- Open Access
A procedure to characterize geographic distributions of rare disorders in cohorts
© Van Meter et al; licensee BioMed Central Ltd. 2008
- Received: 18 October 2007
- Accepted: 28 May 2008
- Published: 28 May 2008
Individual point data can be analyzed against an entire cohort instead of only sampled controls to accurately picture the geographic distribution of populations at risk for low prevalence diseases. Analyzed as individual points, many smaller clusters with high relative risks (RR) and low empirical p values are indistinguishable from a random distribution. When points are aggregated into areal units, small clusters may result in a larger cluster with a low RR or be lost if divided into pieces included in units of larger populations that show no increased prevalence. Previous simulation studies showed lowered validity of spatial scan tests for true clusters with low RR. Using simulations, this study explored the effects of low cluster RR and areal unit size on local area clustering test (LACT) results, proposing a procedure to improve accuracy of cohort spatial analysis for rare events.
Our simulations demonstrated the relationship of true RR to observed RR and p values with various, randomly located, cluster shapes, areal unit sizes and scanning window shapes in a diverse population distribution. Clusters with RR < 1.7 had elevated observed RRs and high p values.
We propose a cluster identification procedure that applies parallel multiple LACTs, one on point data and three on two distinct sets of areal units created with varying population parameters that minimize the range of population sizes among units. By accepting only clusters identified by all LACTs, having a minimum population size, a minimum relative risk and a maximum p value, this procedure improves the specificity achieved by any one of these tests alone on a cohort study of low prevalence data while retaining sensitivity for small clusters. The procedure is demonstrated on two study regions, each with a five-year cohort of births and cases of a rare developmental disorder.
For truly exploratory research on a rare disorder, false positive clusters can cause costly diverted research efforts. By limiting false positives, this procedure identifies 'crude' clusters that can then be analyzed for known demographic risk factors to focus exploration for geographically-based environmental exposure on areas of otherwise unexplained raised incidence.
- True Cluster
- Areal Unit
- Cluster Shape
- Consensus Cluster
- Primary Cluster
Spatial analysis is being used increasingly to generate hypotheses for non-infectious diseases. As such, it has an important role examining putative environmental risk factors for disorders with uncertain etiology. Two types of tests are commonly used to detect spatial clustering, local area clustering tests (LACTs) and global clustering tests. LACTs identify specific points or areas of clustering, while global clustering tests determine a generalized, non-specific pattern of clustering. Given the potential presence of these two cluster patterns, combined with the fact that spatial tests have relatively low power, it is recommended that the two types of tests be used in combination [1–5].
Simulation studies have further questioned the power of spatial cluster detection methods, especially when the cluster is irregularly shaped [6–10], located near borders or areas of low population density  or the elevated relative risk (RR = incidence proportion inside the cluster/incidence proportion outside the cluster) is small [2, 8, 11].
Further test validity issues arise when analyzing rare conditions, where false positive results are more common. Rare disorders require a large study population, which may be created either by aggregating cases over many years and/or using a large geographical area, to provide adequate cases for spatial analysis. (The possible confounding of time trends over a study period can be analyzed separately.) Spatial cohort data are now available for spatial analysis; however, many methods cannot analyze large datasets, necessitating the aggregation of point information into areal units. Problems associated with areal unit-based tests are that, in addition to a loss of information, a large population range across units can create artifactual patterns [12, 13] and tests may fail to identify actual clusters at the extremes of that range .
In an attempt to minimize test inaccuracies, Jacquez et al  suggested using multiple spatial tests at different resolutions, looking for approximate agreement among different tests at different scales. SaTScan , a local area cluster test (LACT) and Maximized Excess Event Test (MEET) , a global clustering test, are among the most powerful and commonly used tests in their respective categories [3, 6, 8]; however, likely due to SaTScan's application of a fixed shape scanning window, it tends to overestimate the true cluster . Two recently developed, flexibly-shaped LACTs, FleXScan  and Episcan , perform comparably to and are based on the same likelihood method as SaTScan.
The aim of this study was to evaluate a procedure, which combined LACTs with a global area test, in an effort to detect clustering of a rare disorder of unknown etiology. In addition, various scan window shapes, acceptance criteria and aggregation unit sizes were examined for alternative cluster shapes. Evaluation criteria were the procedure's sensitivity, specificity and probability of false positives. These results dictated additional criteria included in the procedure described for exploring rare disorders in cohort data.
