disease clusters. Statist. Med. 20, 1443–1460. Cliff, A.D., Ord, J.K., 1972. Testing for spatial autocorrelation among regression residuals. Geographical Anal. 4, 267–284. Cliff, A.D., Ord, J.K., 1981. Spatial Processes: Models and Applications. Pion, London. Cressie, N., Chan, N., 1989. Spatial modeling of regional variables. J. Amer. Statist. Assoc. 84, 393–401. Fotheringham, S., 1997. Trends in quantitative geography: I: stressing the local. Progr. Human Geography 21, 88–96. Fotheringham, S., 1999. Guest editorial: local modeling. Geographical Environ. Modeling 3, 5–7. Getis, A., Aldstadt, J., 2004. Constructing the spatial weights matrix using a local statistic. Geographical Anal. 36, 90–104. Getis, A., Ord, J., 1992. The analysis of spatial association by use of distance statistics. Geographical Anal. 24, 189–206. Griffith, D., 2003. Spatial Autocorrelation and Spatial Filtering. Springer, Berlin. Haining, R., 1990. Spatial Data Analysis in the Social and Environmental Sciences. Cambridge University Press, Cambridge, UK. Kulldorff, M., 1997. A spatial scan statistic. Comm. Statist. Theory Methods 26, 1481–1496. Kulldorff, M., 2006. SatScan user guide. www.satscan.org. (accessed November 2006). Lawson, A.B., Denison, D.G.T., 2000. Spatial Clustering Modeling. CRC Press, New York. Lawson, A.B., Kulldorff, M., 2000. A Review of Cluster Detection Methods in Disease Mapping and Risk Assessment for Public Health. Wiley, New York. Lee, S.I., 2004. A generalized significance testing method for global measures of spatial association: an extension of the Mantel test. Environ. Plann. A 36, 1687–1703. Lin, G., Zhang, T., 2004. A method for testing low-value spatial clustering for rare disease. Acta Tropica 91, 279–289. Moran, P.A.P., 1948. The interpretation of statistical maps. J. Roy. Statist. Soc. Ser. B 10, 243–251. Moran, P.A.P., 1950. A test for the serial independence of residuals. Biometrika 37, 178–181. Oden, N., 1995. Adjusting Moran’s I for population density. Statist. Med. 14, 17–26. Ord, J.K., Getis, A., 2001. Testing for local spatial autocorrelation in the presence of global autocorrelation. J. Regional Sci. 41, 411–432. Rogerson, P.A., Bohning, D., 1999. The detection of clusters using a spatial version of the chi-square goodness of fit statistics. Geographical Anal. 31, 130–147. Schlattmann, P., Bohning, D., 1993. Mixtures models and disease mapping. Statist. Med. 46, 351–357. Schmoyer, R.L., 1994. Permutation tests form correlation in regression errors. J. Amer. Statist. Assoc. 89, 1507–1516. Sen, A., 1976. Large sample-size distribution of statistics used in testing for spatial correlation. Geographical Anal. 9, 175–184. Sokal, R., Oden, N.L., Thomson, B.A., 1998. Local spatial autocorrelation in a biological model. Geographical Anal. 30, 331–351.

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