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Open AcceResearchA comparison of six analytical disease mapping techniques as applied to West Nile Virus in the coterminous United StatesDaniel A Griffith*
Address: Ashbel Smith Professor, School of Social Sciences, University of Texas at Dallas, Richardson, TX, USA

Email: Daniel A Griffith* - [email protected]

* Corresponding author

AbstractWest Nile Virus has quickly become a serious problem in the United States (US). Its extremelyrapid diffusion throughout the country argues for a better understanding of its geographicdimensions. Both 2003 and 2004 percentages of deaths by numbers of reported human cases, forthe 48 coterminous US states, are analyzed with a range of spatial statistical models, seeking tofurnish a fuller appreciation of the variety of models available to researchers interested in analyticaldisease mapping. Comparative results indicate that no single spatial statistical model specificationfurnishes a preferred description of these data, although normal approximations appear to furnishsome questionable implications. Findings also suggest several possible future research topics.

BackgroundWest Nile Virus (WNV [1,2]), first isolated in the WestNile District of Uganda in 1937, is a flavivirus transmittedby a mosquito vector, with a general incubation period of2–14 days following a bite by an infected mosquito, andis closely related to the St. Louis encephalitis virus thatalso is found in the United States (US). WNV can infecthumans, birds, mosquitoes, horses and some other mam-mals, with mosquitoes becoming infected after feeding onthe blood of birds that carry the virus (this virus enters andcirculates in a mosquito's bloodstream for a few daysbefore it settles in the insect's salivary glands); of particu-lar concern is that the adult WNV-carrying Culex species ofmosquito is able to survive through winters. WNV prima-rily results in bird mortality, and human and equineencephalitis. In temperate latitudes, West Nile encephali-tis cases occur primarily in the late summer or early fall;WNV tends to be carried by less than 1 out of every 100mosquitoes residing in geographic regions in which itactively circulates. WNV, with its first detected US case on

Long Island in 1999, has swiftly diffused across the conti-nental US (Figure 1a and Table 1) as well as elsewhere inthe Western Hemisphere. This virus enjoyed a surprisinglyrapid rate of diffusion, spreading from the New York Cityarea to nearby localities contagiously, as well as leapingacross space in a hierarchical fashion through, for exam-ple, bird migration routes (see http://westnilemaps.usgs.gov/). Although presently a person has a low riskof contracting WNV, many people infected with this virus– more than 16,000 have tested positive to date – tend toexperience mild (e.g., flu-like symptoms such as fever,headache, body ache and skin rash) or no symptoms (i.e.,never realizing that they have been exposed to WNV),with less than 1% of those infected developing serious ill-ness (e.g., high fever, severe headache, stiff neck, disorien-tation, tremors, muscle weakness, paralysis and coma),and even fewer dying from the virus (roughly 650 todate). People at higher risk of developing potentially seri-ous conditions include the elderly (age 50 and older) andthose with lowered immune systems, and some categories

Published: 02 August 2005

International Journal of Health Geographics 2005, 4:18 doi:10.1186/1476-072X-4-18

Received: 09 June 2005Accepted: 02 August 2005

This article is available from: http://www.ij-healthgeographics.com/content/4/1/18

© 2005 Griffith; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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of outdoor workers. Meanwhile, because 1999 signifiesthe beginning of a logistic growth curve for WNV cases inthe US, during the very early years of the new millennium,public health officials mostly were concerned aboutwhether or not this virus had diffused to their localities.Presence or absence was the geographic quantification ofchoice. But once WNV appears in an area, infected peoplebegin to die, with human deaths and cases becoming ageographic quantification of choice. In response to thispublic health problem, by 2002 CDC was releasing num-bers of cases and of deaths, for US states; today these dataalso are being released for US counties. (Seehttp:www.cdc.gov/ncidod/dvbid/westnile/surv&control.htm#surveillance.) Data aggregated by state that wereextracted from this web site and used in the paper for anal-ysis purposes appear in additional file 1a (Cases of anddeaths attributed to WNV); the associated geographic con-nectivity matrix (i.e., matrix C) appears in additional file1b (US state geographic connectivity matrix).

Because of the inherently geographic nature of WNV con-tainment and eradication efforts, the objective of thispaper is to summarize some initial spatial statistical anal-yses of the georeferenced virus in its US context using dis-ease map modeling. In equilibrium, such features of WNVas percentage deaths should contain positive spatial auto-correlation (i.e., similar percentages of deaths should clus-ter together on a map) because: common local weatherpatterns tend to spatially cluster and partially govern mos-quito population dynamics; bird populations tend tohave distinct migratory routes and locational preferences;socio-economic/demographic attributes of the humanpopulation cluster geographically; and, by their veryadministrative unit jurisdictional nature, mosquito con-

trol programs tend to be geographically concentrated.One outcome of this work is a set of research hypothesesthat should be analyzed as WNV becomes more stronglyentrenched in the US geographic landscape.