Simulation set 1: Effect of areal unit population size
As with any test, the effectiveness of spatial tests for various data may be evaluated based on its sensitivity (SE = locations in true cluster identified as being in a cluster by a LACT/all locations in the true cluster), specificity (SP = locations not in the cluster that are identified as not being in the cluster by a LACT/all locations not in the true cluster) and whether there are strong influences due to a few cases.
In the first set of simulations, SE and SP were measured both for the actual cluster locations and for all areal units containing part of the cluster. As expected, the exact point-based median SE was higher than the areal unit-based median SE and the opposite was true for the median SP. All of the following results are point-based median SE and SP. Description of the simulation method follows in the Methods section.
The largest increases for median SE with each doubling of nmax were in the poor results at RR = 1.5. At RRs of 2.5 and 5.0, increases in median SE with increased nmax are much smaller, especially in the circular cluster simulations. For the rectangular clusters, the median SE for nmax = 500 (both scans below 0.30) was very low and the median SP for nmax = 4,000 (both scans below 0.965) was low for a rare event.
Simulation set 2: Effect of RR
The important epidemiologic questions of the validity of the observed RR and p value are illustrated in Figure 4. For the circular cluster, the median observed RR converged to the true RR, when the latter was 1.7. For the rectangular cluster, convergence occurred at a true RR of 2.0, though it was near at 1.7. The cluster tests overestimated the RR when the true RR was low and underestimated when it was high. The magnitude of the overestimation at low true RR was not affected by true cluster shapes, though the circular scan overestimation was greater than that of the elliptical scan. When the true RR was above 2.5, the magnitude of the underestimation of the RR was similar and modest for both scan tests on circular clusters; however, when the clusters were rectangular, the underestimation increased, especially in the case of the circular scan test.
In Figure 5 median p values for the lower (< 1.7) true RRs reflect the wide range of observed RR shown in Figure 4. These results informed our spatial testing procedure and the criteria for a positive result, termed Consensus or Potential Clusters, which are described in the methods section.
The mother's address at birth of the 1996 through 2000 State of California Department of Health Services Center for Health Statistics Confidential Birth Files  were geocoded. Cases diagnosed through February 2006 identified from the records of the California Department of Developmental Services (DDS), which assesses eligibility and coordinates services for people diagnosed with various developmental disorders, were matched to the birth records. The DDS divides California into twenty-one catchment regions, each serviced by an independent Regional Center (RC). Two of these were analyzed using the procedure presented in this paper. RC A, with mixed rural and urban areas, has one of the largest geocoded birth populations (219,417) in the state and a relatively low incidence, 28.6/10,000 births (628 cases). RC B serves a mostly urban area with 95,977 births geocoded and a relatively high overall incidence, 70.2/10,000 births (674 cases).
Summary population statistics for Set 1 and Set 2 aggregation units.
Units of analysis
348 Set 1 areal units
175 Set 2 areal units
148 Set 1 areal units
74 Set 2 areal units
Spatial test results
Results of SaTScan tests that allow overlap among identified statistically significant clusters.
Units of analysis
Highly significant clusters, p ≤ 0.01
Significant clusters, 0.01 < p ≤ 0.05
Maximum Relative Risk
Accepted contributing clusters
Results of Episcan tests that allow overlap among identified statistically significant clusters.
Units of analysis
Highly significant clusters, p ≤ 0.01
Significant clusters, 0.01 < p ≤ 0.05
Maximum Relative Risk
Accepted contributing clusters
Results of MEET global clustering test and the primary cluster reported by FleXScan.
FleXScan Primary Cluster
Units of analysis
adj p value
Accepted contributing cluster
Point data test results for RC A
In RC A, there were four statistically significant (p ≤ 0.05) point clusters reported by SaTScan. Two of the three highly significant (p < 0.01) clusters had fewer than ten births each and contained the same four cases. They were not statistically significant when expanded to include one additional case and were not located near the larger clusters. The remaining two large overlapping clusters had 56 cases in 7,569 births and 99 cases in 19,963 births; they formed the basis of a Consensus Cluster as defined earlier.
Areal data test results for RC A
SaTScan areal test confirmed the significant large cluster identified by the point test, identifying two qualifying highly significant clusters, one each in Set 1 and Set 2.
Episcan generated 1,000 clusters in Set 1 and 236 in Set 2 with p ≤ 0.05. Of these, 269 of Set 1 and 126 of Set 2 clusters had RR ≥ 1.7 and qualified for consideration. In Set 1, areal units occurring in at least 54 (20 percent of 269) of the qualifying clusters are mapped in Figure 8. For Set 2, areal units in more than 26 qualifying clusters are mapped. FleXScan's primary cluster also covers this area with p = 0.001 for both Set 1 and 2. With results from all seven LACT applications passing our criteria, RC A has a Consensus Cluster.
