For a stationary point process 𝒫 with intensity λ(ℓ)=λ and a distance r>0, Ripley’s K-function is defined as Kr=λ-1ENcℓ,r-1. where ℓ is any point arising from 𝒫 and c(ℓ,r) is the circle centered at ℓ with radius r [49]. Thus K(r) is the expected number of additional points within a distance of r of any point in 𝒫, re-scaled by the intensity of 𝒫. For a homogeneous Poisson process (realizations of which exhibit complete spatial randomness) on an infinite study area, K(r)=πr2. For a high-level overview of Ripley’s K-function, see [20]; for a more in depth treatment of Ripley’s K-function and related topics in point processes, see [8] or [12].

While the original formulation of Ripley’s K-function assumes a stationary point process, the definition has been extended to handle certain types of nonstationary processes [4], and many related fucntions have been developed to further quantify the behavior of point processes. For instance, the L-function is a rescaling of the K-function, and the K_d-function is a kernel estimator of the probability density function of distances between points [22]. Variants of Ripley’s K-function have also been developed for marked point processes (MPP), i.e., point processes for which each point is associated with a random value called a mark. The D-function is defined for MPP with binary marks, and consists of the difference between the K-function computed on points with one type of mark and the K-function computed on the remaining points [19, 2]. The M-function quantifies the pattern in points with a particular type of mark relative to other points by weighting the number of points of the type in question within a distance r of a given point by the total number of points within distance r [42]. The cross K-function gives the expected number of observed marks within a distance r of a given mark, where the observed mark types must be different than the given mark. The Kmm-function quantifies the correlation between marks of points separated by a distance r [47]. For an effective overview of the practical use and interpretation of these functions, see [41].

Suppose that we have a realization of 𝒫 consisting of n observations, ℓ1,ℓ2,…,ℓn. We can estimate K(r) with Kˆ(r)=λˆ-1∑i=1n∑j=1,j≠inwijn where λˆ=n/A(𝒜), A(𝒜) denotes the area of 𝒜, and wij is a weight associated with points i and j. In the traditional approach, wij=1 if the distance between points i and j is less than r and 0 otherwise [49, 50]. In practice, the traditional estimator generally exhibits some bias due to edge effects. This phenomena arises because Ncℓi,r is often lower than expected for ℓi near the boundary of 𝒜, as some points which would otherwise contributed to Ncℓi,r fall outside of 𝒜. However, many adjusted estimators of Ripley’s K-function exist which modify the wij to account for edge effects [49, 18, 27, 1].

Ripley’s K-function is often used to determine if an observed collection of points exhibits complete spatial randomness (CSR) (i.e., to determine if the points arise from a two-dimensional homogeneous Poisson process). The distribution of Kˆ(r) under CSR for a finite study area 𝒜 can be approximated with Monte Carlo simulations, which are used to perform a hypothesis test with the null hypothesis being that the observed data arises from a homogeneous Poisson process with rate parameter λˆ. In practice, these Monte Carlo simulations are often carried out conditional on a fixed number of observations (n), in which case the null hypothesis is that the data arise from a two dimensional continuous uniform distribution. Large values of Kˆ(r) indicate spatial clustering, that is, the number of points within a distance of r of any given point is larger than would be expected if the data exhibited CSR. Small values of Kˆ(r) indicate dispersion, that is, the number of points within a distance of r of any given point is smaller than would be expected under CSR. While Ripley’s K-function is most commonly used to test a null hypothesis of CSR, it can be used to test more complicated null hypotheses, such as that the data arise from a Neyman-Scott process [18] or a Strauss process [14]. Ripley’s K-function-based hypothesis tests are an attractive tool for spatial data analysis because of their flexibility and interpretability. Ripley’s K-function can simultaneously detect different spatial patterns at different distances (e.g., small distance dispersion combined with large distance clustering), which is a highly desirable and somewhat rare property among cluster detection methods.