Results and discussion: a comparison of disease map modeling resultsGriffith [3, pp. 78–79, 114–116] reports an analysis of thepresence/absence of WNV by state for January throughAugust, 2002 (see Figure 1b). A normal probability modelapproximation cannot be implemented here because theresponse variable is binary. The join count statisticsindicate the presence of strong, positive spatial autocorre-lation. A visual inspection of the map furnished by theCenters for Disease Control (CDC) indicates that only the11 most western states in the coterminous US were absentof WNV. A Markov chain Monte Carlo (MCMC) estimatedauto-logistic model reveals the presence of strong, positivespatial autocorrelation. The spatial filter logistic model (ageneralized linear model specification) furnishes a thirddescription of these data, one involving eigenvector E1.Estimation attempts of both spatial filter and proper con-ditional autoregressive (PCAR) specifications with Baye-sian inference using Gibbs sampling (BUGS) repeatedlyfailed because of encountered phase transitions [4]. Oneinteresting feature of the generalized linear spatial filtermodel is its suggestion that California (a leap across geo-graphic space), New Mexico, Montana and Washingtonshould have had WNV present. The end-of-the-year CDCmap indeed reveals WNV presence in these states. A sec-ond interesting feature of this spatial filter model is that itout-performs both the pseudo-likelihood and the MCMCestimated auto-logistic models.

WNV in the USFigure 1WNV in the US. Left: (a) diffusion of the virus: years of presence is directly proportional to darkness of the gray scale. Right: (b) 2002 presence of WNV by state, through August.


http://www.cdc.gov/ncidod/dvbid/westnile/surv&control.htm#surveillance

http://www.cdc.gov/ncidod/dvbid/westnile/surv&control.htm#surveillance


An analysis of the numbers of deaths attributable to WNVstandardized by the corresponding number of reportedcases (i.e., percentage data) for 2003 (overall 2.7%) and2004 (overall 3.6%), by state (see Figure 2), reveals thepresence of weak-to-negligible spatial autocorrelation(see Tables 2 and 3). An indicator variable has beenincluded to differentiate the states without reported casesfrom the remaining states. Noteworthy findings hereinclude: (1) the intercept term is approximately -3.1,regardless of model specification or year; (2) the normalprobability model approximations highlight the 0-casestates (3 in 2003, and 8 in 2004) as being statistically sig-nificant, whereas the remaining four specifications findthem to be consistent with the other data; (3) the auto-binomial model fails to furnish a reasonable descriptionof these 2004 data; (4) the spatial filter models indicatethat both positive and negative spatial autocorrelationeffects are present in these data – these countervailing geo-graphic trends could be why the global spatial autocorre-lation indices appear to be nonsignificant; and, (5) thePCAR hierarchical generalized linear model (HGLM)uncovers statistically nonsignificant spatial autocorrela-

tion, whereas the spatial filter HGLM uncovers statisticallysignificant spatial autocorrelation – perhaps again reflect-ing the mixture of positive and negative spatial autocorre-lation components.

Even after employing a variable transformation to stabi-lize variance, the Gaussian (i.e., normal probability)approximations continue to display marked overdisper-sion (i.e., excessive variability) in 2003. Residuals forthese models contain only trace amounts of spatial auto-correlation, and the models themselves account for nearlyhalf of the variability in percentage deaths from WNV. Butthese pseudo-R2 values are inflated by the inclusion of theindicator variable denoting states with no cases. And, thesimultaneous autoregressive (SAR) model predicted andobserved values match well for 2003 but are underesti-mated by roughly 30% for 2004, whereas the spatial filterpredicted values tend to be overestimated by roughly40%, on average.

According to the 2004 data (for which pseudo- and maxi-mum likelihood estimates are equivalent here), not only

Table 1: Expansion of WNV across the coterminous US.

Year # states reporting cases # human cases # human deaths

1999 4 62 72000 4 21 22001 10 50 52002 44 4,156 2842003 45 9,862 2642004 40 2,539 100

The geographic distribution of percentages of WNV cases resulting in deathFigure 2The geographic distribution of percentages of WNV cases resulting in death. Left (a): 2003. Right (b): 2004.



does the auto-binomial model furnish a poor descriptionof these data, but it also is plagued by overestimation ofthe predicted values. Meanwhile, the spatial filter general-ized linear model has residuals that contain only tracenegative spatial autocorrelation, furnishes a respectabledescription of these data, lacks overdispersion, and pro-vides predicted and observed values that are well matchedfor 2003 and overestimated for 2004.

The HGLMs have effective degrees of freedom associatedwith them, in order to account for parameter estimates aswell as random effects estimates. Although the PCAR spec-ification furnishes the best overall description of thesedata for 2003, it does so by consuming considerably moredegrees of freedom. This HGLM tends to exhibit consider-able underdispersion (i.e., insufficient variability), a sig-nature of negative spatial autocorrelation. Spatialautocorrelation may well remain in the 2003 HGLMresiduals, with the Moran coefficient (MC) and the Gearyratio (GR) values being inconsistent in their implications.

The PCAR model also dramatically underestimates pre-dicted probabilities. In contrast, the spatial filter HGLMproduces predicted values that match well with their cor-responding observed values, furnishing respectabledescriptions of both data sets.

Overall, no single model specification is superior to allothers. The discrepancies between specification resultsemphasize differences in modeling assumptions, errorstructure, and detailed treatment of latent spatial autocor-relation, resulting in different nuances and idiosyncrasiesof the data being highlighted.

ConclusionSeveral implications can be drawn from the research sum-marized in this paper. Foremost is that while the initialspread of WNV across the US had a prominent geographicdimension, such a dimension has not fully materializedyet for the percentage of deaths from detected cases. Inaddition, the spatial filter model specification is appealing

Table 2: Parameter estimation results for observed 2003 WNV death percentages.