MEET did not detect overall clustering, p ≥ 0.264. In the face of strong LACT evidence of a single, very significant cluster, we interpreted MEET results as support for the existence of no more than one large Consensus Cluster in the study region (Figure 9).
Point data test results for RC B
In RC B, the only significant point cluster identified consisted of four cases in nine births. The next most statistically significant clusters were eight cases in 112 births (p = 0.141), covering the significant cluster, and eight cases in 111 births (p = 0.136).
Areal data test results for RC B
The non-significant results from MEET do not support the existence of global clustering. According to this multiple test procedure, the amount of clustering in this study region was not significantly different from that expected by random.
The size of the dataset SaTScan version 7.0 can analyze is limited by available computer resources. Performing SaTScan on large sets of data points took up to five days per test on Dual 2.4 GHz single-core AMD Opteron 250 Series processors, with eight GB of memory running SUSE Linux 188.8.131.52.
From the simulations, it appears that LACTs do not perform well for a true RR below 1.7. In the second simulation set, the median SE and p values improve dramatically at this point. In the first simulation set, reduced median SE and SP occurred together for RR = 1.5 compared to higher RR levels in all but two of the 32 different areal unit size and cluster shape scenarios. Balancing the increasing median SE and decreasing median SP that occur with increasing unit size, we decided on nmax of 1,000 and 2,000 for testing units in our procedure.
The second set of simulations informed the procedure's cutoffs for observed RRs and p values in the LACTs. The unsettling observation from Figure 4 that the median observed RR is at least 2.0 when the distribution is random (RR = 1.0) in all scenarios demonstrates the limitations of tests with SP of 0.98 when analyzing rare occurrences. For the high observed RR below a true 1.7, we must rely on a maximum p value limit and requiring multiple test agreement to limit these false positives. But as the median observed RR does estimate the median true RR well starting at 1.7, we feel that requiring a minimum observed RR of 1.7 improves the test validity. We did not choose a higher cutoff because the lower limit of the 90 percent range for the circular scan on the rectangular cluster was below 2.0 until the true RR was 4.0.
Since the overestimated observed RRs below true RRs of 1.7 coincide with high median p values, loosening the usual significance limit of 0.05 would bring more of these false positive results into our analysis. At true RR = 1.7, all p values but the circular scan for rectangular clusters were below 0.05. Any lower limit would not detect this level of clustering. Further, if using multiple tests, requiring a p value of 0.05 from each LACT results in a lower overall p value. We therefore kept the significance cutoff for all tests at 0.05.
Incorporating these criteria in our procedure, as we saw in RC B, it characterizes a region as having no clustering although one areal LACT result had a qualifying result. The significant point-based cluster had only four cases and the global test was non-significant. The conclusion of no clustering seems a reasonable inference since the only qualifying result (FleXScan) connected the two small point-based clusters (p = 1.4) with eight cases each. Based on the simulations, it is more likely that the FleXScan result is spurious than that there is a cluster here. In RC A, MEET was also not significant, but a Consensus Cluster was determined nonetheless. As Kulldorff et al  noted, global tests, including MEET, may not detect clustering if it consists of only one or two clusters within the entire study area.
For this low incidence disorder, our results for SaTScan agreed with the findings of Gregorio et al, who compared the results from SaTScan on data both in point form and aggregated by census units. We both found general agreement for cluster detection between the point data and their lowest level of census data aggregation and our smaller areal units. On the other hand, Episcan and FleXScan, using the areal units, sometimes indicated smaller clusters than those indicated by SaTScan with either point or aggregated areal data. Individual data are generally presumed to be more accurate than aggregated data. Perhaps for very low incidence diseases, small, population-defined areal units, such as Epiunts, are more useful in spatial tests such as SaTScan and Episcan. Comparing the accuracy of these two test input forms for a very low incidence disease would be an interesting simulation study.
As both SaTScan and FleXScan are available with graphical user interfaces, they are more likely to be used than the less-user friendly Episcan and MEET. This is unfortunate as Episcan yields more spatial information than the other LACTs, as was shown in Figure 8, and it is an integral part of this procedure. Episcan probably has higher SE than our simulations demonstrated; it incorporates more than just the most likely cluster in its result.