In this work, we assume we have a set of spatial regions, referred to as areal units, whose boundaries are fixed and known (e.g. the census tracts in a particular state). We observe a binary response variable for each of these areal units (e.g. whether the census tract is a food desert) which we refer to as ‘binary areal data’ to distinguish it from areal data that are associated with one or more non-binary numeric attributes. For brevity, we will refer to areal units for which this binary random variable is equal to 1 as positive units, as they are positive for the characteristic of interest. The location and boundaries of the areal units are considered fixed, with the binary random variable serving as a random ‘mark’. To parallel the point process case, we loosely define clustering for binary areal data as an excess of positive units in a particular area and dispersion as positive units occurring a semi-regular intervals. The terms ‘clustered data’ or ‘clustering’ are sometimes used for data with positive spatial autocorrelation. For example, census tracts with a high rates of food insecurity might tend to be closer to other tracts with high rates. In binary areal data, there is no meaningful distinction between spatial clustering (an excess of positive units) and spatial autocorrelation (a higher concentration of similar values of the random variable).

We will define an areal process 𝒜 on a Borel subset 𝒜⊆R2 as a collection of N disjoint Borel sets a1,a2,…,aN whose union covers 𝒜 and a vector Y=Y1,Y2,…,YN′ where Yi is a binary random variable associated with ai. We will refer to the ai as areal units, and the set of ai for which Yi=1 as positive areal units. For any Borel subset 𝒮 of 𝒜, define 𝒩(𝒮)=∑i=1NA𝒮∩aiYi. Thus 𝒩(𝒮) is the amount of area in 𝒮 which falls into positive areal units. Note that 𝒩(⋅) maybe thought of as the areal analog of N(⋅) in the point process case: N(⋅) counts observed points, 𝒩(⋅) counts observed area.

Given an areal process 𝒜, define the positive area proportion function for an areal unit ai at a distance r>0 as Mi(r)=E𝒩cℓi,r∩aic+Acℓi,r∩aiAcℓi,r∩𝒜⋅𝒩(𝒜)A(𝒜)-1∣nY>0. where ℓi is the centroid of ai and nY is the number of positive units. Note that conditioning on nY is necessary for the expectation to be defined, as 𝒩(𝒜)=0 if nY=0. The first term in the expectation is the proportion of positive area within a distance of r of ℓi where the numerator captures area that either falls in ai(A[⋅]) or is positive (𝒩[⋅]). The second term is the inverse of the total proportion of the study area which is positive.

Binary areal data is clustered if positive areas tend to occur near other positive areas. Here, Mi(r) is used to quantify the expected amount of additional positive area near ai. This quantity only meaningfully assesses clustering near ai if ai is positive. Just estimators of Ripley’s K-function are only computed at observed points, estimators of Mi(r) are only computed for positive units ai. The inclusion of Acℓi,r∩ai in the numerator of the first term ensures that sample based estimators, which are only computed if ai is positive, are unbiased for Mi(r). Just as Ripley’s K-function quantifies the number of points within a certain distance of an observed point, the positive area proportion function quantifies the amount of positive area within a certain distance of a positive unit’s centroid. See Figure 1 for an illustration of the quantities involved in Mi(r).

Heuristically, Mi(r) is loosely analogous to an edge-corrected Ripley’s K-function, with positive area playing the role of observed points. However, the inherent areal structure makes Mi(r) dependent on the choice of the positive unit ai, while Ripley’s K-function does not depend on the choice of the observed point. To remove this dependence on the choice of ai, we can average the Mi(r) values over the positive units. Define the positive area proportion function at a distance r by M(r)=E1nY∑i=1NMi(r)Yi∣nY>0 where the expectation is taken over Y. Thus, M(r) is the expected value of the average of the positive area proportion function of the positive units. Allow M0i(r) and M0(r) to denote Mi(r) and M(r) for a stationary independent process, respectively. It can be shown that M0(r)=1N∑i=1NM0i(r). See Web Appendix A for more details.