Gaussian SAR Gaussian spatial filter binomial spatial filter

Statistic estimate se estimate se estimate se

Spatial autocorrelation 0.08 0.101 1 (spatial filter MC = 0.243) 0.144 1 (spatial filter MC = 0.359) 0.090intercept -3.16 0.164 -3.15 0.105 -3.11 0.078I0 2.32 0.629 2.32 0.419 -18.21 25149approximate dfs 45 39 40Residual P(S-W) 0.029 0.212 0.522Residual MC 0.0433 -0.0431 (z = -0.2) -0.0927Residual GR 1.0295 0.9720 0.9782deviance 6.1357 4.2542 1.0198pseudo-R2 0.425 0.643 0.292predicted-observed regression slope

1.05 0.76 0.69

HGLM

proper CAR spatial filter

Statistic estimate se estimate se

spatial autocorrelation 0.09 0.104 0.999 (spatial filter MC = 0.359) 0.094intercept - 3.25 0.187 -3.12 0.072I0 -80.11 57 -78.86 60approximate dfs 26 40residual P(S-W) 0.018 0.001residual MC 0.0119 0.0350residual GR 0.7141 0.7494deviance 0.4089 1.1660pseudo-R2 0.697 0.442predicted-observed regression slope

1.82 0.97



because of its simplicity, and because it is able to sense thepresence of a mixture of positive and negative spatialautocorrelation components latent in the geographic dis-tribution of WNV deaths. This model feature is not sharedby the more conventional specifications. These findingssuggest the following research hypothesis: spatial autocor-relation contained in a disease map has its negative com-ponents fade, and its positive components strengthen,through time. This hypothesis can be tested by evaluatingthe regression coefficients of spatial filter eigenvectors asannual WNV data become available. The following is asecond research hypothesis suggested by Tables 2 and 3:half of the variability in percentages of WNV deaths maybe accounted for with yet-to-be-determined socio-eco-nomic/demographic attribute variables. Again, thishypothesis can be tested by computing the pseudo-R2

model values as annual WNV data become available. Italso can be assessed by uncovering the unknowncovariates. Although this is a descriptive feature of WNVhere, this characterization is expected to typify expansiondiffusion following an initial invasion of some territory

[5]. And, the following is a third hypothesis: stateswithout reported WNV cases are from the same statisticalpopulation as states with reported cases. This hypothesiscan be tested by evaluating the zero-case indicator varia-ble generalized linear model regression coefficient asannual WNV data become available.

Because WNV risk factors include being elderly, the size oflocal bird populations, and exposure to certain species ofmosquitoes, geographic distributions of these groups ulti-mately should impact upon the geographic distribution ofWNV deaths. The elderly tend to be clustered nationally,with this effect perhaps being controllable by age-stand-ardization of case counts. The use of door and windowscreens, mosquito repellant, and adult, larvae and breed-ing site mosquito control programs tend to have socio-economic/demographic dimensions with spatial expres-sions. All of these factors tend to impact upon contagiondiffusion, inducing positive spatial autocorrelation.Meanwhile, factors such as migratory bird routes result inleaps across geographic space (i.e., hierarchical diffusion),

Table 3: Parameter estimation results for observed 2004 WNV death percentages.

Gaussian SAR Gaussian spatial filter binomial spatial filter

Statistic estimate se estimate se estimate se

Spatial autocorrelation 0.01 0.101 1 (spatial filter MC = 0.030) 0.135 6.07 4.701intercept -3.30 0.172 -3.31 0.124 -3.48 0.182I0 2.63 0.491 2.62 0.323 -20.01 41223approximate dfs 45 39 45Residual P(S-W) 0.249 0.256 < 0.0001Residual MC -0.0224 -0.0753 (zMC = -0.3) -0.073Residual GR 1.1361 1.0788 1.088deviance 0.9001 0.4686 1.0346pseudo-R2 0.494 0.754 0.07predicted-observed regression slope 1.31 0.72 0.87

HGLM

proper CAR spatial filter

Statistic estimate se estimate se

spatial autocorrelation -0.08 0.148 0.999 (spatial filter MC = 0.762) 0.226intercept -3.28 0.122 -2.99 0.122I0 -81.63 62 -80.94 60approximate dfs 45 44residual P(S-W) <0.0001 0.192residual MC 0.0596 0.0130residual GR 0.9417 0.9270deviance 0.9715 0.4992pseudo-R2 0.328 0.411predicted-observed regression slope 30.60c 0.99



which initially introduce a negative spatial autocorrela-tion dimension into the geographic distribution of WNVdeaths (i.e., a location with cases being surrounded byneighboring locations with no cases). As these diffusionpaths become reinforced through cyclical repetition overtime, accompanied by repeated annual waves of local con-tagion diffusion, a more uniform geographic distributionof WNV reservoirs should materialize, causing the nega-tive spatial autocorrelation dimension to fade away. In theend, WNV should be characterized by positive spatialautocorrelation reflecting annual weather map patternsthat promote, and the effectiveness of public health pro-grams that attempt to minimize, the size of mosquitopopulations. Given the figures reported in Table 1, thesepatterns should contain a fair degree of variability. A sim-ilar argument pertains to horse deaths from WNV, too.

The principal covariate in the spatial statistical modelspecifications estimated in this paper is spatial autocorre-lation. Based on the pseudo-R2 values reported in Tables 2and 3, this covariate tends to account for about 50% of thevariability in the percentages of WNV deaths. This findingis a clue that socio-economic/demographic attributes –presumably associated with the use of door and windowscreens, mosquito repellant, and adult, larvae and breed-ing site mosquito control programs – should be exploredin order to identify those that statistically describe this var-iability. Of course, part of this covariation may disappearby using age-standardized figures. Part may disappearover time as health professionals increasingly gain experi-ence in detecting and treating WNV cases. And, part maydisappear as the hierarchical diffusion component ofWNV expansion disappears.