This procedure is designed to minimize false positive results without compromising power unnecessarily. For an exploratory spatial analysis, a false positive can be a false alarm, diverting research efforts and funds by generating false hypotheses from the many possible environmental risk factors to be assessed in any identified area. Different situations may warrant different procedures. Our simulations looked at the observed and known RRs for a variety of cluster shapes. The overestimation of the low (1.0 – 1.7) true RRs cautions against accepting higher p values to compensate for low power. Limiting the p values to 0.05 for significance will remove much of these exaggerated results. Results from a cluster larger than the five percent one we simulated could have a narrower 90 percent range, though the average estimate might not change.
We were surprised to see the magnitude of the downward bias of observed RRs for true RRs over 3.0, especially for circular scans with the rectangular clusters. Variable elliptical scans showed less of a downward bias meaning Episcan would therefore be the more reliable test for unknown cluster shapes. This does warrant a caveat for inferences from LACTs. The RR from a LACT will be lower for true RRs over 3.0 when compared to another association obtained from a different statistic.
To our knowledge, this is the first report combining multiple spatial tests into a systematic approach for a large cohort of individual locations. In an area like California, with county population densities ranging from 2 to 8,714 per square mile , only a cohort study provides accurate spatial distributions. While any spatial analysis of events with prevalence less than 0.005 remains challenging, for truly exploratory research, as our simulations demonstrate, this procedure is an improvement over the common practice of using one of the available LACTs and a global test with no limits on acceptable RR. By requiring confirmation from multiple tests and setting RR and p value limits for LACTs, we expect to reduce the percentage of false positive results. By considering Potential Clusters identified in only one set of partitioning areal units, we expect to reduce the false negative results due to aggregation units.
Our simulations illustrated median SE < 0.8 for rectangular clusters with true RR < 3.0. But Figure 3 shows that one-third of simulations are RR = 2.0 had SE ≥ 8.0. The lower SE values are associated with the more elongated clusters. For these shapes, Episcan and FleXScan may be more powerful than the single maximum likelihood cluster. In use, the three different LACTs showed a surprising agreement on an area for a cluster with a significant p value and a RR over 2.0 in almost half of the 21 DDS RC catchment areas studied. Since these clusters easily pass our criteria using parallel multiple tests, we feel we have not been too stringent. This procedure identified 10 Consensus Clusters and 2 Potential Clusters within the 21 RC areas.
Further validation for this procedure came from demographic analysis of the Consensus Clusters. In multiple regressions using individual demographic data on known risk factors, almost all of the defined cluster populations showed strong, significant associations with one of these variables, indicating actual clusters. In such clusters that appear to be explained mainly by demographics, either environmental risks occur in a non-geographical distribution, i.e. in the home or workplace, or there is some risk attached to demographic status. If so, then environmental exposure research is likely to be more fruitful if focused on the home and workplace. If not, then we can search for geographically distributed environmental risks within the remaining unexplained clusters.
Simulations of the study questions
Simulations were based on the geocoded maternal address at birth for all births during a period of five years in a two county area of California, USA, with 219,417 locations in a land area of 8,375 square miles. The background prevalence was chosen as 0.004, which was the overall background incidence in our statewide study dataset. Each simulation had one randomly located cluster containing approximately five percent of the locations. For each simulation, Bernoulli experiments were performed at all locations with the risk given by the scenarios below.
For every scenario, two sets of 100 simulations each were performed. One set used a circular shape for the cluster; the second set used a rectangular shape with aspect ratio varying from 2 to 10, simulating elongated clusters, and a random orientation.
For all simulations, we used the Episcan procedure to produce the most likely cluster for each simulation using (a) only circular (representing SaTScan) and (b) elliptical cluster scan shapes.
Scenario 1. To explore the question of the effect of unit population size, assess all combinations of:
a. RR of 1.5, 2.5, and 5.0
b. nmax of 500, 1000, 2000, and 4000
Scenario 2. To explore the question of the effect of the true cluster RR on the LACT results, the unit nmax was fixed at 1000 and the RR of the actual cluster was varied from 1.0 (random distribution) to 5.0 in steps of 0.1. Simulations and all additional programming were done in R version 2.n .
SaTScan uses a specified shape, usually a circle, to create a series of clusters with each data point or every areal unit's centroid as a starting point, assigning the most likely cluster's empirical p values based on 999 or 9,999 Monte Carlo randomized case distribution simulations over the study population distribution . The test is based on the binomial distribution for individual data points and the Poisson distribution for data in areal units of aggregated populations. FleXScan  and Episcan , build upon this method to highlight clusters of indeterminate shape. Episcan uses the binomial probability distribution on either the centroids of aggregated areal units or disaggregated point data , while FleXScan uses the Poisson probability distribution on aggregated areal units . Since we are concerned with rare events with prevalence under one percent, the two distributions produce a similar model for testing clusters.