For an areal process realization with Y=y and ny>0, we can estimate Mi(r) with M^i(r,y)=𝒩cℓi,r∩aic+Acℓi,r∩aiAcℓi,r∩𝒜𝒩(𝒜)A(𝒜)-1. Note that by definition E[M^i(r∣Y)∣nY>0]=Mi(r). Further, define M^r,y=1ny∑i:yi=1M^ir,y. as the sample mean of the M^i(r,y) s for each positive unit. It can be shown that if 𝒜 is a stationary, independent process, then E[M^(r,Y)∣nY>0]=M0(r). This fact forms the crux of the hypothesis testing procedure presented in the next section. See Web Appendix A for additional details.

Suppose we have a realization y from an areal process 𝒜, and we wish to test the null hypothesis that 𝒜 is a stationary, independent areal process against the alternative hypothesis that 𝒜 exhibits clustering and/or dispersion. To reduce the variability of the assumed null distribution, we will condition on the number of positive units in the realization. Conditional on nY=n, the null hypothesis of a stationary, independent process implies that P(Y=y)=Nn-1 for y:∑i=1Nyi=n and P(Y=y)=0 otherwise. The null hypothesis is thus equivalent to the so-called ‘random labeling hypothesis’, under which all configurations of n positive units are equally likely.

A clustered areal process might violate the stationarity assumption, or the independence assumption, or both. Formally, we consider an areal process to exhibit excess-clustering if there exists a subset of contiguous areal units indexed by 𝒞 such that for all i∈𝒞,PYi=1>PYj=1 for j∉C. That is, 𝒜 exhibits excess clustering if the probability of any areal unit in 𝒞 being positive is higher the probability of a unit out of 𝒞 being positive. An areal process exhibiting excess-clustering is not stationary, though it could be independent.

We will consider an areal process to exhibit autocorrelated-clustering if there exists at least one areal unit ai with at least one neighbor aj such that Yj=1∣Yi=1>PYj=1, that is Y exhibits positive spatial autocorrelation. Here the definition of neighbor can be taken to be any desired measure of proximity (shared border, centroids within a certain distance, etc.). Autocorrelated-clustering violates the independence assumption, but not necessarily the stationarity assumption. For the purposes of developing a hypothesis testing procedure, we will consider an areal process to be clustered if it exhibits excess-clustering or autocorrelated-clustering or both. Note that unless more than one realization of the same areal process is observed (which is rare), it will not be possible to distinguish between the two types of clustering.

Consider the following test statistic Tn(r,y)=M^(r,y)-M0(r,n) where n=∑i=1Nyi, M0(r,n)=E[1nY∑i=1NMi0(r,n)Yi∣nY=n] and Mi0(r,n)=E𝒩cℓi,r∩aic+Acℓi,r∩aiAcℓi,r∩𝒜⋅𝒩(𝒜)A(𝒜)-1∣nY=n for a stationary, independent process. Under the null hypothesis, ETn(r,Y)∣nY=n=0. Positive values of Tn(r,y) suggest clustering at distance r. Data which exhibits excess-clustering will have a larger-than-expected number of positive units in 𝒞, which will tend to inflate the values of M^i(r,y) for i∈𝒞 and thus inflate M^(r,y). Data which exhibits autocorrelated-clustering will have a larger than expected number of positive units with other positive units nearby, which will also inflate M^(r,y). When the null hypothesis is rejected in favor of clustering at distance r, we conclude that there is more area within a distance of r of each positive unit centroid than expected under a stationary, independent process.

The null distribution of Tn(r,⋅) can be estimated using a Monte Carlo procedure in which data is generated under the random labeling hypothesis, that is, the Y values are generated by selecting n positive areal units with equal probability. Note that M0(r,n) can be computed exactly, but doing so requires evaluating NNn-dimensional sums. Additionally, obtaining the terms in these sum requires computing the area of polygon intersections, which is generally computationally intense. As an alternative, we propose approximating M0(r,n) by averaging the M^(r,y) values from the same Monte Carlo procedure used to estimate the null distribution of Tn(r,⋅). An α-level test for clustering at distance r can then be conducted by rejecting the null hypothesis if Tn(r,y) exceeds 1-α quantile estimated from the simulated null distribution.