Now that WNV has appeared at one time or another ineach of the coterminous US states, years in which no casesare detected in a given state have become a feasible out-come within the same statistical population. These zero-cases can naturally occur when weather patterns suppressmosquito populations, or can result from temporaryeffectiveness of human uses of screens and mosquitorepellants, as well as governmental adult, larvae andbreeding site mosquito control programs. They also canresult from cyclical biological processes in the various ani-mal populations involved in WNV transmission. Wheninspecting Table 1, one should not be surprised that themore severe outbreak year of 2003 is accompanied byonly 3 states having zero cases, while the less severe out-break year of 2004 is accompanied by 8 states having zerocases. Of note is that this scenario is somewhat incom-plete, since the spatial diffusion of WNV was reaching geo-graphic saturation during these two years. Nevertheless,more severe outbreak years should tend to be accompa-nied by a more widespread geographic distribution of thevirus.

Another general finding is that the normal probabilityapproximation model specification tends to overempha-size the statistical significance of those states with nocases. In part this result may link to the use of translationparameters that convert the odds ratio for these states toroughly 0.5. Meanwhile, one implementation data adjust-ment made for the other model specifications involvedrecoding 0s to 1s for the cases variable, which does notalter the corresponding percentage (i.e., the number ofdeaths remained 0). This data adjustment will be dis-pensed with once WNV becomes more prevalent acrossthe US, and hence states will not be without cases, butmost likely will have to be retained for initial county-levelgeographic resolution studies. And, because of the size ofits autoregressive parameter, the auto-binomial modelproves to be of less interest for model comparisonpurposes.

Overall general findings suggest several rules of thumbthat should help guide an analyst in his/her disease mapmodeling efforts. Foremost, switching between modelspecifications should yield similar intercept values; ifmarkedly different values are obtained, an analyst shouldbe suspicious and ascertain why. Second, non-normaldata are best described with non-normal probability mod-els; an analyst always should be aware of nontrivial spec-ification error. Third, a Gaussian approximation spatialfilter model can be used to more quickly explore whetherboth positive and negative spatial autocorrelation compo-nents underpin a disease map; a spatial filter modelspecification enables a detailed understanding of latentspatial autocorrelation. And, fourth, a spatial filter can beused to more quickly explore spatial structuring of randomeffects in a Bayesian analysis involving a large n; a spatialfilter model specification dramatically reduces the numer-ical intensity of MCMC computations.

Finally, popular individual observation diagnostic statis-tics may be evaluated in terms of their covariations withspatial autocorrelation by regressing them on thecandidate spatial filter eigenvectors. DFBETA diagnosticstatistics, one for each attribute variable, specify the stand-ardized differences in regression estimates for assessingthe effects of individual observations on the estimatedregression parameters in a fitted model. And, the HIdiagnostic statistic specifies the diagonal element of thehat matrix for detecting extreme points in a regressorattribute variable matrix. For the 2003 data and thereduced-form logistic regression model, eigenvectors E1,E7, E12 and E26 account for roughly 40% of the variance inthe intercept DFBETA statistic, as do eigenvectors E3, E7,E18, E27 and E29 for the indicator variable DFBETA statistic.Eigenvectors E1 and E27 account for roughly 20% of thevariance in the HI statistic. These three sets of eigenvectorsoverlap with those selected for the affiliated generalized



linear spatial filter model only in E1 and E18. Meanwhile,for the 2004 data and the reduced-form logistic regressionmodel, eigenvectors E1, E3, E10, E15 and E22 account forroughly 45% of the variance in the intercept DFBETA sta-tistic, as do eigenvectors E1, E5, E14, E19, E21, E27 and E29 forthe indicator variable DFBETA statistic. Eigenvectors E4,E4, E7 and E24 account for roughly 30% of the variance inthe HI statistic. These three sets of eigenvectors overlapwith those selected for the affiliated generalized linearspatial filter model only in E1 and E15. Future researchshould include scrutinizing the full battery of such diag-nostics in this manner, in order to better articulate rela-tionships between local and global information.Subsequent research also needs to address the problem ofstates with very few cases, whose predicted values willtend to have excessive variability, and states with verylarge numbers of cases, whose predicted values will tendto be significant by default.

Together these findings suggest several implications aboutthe diffusion of WNV, whose initial spread across the cot-erminous US at the state level of geographic resolutionnow is complete. In years to come, diffusion across stateswill be in terms of waves of re-infection. But infill conta-gious diffusion still is occurring at the county and finerresolutions. Findings summarized in this paper imply thatfor this level of diffusion, the geographic distribution ofcounty populations should introduce both positive andnegative spatial autocorrelation components into theresulting map patterns. Furthermore, one prominentsocio-economic covariate should be the differencebetween rural and urban locations. Others should be cov-ariates of the willingness of local populations to acceptand fund aggressive mosquito control programs, as well asindividually adopting measures to prevent mosquitobites.