FleXScan takes a nearest-neighbor approach to building clusters so it is free to connect up to a prespecified maximum number of areal units with elevated prevalence in indeterminately-shaped non-overlapping clusters . Episcan generates sets of elliptical clusters of varying sizes for every data point using each of its neighboring data points as the second elliptical focus . Episcan reports a prespecified number of the most significant clusters, usually at least 1,000. Using these clusters, the areal units occurring in the largest number of, often overlapping, clusters with p values below a specified maximum are highlighted rather than a single most likely cluster. In this way, Episcan is not limited to its elliptical cluster design. Both FleXScan and Episcan can also be restricted to the same set of circular clusters used in SaTScan as a special case. Episcan always includes the circles as part of its analysis while FleXScan requires a separate analysis.
MEET is a global clustering test that searches for the most likely cluster size for the areal data being studied by varying a scale parameter, λ, over a default range. It then tests the hypothesis of no clustering of cases given the background population distribution versus the alternate hypothesis that there is some clustering . It is most powerful in detecting multiple clusters of similar size. A single adjusted overall p value is reported for the computed clustering statistic, C, along with the distribution of p values over the range of λ. MEET also indicates which areal units contributed the most to C. If C is statistically significant, these areal units are likely cluster candidates.
Parameter settings for spatial tests
Maximum allowed size for detecting clusters
Each LACT has recommended default parameter settings. The default maximum cluster size to be detected for SaTScan is 50 percent of the locations; for Episcan it is 15 percent.
To maintain consistency among tests, a single parameter setting was used for all analyses. For Episcan and SaTScan, we found higher agreement between the point and areal tests with a maximum cluster size of 15 versus 50 percent for low incidence data and therefore used 15 percent in our procedure. Using a maximum of 15 or 20 areal units, FleXScan typically identified the same clusters; however, occasionally (< 5 percent), it identified two different clusters, anchored at the same unit, but continuing in different directions. We used the default setting of 15 percent, as we found no compelling reasons to not do so.
SaTScan secondary cluster reporting limit
After reporting the primary most likely cluster, SaTScan's cluster reporting setting dictates what secondary clusters, with lower p values, will be reported. The default setting of "no geographical overlap" reports only secondary clusters that share no locations with the primary cluster. Other settings allow various levels of primary cluster overlap for the secondary clusters. Using these settings for both point and areal data, we found some overlapping clusters with higher, yet still very significant p values that had a higher RR than the primary cluster. In densely populated areas, these overlapping clusters may indicate a non-circular cluster, which can be compared to those identified using the more flexible LACTs. Using any of the cluster reporting settings that were more restrictive than "no pairs of centers both in each other's clusters" SaTScan failed to detect some statistically significant overlapping clusters; however, less restrictive settings examined gave no additional useful information.
Data format considerations
Epiunit areal units
The areal units used in the simulations were produced using Epiunits, the geo-aggregation tool for spatial point data developed for this study (Christiansen, unpublished), to minimize the number of units with very low population and/or zero case counts.
By specifying a maximum population per areal unit, the user effectively sets the minimum unit population at one-half of nmax. This creates a set of areal units with more homogeneous populations without restricting the range of geographical sizes, which can be large.
For this procedure, two sets of areal units were created for testing by using the Epiunit NHalf algorithm with nmax set at either 1,000 (Set 1) or 2,000 (Set 2). If a study region population density is less than two per square mile, fewer than 30 areal units are generated for Set 2, or 2,000 is more than the maximum population for the Episcan or SaTScan scanning window, a Set 3 with nmax of 500 should replace Set 2 in the analysis.
SaTScan is the only LACT that can currently analyze a large population without aggregating point locations into areal units. With low prevalence, smaller clusters are very sensitive to the change of one or two cases. Logically, the lower the prevalence, the larger the cluster population needed for a stable result.
We found SaTScan could identify highly significant point-based clusters, with an extremely high RR, that were composed of two cases in less than 50 births over five years. However, these clusters were not contained within larger clusters of raised prevalence. Particularly with a rare event with a genetic component, two people in one location should not be regarded as a cluster if there are no other events in the surrounding group of some specified minimum size.
We chose to require a minimum cluster size of 1,000 population as a very liberal limit since areas we analyzed separately had a range of overall incidence of 0.0020 to 0.0088.