The same test statistic Tn(r,⋅) can be used to detect dispersion. Dispersion is a difficult phenomena to precisely quantify. Intuitively, an areal process is dispersed if positive areal units tend to be located further away from other positive units than would be expected for a stationary, independent process. To produce a working definition of dispersion, suppose we have defined a neighbor structure on the areal units. We will consider an areal process to exhibit buffered-dispersion if there exists a set of non-neighboring unit(s) indexed by 𝒟 such that for all i∈𝒟 and all neighbors aj of ai,PYi=1>PYj=1. Here, the units in 𝒟 are surrounded by ‘buffer units’ with a lower probability of being positive. An areal process which exhibits buffer-dispersion is not stationary, but it could be independent. We will consider an areal process to exhibit autocorrelated-dispersion if there exists at least one areal unit ai with at least one neighbor aj such that PYj=1∣Yi=1<PYj=1. Autocorrelated dispersed areal processes are not independent, though they may be stationary. For the purposes of hypothesis testing, we will consider an areal process dispersed if it is either buffer-dispersed or autocorrelated-dispersed. While this definition likely does not cover all processes which could give rise to dispersed areal data, it does cover two most common cases of areal unit(s) with a high probability of being positive surrounded by areal units with lower probability of being positive and the case of negative spatial autocorrelation.

If an areal process is dispersed, then at least some of the positive units have fewer positive units nearby than we would expect for a stationary, independent process. These units will tend have to have lower than expected M^i(r,y) values, decreasing M^(r,y). We can perform an α-level test of the null hypothesis that 𝒜 is a stationary, independent process against the alternative hypothesis that 𝒜 exhibits dispersion by rejecting the null hypothesis if Tn(r,y) falls below the α-quantile of the null distribution of Tn(r,⋅), which can be estimated using the Monte Carlo procedure described previously. If we reject the null hypothesis in favor of dispersion, we conclude that the amount of positive areal within a distance r of positive unit centroids is less than expected for stationary independent process. Finally, an α-level two-tailed test for either clustering or dispersion can be conducted by rejecting the null hypothesis if Tn(r,y) is less than the estimated α/2 quantile of the null distribution or if Tn(r,y) is greater than the 1-α/2 quantile.

In many applications, the ideal distance at which to test for spatial patterns may not be known. Computing the PAPF test statistic at a variety of radii induces a multiple testing problem and the potential for an inflated type I error rate. To control the overall type I error rate associated with such a procedure, we propose the following global test for clustering over a range of distances r∈ℛ=r1,…,rR. Define the global test statistic TnC(y)=maxr∈ℛTn(r,y)/Snr where Snr=var⁡Tn(r,⋅)1/2. We can estimate the null distribution of TnC(⋅) within the Monte Carlo procedure discussed above. That is, Tn(r,y) is computed for each r∈ℛ and TnC(⋅) is computed for each dataset. Note that we can estimate var⁡Tn(r,⋅) from the same Monte Carlo procedure. An upper tailed test is indicative of clustering. To test for dispersion, define the test statistic TnD(Y)=minr∈ℛTn(r,y)/Snr. The null distribution of TnD(⋅) can be estimated analogously to that of TCn(⋅) and a lower tailed test is indicative of dispersion. To simultaneously test for either clustering or dispersion, both tests can be performed and a Bonferroni correction for two tests applied. Similar methods have been proposed to derive global tests from Ripley’s K-function and Ripley’s D-function [19].

In this section, we perform an extensive simulation study to compare the performance our proposed PAPF hypothesis testing method to the performance of the global Moran’s I statistic, the Getis-Ord general G statistic, and the spatial scan statistic. We also consider the performance of the misapplication of three point process methods to the areal unit centroids: a edge-corrected Ripley’s K-function test, a related test based on Ripley’s D-function [19], and the average nearest neighbor method [13]. These misapplications are included to highlight the importance of analyzing areal data only with methods designed for areal data. After describing our data generation procedures, we provide details on the implementation of each method.