Methods: spatial statistical modeling approachesAn analyst can choose from a variety of analytical spatialstatistical tools to study a disease map. The first of these tobe developed historically is the spatial autoregressivemodel, which is based upon a normal probability model,and hence requires disease map data to conform to a bell-shaped curve; often this requirement necessitates the useof a Box-Cox type of power transformation. Recent quan-titative geography methodological developments havesupplemented this approach with the spatial filter modelspecification [6]. One advantage of a spatial filterapproach is that it also enables use of a generalized linearmodel specification [7,8], which for disease mapping pur-poses is based upon the binomial, Poisson, or negativebinomial probability models (depending upon whether adisease map is expressed in terms of a binary, a percentageor a count variable). Recent MCMC methodology alsoenables the use of the binomial, Poisson, or negative

binomial probability models with a spatial autoregressivespecification [9-11]. In addition, MCMC has made Baye-sian analysis implementable and hence more accessible,enabling researchers to estimate both conditional autore-gressive [12,13] and spatial filter HGLM specifications.Because these modeling approaches involve different dataassumptions, especially in terms of error, researchersinterested in analytical disease mapping need a fullerappreciation of the variety of models they can employ intheir analyses; six specifications are treated here, namelythree conventional and their three spatial filter counter-parts. Furnishing the basis for this appreciation is one ofthe purposes of this paper.

A sizeable part of spatial statistics is concerned withaccounting for observation correlational effects arisingfrom the geographic configuration of data. Quantitativelycharacterizing this configuration commonly is achieved inone of two ways: (1) establishing a geocoded coordinatefor each areal unit, and then computing inter-point dis-tances; and, (2) establishing a surface partitioning, andthen constructing an n-by-n binary matrix C, whose cellentries are cij = 1 if areal units i and j share a boundary(employing analogies with chess: if it is non-zero inlength, then the linkages are referred to as the rook's case;if it is both zero and non-zero in length, then the linkagesare referred to as the queen's case), and cij = 0 otherwise.This queen's adjacency formulation is employed here.

The traditional spatial autoregressive modelSpatial statistics addresses the issue of observational cor-relation amongst georeferenced observations, which isknown as spatial autocorrelation; this type of correlationcan be indexed with a MC or a GR. This autocorrelationoften is positive in nature, with most phenomena exhibit-ing a moderate tendency for their similar values to clusterin geographic space. Occasionally, the tendency is for dis-similar values to cluster in geographic space, representingnegative spatial autocorrelation.

An autoregressive model specification accounts for spatialautocorrelation by including a variable on the right-handside of an equation that is a function of the neighboring Yvalues; in other words, the disease map variable, Y,appears on both sides of an equation. When coupled withregression and the normal probability model, this specifi-cation results in a covariation term characterizing spatialautocorrelation in one of two popular ways. Denoting theautoregressive parameter that captures spatial autocorrela-tion with ρ, a conditional autoregressive (CAR) covari-ance specification involves the matrix (I - ρ C), where I isan n-by-n identity matrix. Because matrix C is raised to thepower 1 (i.e., only adjacent neighbors are involved in theautoregressive function), this expression is considered afirst-order specification, with the autoregressive term



being CY. Because I is the identity matrix, individual arealunit variance is conditionally constant.

An important matrix can be constructed from C1, whichis the vector of number of neighbors. If the inverse of theelements of C1 are inserted into the diagonal of a diago-nal matrix, say D-1, then W = D-1C becomes a stochasticmatrix (i.e., each of its row sums equals 1). One appealingfeature of this matrix is that the autoregressive termbecomes WY, which renders averages, rather than sums, ofneighboring values. Because a covariance matrix must besymmetric, a matrix W specification can be used with aCAR model only by making the individual areal unit var-iance nonconstant: (I - ρ D-1C)D-1 = (D-1 - ρ D-1CD-1).One appealing feature of this version is that it restrictspositive values of the autoregressive parameter to the

more intuitively interpretable range of 0 ≤ ≤ 1.

The SAR model (see additional files 2b: Data input andpreparation for the SAR model estimation with SAS; 2b:SAR model estimation with SAS) furnishes an alternativespecification that frequently is written in terms of matrixW. As such, its spatial covariance is a function of thematrix (I - ρ CD-1)(I - ρD-1C) = (I - ρ WT)(I - ρ W), whereT denotes matrix transpose. The resulting matrix is sym-metric, is considered a second-order specification becauseit includes the product of two spatial structure matrices(i.e., WTW) – adjacent areal units as well as those having asingle intervening unit are involved in the autoregressivefunction – and also restricts positive values of the autore-gressive parameter to the more intuitively interpretable

range of 0 ≤ ≤ 1.

For the percentage of deaths associated with diagnosedWNV cases, the log-odds ratio requires the following Box-Cox type of transformations in order to conform to a bell-shaped curve:

Non-zero translation parameters primarily are due to thepresence of 0 cases, and partially due to the presence of 0deaths for some non-zero cases. WNV deaths are a rareevent, with the probability of death being highly skewed.These constants shift the locations of the empirical prob-abilities within the interval [0, 1] in order to have the databetter conform to a normal frequency distribution.Symmetricizing effects of these transformations are por-trayed in Figure 3. Zero-zero states, whose ratio becomesroughly 0.5, can be differentiated from the remainingstates with an indictor variable designating them as poten-tially coming from a different statistical population.Because these transformations to normality should stabi-lize variance, arguably estimation can be done without a

weighting scheme [e.g., following the variability of a bino-mial probability, the appropriate weighting schemewould involve division by

]; compari-

sons with and without the use of weights revealed no realdifferences in results.