Test procedure of parallel tests and their parameter settings
SaTScan, Episcan, FleXScan and MEET were applied to two different sets of Epiunit-created areal units. SaTScan was also applied to the disaggregated point data. Together they gave a more accurate assessment of possible geographic clustering of a low incidence disease. The following settings were used:
For both areal units and point locations, SaTScan identified high risk clusters only, with maximum cluster size of 15 percent of the study population. The cluster reporting parameter setting was "no pairs of centers both in each other's clusters". Point location clusters were assessed with the binomial distribution and areal unit clusters with the Poisson distribution.
Episcan was run with maximum cluster size of 15 percent of the locations and maximum of 10 percent of other areal units used as a second focus for each initial point. The number of clusters to be created and evaluated was the lower of 1,000 or the maximum possible for the set of areal units. Episcan displays the areal units occurring in more than a designated percentage, here 20, of the qualifying clusters with heat colors indicating their frequency.
FleXScan was run with the default settings, including maximum cluster size of 15 areal units. FleXScan reports only non-overlapping clusters, usually producing one or two clusters with the lowest p value of any LACT and statistically very non-significant secondary clusters.
The RR that is included in the FleXScan output uses the incidence in the entire study region as the denominator and so was recalculated with the incidence in the study region outside of the cluster as the denominator to conform to RR measures produced by the other LACTs.
MEET was run with latitudes adjusted by cos(mean(degrees longitude)*pi/180) instead of a fixed number in the distance measure, dij, when computing neighbor distances, and the default one to twenty range for λ, the scale parameter.
Defining Consensus Clusters of high incidence
We defined a Consensus Cluster as occurring when the point LACT identified a cluster of at least 1000 births and all three areal LACTs on both sets of areal units indicated this same area. All seven LACT results must meet criteria of p ≤ 0.05 and RR ≥ 1.70. Therefore, if five cases per thousand are expected, the minimal qualifying point cluster would contain nine cases. The results of this composite test procedure, having passed multiple tests and rules, were referred to as Consensus Clusters to distinguish them from the statistically significant clusters defined by the individual tests. A Consensus Cluster did not require MEET to be significant. Such a result implied that there was no significant clustering in the remainder of the study area outside the defined cluster.
The actual cluster estimate was defined as the total of the areal units in the FleXScan cluster and areal units occurring in more than 75 percent of qualifying Episcan-defined clusters approximately covering the same area from either Set 1 or 2, whichever cluster estimate is smaller. There could be more than one non-overlapping cluster so defined in a study region.
Defining Potential Clusters less stringently
As noted before, a cluster could be masked by the boundaries of one set of partitioning areal units and not another. So we defined a Potential Cluster as an area that qualified as a Consensus Cluster in the results of the point test and only Set 1 or Set 2. The defined area is as above.
Assessing non-environmental contribution to the defined clusters
If there are known demographic risk factors for the disorder that could cluster geographically, i.e. socioeconomic status, their contribution to the Consensus Clusters must be explored. For areal units analyzed using the Poisson distribution, both SaTScan and MEET can evaluate categorical population covariates by stratification within the units. With a rare disorder, this is not feasible for more than one or two variables because the stratification would lead to most strata having no cases. Instead, logistic or Poisson regression should be performed, using individual level demographic information comparing locations within the cluster with those outside, regardless of case status. The strength and size of these associations will affect the likelihood that the cluster has a geographically distributed environmental association.
The authors are grateful for the data management that Lora Delwiche, Dept of Public Health Sciences, School of Medicine, UC Davis, has done for the project.
The authors would also like to thank the reviewers for their constructive comments that have greatly improved this article.
This study was supported by the Center for Animal Disease Modeling and Surveillance (CADMS) and a pilot grant from the M.I.N.D. Institute, both of UC Davis, and grants from NIH #P01 ES11269 and EPA STAR grant #R829388. Van Meter was supported by CADMS, the Office of Graduate Studies and the Graduate Group in Epidemiology, all of UC Davis. Christiansen was supported by CADMS. Hertz-Picciotto was supported by NIH #P01 ES11269 and R01 ES015359, and EPA STAR grant #R829388. Carpenter was supported by CADMS. Hertz-Picciotto, Azari and Carpenter are also funded as faculty of UC Davis. Manuscript preparation was funded by CADMS. Funding bodies had no role in study design; in the collection, analysis, and interpretation of data; in the writing of the manuscript; nor in the decision to submit the manuscript for publication.
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