We consider the performance of our proposed hypothesis testing procedure using two study areas which are shown in Figure 2:

𝒜1 A 20 by 20 regular grid of N₁ = 400 cells

𝒜2 The N₂ = 3, 108 counties (and county-equivalents) in the contiguous US.

We define three distributions that will be used to sample the observed units. For each, areal unit ai is positive when Yi=1. First, Y~SWoR⁡(N,k,p) indicates that the random variable Y=Y1,…,YN arises by selecting k elements from {1,2,…,N} via sampling without replacement (SWoR) where p=p1,…,pN′ gives the probability of selecting each element, and Yi=1 if i was selected and 0 otherwise. Second, Y~C(k,m,q) denotes that Y is generated using the following two-step process where Yi=1 if element i was selected in either step:

m elements of {1,2,…,N} are randomly selected via SWoR⁡N,m,p1 where p1i=N-1 for all i.

Third we define Y~M(n,m,q,k) if Y is generated as follows:

Divide the study area 𝒜 into three regions: a clustered region Rc, a dispersed region Rd and a random scatter region Rr.

For the first distribution, p can be used to to generate data under the null hypothesis or data with clustering in certain locations where the pi’s are larger. For the second distribution, q controls the degree of clustering or dispersion where larger q>1 lead to more clustering while smaller q<1 lead to more dispersion. The third distribution produces an area of small-scale clustering in Rc and an area of small-scale dispersion in Rd. When k is sufficiently large relatively to Nc, this process will also produce clustering at larger distances in Rc.

For each study area 𝒜j,j=1,2, we generate data under 21 different scenarios. For brevity, we remove the subscript j when describing these scenarios (all depend on Nj). First we consider the null hypothesis of a stationary independent areal process (e.g. no spatial pattern in the positive units) via the following three scenarios: I1:Y~SWoR⁡(N,⌈N/10⌉,p) I2:Y~SWoR⁡(N,⌈N/4⌉,p) I3:Y~SWoR⁡(N,⌈N/2⌉,p) where p=p1,…,pN′=(1/N,…,1/N)′.

We also consider 12 scenarios in which the locations of observed units are clustered. The following 6 scenarios assess the ability to excess-clustering: C1:Y~SWoR⁡N,⌈N/10⌉,p1 C2:Y~SWoR⁡N,⌈N/10⌉,p2 C3:Y~SWoR⁡N,⌈N/4⌉,p3 C4:Y~SWoR⁡N,N4,p4 C5:Y~SWoR⁡N,⌈N/2⌉,p5 C6:Y~SWoR⁡N,⌈N/2⌉,p6. The N-dimensional vectors pl=pl1,…,plN′ for l=1,…,6 are defined as follows: an entry of pl is equal to q/D if the unit is shown in blue in Figure 3 and equal to 1/D otherwise where D is such that ∑ipli=1. For l∈{1,3,5} we take q=5 and for l∈{2,4,6} we take q=10. Thus the blue units are 5 times more likely to be observed than the white units under C1,C3 and C5, and 10 times more likely to be observed under C2,C4 and C6.

Next, we assess the ability of our hypothesis test to autocorrelated-clustering by considering the following 6 scenarios: C7:Y~C(⌈N/10⌉,⌈N/100⌉,5) C8:Y~C(⌈N/10⌉,⌈N/100⌉,10) C9:Y~C(⌈N/4⌉,⌈N/40⌉,5) C10:Y~C(⌈N/4⌉,⌈N/40⌉,10) C11:Y~C(⌈N/2⌉,⌈N/20⌉,5) C12:Y~C(⌈N/2⌉,⌈N/20⌉,10) Note that C7,C9 and C11 correspond to ‘weaker’ clustering, in the sense that they tend to select fewer adjacent units than mechanisms C8,C10 and C12

Next, we assess the ability of our hypothesis testing procedure to detect spatial dispersion under the following 4 scenarios. D1:Y~C⌈N/10⌉,⌈N/100⌉,110 D2:Y~C(⌈N/10⌉,⌈N/100⌉,0) D3:Y~C⌈N/6⌉,⌈N/100⌉,110 D4:Y~C(⌈N/6⌉,⌈N/100⌉,0) Here, in D1 and D3 adjacent units are one tenth as likely to be observed as non-adjacent units, creating a mild dispersion effect. Under D2 and D4 adjacent units cannot be selected at all, creating a stronger dispersion effect. Finally, we consider only two samples sizes when assessing dispersion (N/10 and N/6) because it becomes increasingly difficult or impossible to select only non-adjacent units as number of selected units increases.