The spatial filter modelSpatial filtering involves regressing a disease map variableon a set of synthetic variates representing distinct map pat-terns that accounts for spatial autocorrelation; each of thethree preceding spatial statistical model specifications canbe replaced with a spatial filter model specification. Grif-fith [3] develops one form of spatial filtering whose syn-thetic variates are the set of n eigenvectors extracted frommatrix (I - 11T/n)C(I - 11T/n), the matrix appearing in thenumerator of the MC index of spatial autocorrelation,where 1 is an n-by-1 vector of ones (see additional file 3:Minitab 14.13 code for computing spatial filter eigenvec-tors). This procedure is similar to executing a principalcomponents analysis in which the covariance matrix isgiven by (I - 11T/n)C(I - 11T/n). But rather than using theresulting eigenvectors to construct linear combinations ofattribute variables, the eigenvectors themselves (instead ofprincipal components scores) are the desired syntheticvariates, each containing n elements, one for each areal

unit. The extracted eigenvector relates to the mean

response, and the remaining (n-1) extracted eigenvectorsrelate to distinct map patterns characterizing latent spatialautocorrelation – whose MCs are given by standardizingtheir corresponding eigenvalues [14] – that can material-ize with matrix C. Furthermore, for a given geographiclandscape surface partitioning, the eigenvectors representa fixed effect in that matrix (I - 11T/n)C(I - 11T/n) doesnot, and hence they do not, change from one attribute var-iable to another.

Because this eigenfunction decomposition yields n eigen-vectors, a disease map analyst needs to restrict attention toonly those eigenvectors describing substantive positive/negative spatial autocorrelation (e.g., MC > 0.25 – a valuethat tends to relate to about 5% of the variance in Y beingattributable to redundant information arising from latentspatial autocorrelation, given a particular areal unit neigh-borhood configuration), reducing the candidate set to amore manageable number for describing a given diseasemap. Supervised stepwise selection from this set of eigen-vectors is a useful and effective approach to identifying thesubset of eigenvectors that best describes latent spatialautocorrelation in a particular disease map. This proce-dure begins with only the intercept included in a regres-sion specification. Next, at each step an eigenvector isconsidered for addition to the model specification. For

ρ̂

ρ̂

2003 : ; LN# deaths + 0.14

# cases - # deaths + 0.30 an

dd, 2004: LN# deaths + 0.15

# cases - # deaths + 0.29

..

(# deaths + 0.14)(# cases - # deaths + 0.30)

1

n1



the stepwise linear Gaussian model, commonly the eigen-vector having the largest partial correlation with variableY is selected, but only if its corresponding F-ratio achievesor surpasses a prespecified level of significance; this is thecriterion used to establish statistical importance of aneigenvector. Meanwhile, in stepwise generalized linearmodeling regression, the eigenvector that produces thegreatest reduction in the log-likelihood function chi-square test statistic is selected, but only if it produces atleast a prespecified minimum reduction; as before, this isthe criterion used to establish statistical importance of aneigenvector. In each statistical procedure, at each step alleigenvectors previously entered into a spatial filter equa-tion are reassessed, with the possibility of removal of vec-tors added at an earlier step. The forward/backwardstepwise procedure terminates automatically when someprespecified threshold values (respectively for F-ratios andchi-square statistics) are encountered for entry andremoval of all candidate eigenvectors. The ultimate

inclusion criterion is determined by the MC value of theresiduals, which should indicate an absence of spatialautocorrelation. Satisfying this MC condition sometimesrequires supervised backward elimination of marginallyselected eigenvectors because their inclusion has forcedthe residual MC value to decrease too far below 0. Thisfinal stopping criterion for the linear Gaussian model isrelatively easy to implement because MC distributionaltheory is known for linear regression residuals; a corre-sponding stopping rule for generalized linear modelingregression is far more difficult to implement because of alack of such distributional theory.

Spatial filters for both 2003 and 2004 maps of WNVdeaths are a mixture of eigenvectors representing positiveas well as negative spatial autocorrelation. The 2003 Gaus-sian analysis (see additional file 4: Data input, prepara-tion, and estimation of the Gaussian spatial filter modelwith SAS) identifies the following as prominent eigenvec-

Boxplots of raw and transformed percentage deaths from diagnosed WNV casesFigure 3Boxplots of raw and transformed percentage deaths from diagnosed WNV cases.



tors, with the nature of their respective spatial autocorre-lation denoted in parentheses:

E1 (+), E2 (+), E6 (-), E10 (+), E15 (-), E21 (-) and E25 (-).

Meanwhile, the 2003 generalized linear model logisticregression analysis (see additional file 5: Data input, prep-aration, and estimation of the logistic spatial filter modelwith SAS) identifies the following as prominenteigenvectors:

E1 (+), E6 (-), E11 (-), E15 (-), E18 (+) and E25 (-).

Eigenvectors E1, E6, E15 and E25 are common to these twosets. The 2004 Gaussian analysis identifies the followingas prominent eigenvectors:

E1 (+), E3 (+), E15 (-), E24 (-), E26 (-), E27 (+) and E28 (-).

Meanwhile, the 2004 generalized linear model logisticregression analysis identifies the following as prominenteigenvectors:

E1 (+), E3 (+) and E15 (-).

Eigenvectors E1, E3 and E15 are common to these two sets.Maps of eigenvectors E1 and E15, common to all four ofthese sets, appear in Figure 4; these represent prominentmap patterns underlying the geographic distribution ofWNV death percentages.

The MCMC techniqueMCMC provides a mechanism for taking spatially depend-ent samples from probability distributions in situations

where the usual sampling is difficult, if not impossible.Many auto-models fall into this category, particularlybecause the normalizing constants for their joint or poste-rior probability distributions are either too difficult to cal-culate or analytically intractable. MCMC is used tosimulate from some n-by-1 joint probability distributionp known only up to a constant factor, c. That is, p = cq,where q is known but c is unknown and an intractablemathematical expression [see [15], pp. 428 and 431, formathematical statements of c for auto-Poisson and auto-binomial models]. MCMC sampling begins with condi-tional (marginal) probability distributions, and parame-ter estimates that are obtained using pseudo-likelihoodestimation (i.e., an autoregressive term is estimated with aconventional regression procedure; see additional file 6:Data input, preparation, and pseudo-likelihood estima-tion of the autologistic model with SAS). This involvesestimating covariate coefficients (β) and ρ as thoughobservations are independent. MCMC outputs a sampleof values for each parameter drawn from the joint poste-rior probability distribution.

Gibbs sampling is a MCMC scheme for simulation from pwhere the Markov chain transition matrix (M) is definedby the n conditional probability distributions of p. It is astochastic process that returns a different result with eachexecution, a method for generating a joint empirical dis-tribution of several variables from a set of modeled condi-tional distributions for each variable when the structure ofdata is too complex to implement mathematical formulaeor directly simulate. It is a recipe for producing a Markovchain that yields simulated data that have the correctunconditional model properties, given the conditionaldistributions of those variables under study [16]. The

Maps of selected eigenvectorsFigure 4Maps of selected eigenvectors. Vector element values are directly proportional to darkness of the gray scale. Left (a): marked positive spatial autocorrelation (MC = 1.06). Right (b): strong negative spatial autocorrelation (MC = -0.50).



principal idea behind it is to convert a multivariate prob-lem into a sequence of univariate problems, which thenare iteratively solved to produce a Markov chain. The fol-lowing Gibbs sampling algorithm description [17] is for aselected auto-model and uses the pseudo-likelihoodparameter estimates of the parameters β and ρ:

Step 1: Initialize a map (k = 0) by taking i = 1, ..., n inde-pendent random samples {yi,k=0} from a chosen probabil-ity model (e.g., a binomial model).

Step 2: Obtain new values (initially k = 1) yi,k by sequen-tially moving from one location (i) to another (j) on theinitial map and randomly sampling from the appropriateauto-model (e.g., the auto-binomial model) using thepseudo-likelihood parameter estimates. Site selection forthis process of obtaining {yi,k=1} from {yi,k=0} can followrandom permutations of location sequences. The value ateach location is updated immediately after it is computed.

Step 3: Obtain new values (initially k = 2) yi,k+1 by sequen-tially moving from one location to another on the kth

map, again randomly sampling from the appropriateauto-model, and immediately updating the value at eachlocation.

Step 4: Repeat step 3 for iterations k = 3, 4, ..., until con-vergence of the sufficient statistics of the parameters ofinterest occurs.

The final output then can be used to compute maximumlikelihood estimates of parameters.

Once a Markov chain transition matrix is constructed, asample of (correlated) drawings from a target distributioncan be obtained. This is done by simulating the Markovchain a large number of times (say, 525,000), removing a"burn-in" set of iterations (say, 25,000), weeding it (selectonly every, say, 100th result), and recording its sufficientstatistics. Convergence needs to be monitored, and hencethe sufficient statistics need to be recorded. This recordingshould be done after the completion of each iteration. Asuitable burn-in period is needed in order to generate M,the limiting Markov chain transition probability matrix,and hence before collecting statistics, and because sam-ples are serially correlated, the chain needs to be weeded.

One difficulty with estimation of an auto-binomialmodel, which is supported by MCMC techniques, is thatthe relationship between its intercept and autoregressiveparameter is established by the global percentage ofdeaths for a disease map. The ideal situation occurs whenρ = -α/2 [[18], p. 3]; this point is illustrated here in Table3. Because this parameter estimate is inconsistent with theother model specifications, this model is not treated in

great detail here. Of note is that the WinBUGS softwarepackage cannot be used to estimate this model becausethe spatial lag variable must be recomputed at eachMCMC step.

The impact of spatial autocorrelation on the frequencydistribution of a binomial random variable is illustratedin Figure 5. For a 100-by-100 lattice, pseudo-randombinomial counts were generated for N = 1,000 and p =0.05; each count then was divided by 10,000. As conven-tional statistical theory states, the distribution of these val-ues is approximately normally distributed with mean 0.05and variance (0.05)(1-0.05)/100. The principal impact ofspatial autocorrelation is to reduce the more central fre-quencies and increase the tail frequencies. As spatial statis-tical theory states, the mean remains 0.05, but thevariance increases (here by a factor of 1.55), regardless ofwhether only positive or only negative spatial autocorre-lation is contained in the probabilities. The effect of mix-ing an equal amount of positive and negative spatialautocorrelation is to have some of the autocorrelationeffects cancel out; now the variance is inflated by a factorof 1.35, with the equal mixture better preserving the orig-inal kurtosis.