Finally, we assess the ability of the PAPF method to detect the simultaneous presence of spatial clustering and dispersion at different distances using the following two scenarios: M1:Y~M⌈N/10⌉,mj1,10,3 M2:Y~M⌈N/6⌉,mj2,10,3 We take m11=9,m21=13,m12=60 and m22=100. The regions Rc,Rd and Rr defined for our two study areas are shown in Figure 3. Examples of data generated each scenario for 𝒜1 are shown in Figure 4. Web Figure 1 provides similar examples for 𝒜2.

All hypothesis tests were performed with at a level of α=0.05. For each study area, both global and radii-specific PAPF tests, Ripley’s K tests and Ripley’s D tests were performed. The radii-specifc PAPF tests were performed for 10 radii, r∈ℛj=rj1,…,rj10. The set of radii used for each study area are shown in Figure 2. The smallest radius was equal the smallest distance between any areal unit centroids, and the radii were increased incrementally, with the largest radii being approximately one fourth of the width of the study area.

Tables 2 and 3 summarize selected results from study area 𝒜1 (the regular grid) and study area 𝒜2 (the US counties), respectively. The remaining results can be found in Web Tables 1 and 2. Each table reports the empirical rate of rejection for the null hypothesis. For scenarios I1,I2 and I3, this quantity is the empirical type I error rate; for the other scenarios, this quantity is the empirical power. As the PAPF, Ripley’s K and Ripley’s D tests were performed at 10 different radii, the rejection rate for each radius is reported separately.

Under the null scenarios (I1,I2 and I3), the empirical type I error rates of the global and radii-specific PAPF methods, the global Moran’s I statistic, the Getis-Ord general G statistic, the spatial scan statistic and the global and radii-specific Ripley’s D methods are within the Monte Carlo margin of error of their nominal levels. However the empirical type I error rate of global and radii-specific Ripley’s K methods and the average nearest neighbor method were highly inflated for most scenarios. This is presumably due to the incongruities between the assumed null hypothesis of these tests and the null hypothesis under which data was generated.

Under the excess clustering scenarios C1-C6, the global PAPF test had 100% empirical power to detect clustering, with the exception of scenario C1 under the regular grid. Additionally, the power of the radii-specific PAPF tests is fairly consistent across radii. For these scenarios, the performance of the global PAPF test is comparable to or better than the performance of the Moran’s I statistic, the Getis-Ord general G statistic and the spatial scan statistic. Interestingly, while the performance of the global and radii-specific Ripley’s D tests is comparable to or better than that of the global and radii-specific PAPF test for DGMs C1-C4, the Ripley’s D test performs quite poorly for the larger sample sizes (DGMs C5-C6) on the regular grid, while the PAPF test continues to perform well.

Under the autocorrelated-clustering DGMs C7-C12 the empirical power of the global PAPF test is generally high and the performance of the PAPF is comparable to that of Moran’s I, Getis-Ord general G, and the scan statistic, with the exception scenarios C7 on the regular grid and C11 on the US counties, for which Moran’s I and Getis-Ord are the top performers. The performance of the PAPF test and the Ripley’s D test are generally comparable for scenarios C7-C10 on the regular grid, but the performance of the Ripley’s D test deteriorates drastically for scenarios C11-C12, while the PAPF test continues to exhibit good performance. Both the global and local PAPF tests tends to outperform their Ripley’s D counterparts for the US county scenarios.