The spatial Bayesian HGLMMeanwhile, Bayesian random-effects HGLM specifica-tions also can be used to deal with non-normal data. Oneappealing feature of this approach is that spatial autocor-relation in a non-normal georeferenced random variablecan be captured without having to derive an explicit mul-tivariate generalization of its distributional form. Win-BUGS [12] and GeoBUGS [13] support implementationof this model for disease mapping purposes. When ana-lyzing maps of disease, the random effects can be spatiallystructured and/or unstructured. The CAR model is oneway spatial structuring is included; the spatial filter isanother way. If the autoregressive parameter ρ is esti-mated, the specification is called a PCAR model. If theautoregressive parameter value is set equal to 1, then a sec-ond unstructured random effect term is included, and thespecification is called an improper CAR (ICAR) model.Parameters are estimated with MCMC techniques.

Because Bayesian statistical analysis is involved, prior dis-tributions need to be posited for each varying quantity:the response variable, each variable coefficient, the spatialautoregressive parameter, the error variance, and the ran-dom error term. The response variable prior distributionincludes the model statement. The random error termmay be posited as a PCAR or an ICAR specification. Ifeither a ICAR or a spatial filter term is included, then aprior distribution for an unstructured random effect mustbe included. For a disease map, the response variable priordistribution frequently will be Poisson (for counts) or



binomial (for presence/absence or percentages). Theaccompanying CAR/PCAR frequently involves an auto-normal specification. The spatial autoregressive parameterprior distribution usually is uniform. The normal distribu-tion furnishes a feasible prior distribution for covariatecoefficients, including the spatial filter. And, the error var-iance prior distribution often is the gamma distribution.

The following binomial HGLM involving the PCARmodel was estimated using the US WNV data andGeoBUGS (see additional file 7: 2003 data input andestimation of the PCAR logistic spatial filter model withGeoBUGS):

where α is the intercept, I0 is the binary 0–1 indicator var-iable for 0-case states, β1 denotes its regression coefficient,and νi denotes unobserved US state-specific random

effects. The prior distributions attached to this log-meanresponse equation are:

Di ~ binomial(pi, Ci),

α ~ normal(0, 0.0001),

β1 ~ normal(0, 0.0001),

νi ~ auto-normal , with a conditional

autoregressive model specification corresponding to aproper multivariate Gaussian distribution with a full-rankcovariance matrix (I - ρ C),

~ gamma(0.5, 0.0005), and

ρ ~ uniform(1/λ48, 1/λ1), where λ48 and λ1 respectively arethe smallest and largest eigenvalues of matrix C,

Frequency distributions for simulated binomial probabilities (N = 10,000, p = 0.05, 100-by-100 lattice)Figure 5Frequency distributions for simulated binomial probabilities (N = 10,000, p = 0.05, 100-by-100 lattice). Top left (a): random simulated data. Top right (b): simulated data with marked positive spatial autocorrelation embedded (MC = 1.01). Bottom left (c): simulated data with marked negative spatial autocorrelation embedded (MC = -1.01). Bottom right (d): simu-lated data with an equal mixture of marked positive and marked negative spatial autocorrelation embedded.

pe

i + I1 0 i= −

+ +11

1 α β ν ,

( , )cij jj=1

nν σε

−∑ 2

σε−2



where ~ denotes "distributed as," Di and Ci respectivelydenote the number of deaths and the number of cases instate i, and σε is the standard deviation of the randomeffects term.

The spatial filter version (see additional file 8: 2003 datainput and estimation of the logistic spatial filter modelwith WinBUGS) removes the CAR specification, and addsa spatial filter term together with its regression coefficientβ2 ~ normal(0, 0.0001); the random effects now are inde-

pendent normal(0, ).

Additional material

AcknowledgementsThis research was supported by NIH grant #P20RR20770, and presented to the P20 Exploratory Centers Program Workshop, University of Miami, May 22–25, 2005.

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Additional File 1Cases of and deaths attributed to WNV; US state geographic connec-tivity matrix. Tabulated state-by-attribute data (a), and tabulated state-by-state binary geographic connectivity matrix data (b).Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S1.pdf]

Additional File 2Data input and preparation for the SAR model estimation with SAS; SAR model estimation with SAS. SAS computer code, in which the input data file paths and file names may need to be changed (a), for estimating a simultaneous spatial autoregressive model (b).Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S2.pdf]

Additional File 3Minitab 14.13 code for computing spatial filter eigenvectors. Minitab computer code, in which the input data file paths and file names may need to be changed; the number of areal units is stored in K1, the geographic connectivity matrix is stored in M1, and the spatial filter eigenvectors are stored in M2.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S3.pdf]

Additional File 4Data input, preparation, and estimation of the Gaussian spatial filter model with SAS. SAS computer code, in which the input data file paths and file names may need to be changed, for estimating a linear regression spatial filter model.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S4.pdf]

Additional File 5Data input, preparation, and estimation of the logistic spatial filter model with SAS. SAS computer code, in which the input data file paths and file names may need to be changed, for estimating a generalized lin-ear (logistic) regression spatial filter model.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S5.pdf]

σε−2

Additional File 6Data input, preparation, and pseudo-likelihood estimation of the auto-logistic model with SAS. SAS computer code, in which the input data file paths and file names may need to be changed, for estimating a generalized linear auto-logistic regression model.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S6.pdf]

Additional File 72003 data input and estimation of the PCAR logistic spatial filter model with GeoBUGS.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S7.pdf]

Additional File 82003 data input and estimation of the logistic spatial filter model with WinBUGS.Click here for file[http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S8.pdf]


http://www.biomedcentral.com/content/supplementary/1476-072X-4-18-S1.pdf








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http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/geobugs12manual.pdf


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