Under the dispersion DGMs D1-D4 the performance of all methods was roughly comparable, except for the smaller sample sizes for the regular grid, for which the performance of the global PAPF statistic was noticeably worse than the others.

Under the mixture of clustering and dispersion scenarios M1-M2, the ability of the global PAPF test and global Ripley’s D tests to detect clustering is quite good for the regular grid, but performance deteriorates noticeably for the US counties. The opposite is true for the spatial scan statistic, which exhibits poor performance (0% power) for the regular grid but excellent performance (100% power) for the US counties. The performance of the global Moran’s I and Getis-Ord general G statistics to detect clustering is poor for all scenarios. The ability of the PAPF test, the Ripley’s D test, and the global Moran’s I statistic to detect dispersion is poor for the regular grid. However the ability of the PAPF test to detect dispersion on the US counties improves dramatically, while the performance of the other methods remain poor.

In summary, of the nine methods considered, only seven (the global and radii-specific PAPF tests, the global Moran’s I statistic, the Getis-Ord general G statistic, the spatial scan statistic and the global and radii-specific Ripley’s D tests) maintained their nominal type I error rates. The empirical power of the PAPF test was generally comparable to or better than all other methods, though there were a few exceptions to this rule.

Only Ripley’s K-function, Ripley’s D-function and the PAPF method allow one to detect distance-specific clustering or dispersion. Thus, if one wishes to test for spatial patterns at a single, specific distance, the radii-specific Ripley’s K, Ripley’s D and PAPF methods are the only potential options. However, the Ripley’s K methods exhibited a severely inflated type I error rate and are thus unreliable. Notably, the performance of the global PAPF test is comparable to or better than that of the global Ripley’s D test for all scenarios. In the some scenarios, the performance of the global PAPF test was 4–6 times better than that of the global Ripley’s D test. The Ripley’s D test is designed for point process data and is here being misapplied to areal data, which may explain some of the strange behavior in it’s results (e.g. worsening performance as the number of positive units increases). The Ripley’s D test compares Ripley’s K-function on the positive units to Ripley’s K-function on the negative units. In scenarios C5 and C6 on the regular grid, the global Ripley’s D test performed very poorly. In these scenarios, almost all of the 200 negative units occur outside the region of excess clustering (shown in blue in Figure 3). As there are 300 total units outside this region, this implies that probability that a unit in this region is negative is roughly 0.66, as opposed to 0.5 under the random labeling hypothesis. This implies that the negative units are also clustering- that is, they are more likely to be in the non-blue region than expected under the null hypothesis. Since Ripley’s D-function compares the degree of clustering in the positive units to the degree of clustering in the negative units, the presence of clustering in both types of units may explain the lack of statistical significance in the global Ripley’s D test. While this clustering of negative units also occurs in scenarios C1-C4, it is less extreme. For example, in scenario C2, the probability of a unit outside the blue region being negative is approximately 0.97, versus 0.9 under the null hypothesis. The generally superior performance of the global PAPF method combined with the fact that Ripley’s D-function is a point process method being misapplied to areal data compel use to recommend the use of the PAPF rather than Ripley’s D-function to detect distance-specific patterns.

To assess the performance of the PAPF method under extremely small sample sizes, a simulation study was conducted using a 4×5 regular grid. In general, the global Ripley’s D test had the highest power to detect clustering and the ability of all global methods to detect dispersion was roughly the same. However power of the radii-specific PAPF tests to detect both clustering and dispersion was generally higher than that of the radii-specific Ripley’s D tests. The full simulation results can be found in Web Table 3.

Another simulation study was conducted to assess the sensitivity of the PAPF method to the dependence on areal unit centroids. In brief, the role of each areal unit’s centroid in the definition of the PAPF was replaced with a randomly choosen point from each areal unit. For full details, see Web Appendix B of the Supplementary Material. The results of this simulation study are found in Web Tables 4 and 5 in the Supplementary Material. The performance of this alternative PAPF method was comparable to that of the standard method.

CEs are a private and generally perpetual form of land conservation that legally severs aspects of private landownership (e.g., development rights, resource extraction, etc.) from a parcel of land [44]. Although a landowner makes an individual decision to place a CE, prior research has indicated spatial clustering of CEs over time, throughout the US [36]. Cumulative and clustered CE use may impact regional ecosystem character by altering the degree of CE parcels’ isolation or connectivity with other ecologically valuable parcels and may change ecologic quality on the CE parcel itself [29]. The greater the mass of clustering and ecological systems integrity, the more impact there may be on the land conversion rates at the county level, and on the decision to leave a parcel in open space (or not), potentially affecting placement of other socially valuable land uses. Recognizing if and where CEs are clustered and linking the social, political, biological, and geographical characteristics to the clustered areas may help elucidate the factors driving CE placement [7].

The Boulder County data consists of 112,819 land parcels in place in 2008. Of these land parcels, 817 were held as CEs. A parcel was considered to be part of a CE if any part of the parcel was part of an easement. Figure 5 depicts the land parcels; CE parcels are shown in blue. Global PAPF tests for clustering and dispersion were applied to the dataset using a Bonferroni correction to maintain an overall significance level of α, along with two-tailed radii-specific tests at 10 different radii, depicted in Figure 5.

In order to apply our method, the distribution of the global and radii-specific PAPF test statistic under the null hypothesis was estimated using 200 Monte Carlo simulations. In each Monte Carlo simulation, 817 parcels were selected via simple random sampling without replacement. The observed global PAPF test statistic for clustering T817C(y) was larger than the 97.5th quantile of it’s estimated null distribution. The radii-specific test for radius r2 (approximately 1.1 miles) was also larger than the 97.5th quantile of it’s null distribution. No other radii-specific tests were significant. These results indicate that parcels which contain CEs are significantly clustered at small distances. The exact test statistics and estimated quantiles can be found in Web Table 6 in the Supplementary Material.

In the context of CEs with a purpose of biological conservation, clustering easements close to one another is one reserve design principle to improve landscape connectivity and combat the adverse effects of habitat fragmentation from human land conversion [17, 31]. Larger and higher quality habitats (particularly on CEs) increase the size and stability of source populations and subsequently increase species dispersal capabilities [32]. Clustering and structural connectivity between conservation areas are not always positive, however, as clustering may also leave these areas vulnerable to spatially autocorrelated extinction pressures, such as diseases, invasive species, stochastic environmental events, or negative effects from localized urban growth [21]. Given that the PAPF method indicated the spatial clustering of CEs at short distances in Boulder County, more detailed landscape connectivity studies focused on functional connectivity may be warranted [6, 54].

Next, we use the PAPF method to determine if US counties with high childhood overweight/obesity rates are spatially clustered. County-level childhood overweight rates were estimated from data collected in the 2016 National Survey of Children’s Health using a multilevel small area estimation approach as described in [56]. A county was considered to have a high overweight rate if its estimated rate exceeded the 75th percentile of all county overweight rates. There are 3,108 counties and county-equivalents in the contiguous US, and 786 of these counties were found to have a high rate of childhood overweight. These counties are shown in blue in Figure 5, along with the radii at which PAPF was applied. As for the CEs data application, an α=0.05 global test for clustering or dispersion was conducted using T786C(y) and T786D(y) and applying a Bonferroni correction. A two-tailed α=.05 level test was also conducted for each radii.

The distribution of the global and radii-specific PAPF test statistics under the null hypothesis was estimated using 200 Monte Carlo simulations. In each Monte Carlo simulation, 786 counties were selected via simple random sampling without replacement. The observed global PAPF test statistic for clustering T786C(y) was larger than the 97.5th quantile of it’s estimated null distribution. All 10 radii-specific test statistics were also greater than the 97.5th quantiles of their null distributions, indicating that counties with high rates of childhood overweight are significantly clustered at small and large distances. As Southeastern and Midwest states tend to have higher overweight and obesity rates than the rest of the country [25, 11], these results are not surprising. The exact test statistics and estimated quantiles can be found in Web Table 7 in the Supplementary Material.