methodology open access adjusting for sampling …keywords: disease mapping, sampling variability,...

METHODOLOGY Open Access

Adjusting for sampling variability in sparse data:geostatistical approaches to disease mappingKristen H Hampton1, Marc L Serre2*, Dionne C Gesink3, Christopher D Pilcher4 and William C Miller1,5

Abstract

Background: Disease maps of crude rates from routinely collected health data indexed at a small geographicalresolution pose specific statistical problems due to the sparse nature of the data. Spatial smoothers allow areas toborrow strength from neighboring regions to produce a more stable estimate of the areal value. Geostatisticalsmoothers are able to quantify the uncertainty in smoothed rate estimates without a high computational burden.In this paper, we introduce a uniform model extension of Bayesian Maximum Entropy (UMBME) and compare itsperformance to that of Poisson kriging in measures of smoothing strength and estimation accuracy as applied tosimulated data and the real data example of HIV infection in North Carolina. The aim is to produce more reliablemaps of disease rates in small areas to improve identification of spatial trends at the local level.

Results: In all data environments, Poisson kriging exhibited greater smoothing strength than UMBME. With thesimulated data where the true latent rate of infection was known, Poisson kriging resulted in greater estimationaccuracy with data that displayed low spatial autocorrelation, while UMBME provided more accurate estimatorswith data that displayed higher spatial autocorrelation. With the HIV data, UMBME performed slightly better thanPoisson kriging in cross-validatory predictive checks, with both models performing better than the observed datamodel with no smoothing.

Conclusions: Smoothing methods have different advantages depending upon both internal model assumptionsthat affect smoothing strength and external data environments, such as spatial correlation of the observed data.Further model comparisons in different data environments are required to provide public health practitioners withguidelines needed in choosing the most appropriate smoothing method for their particular health dataset.

Keywords: disease mapping, sampling variability, spatial distribution, spatial analysis, epidemiological methods

BackgroundDisease maps that summarize the spatial and spatio-temporal variation in rates of disease have a wide rangeof applications, from hypothesis generation to publichealth surveillance. Identification of areas with unusuallyhigh or low rates may indicate clusters or ‘hot spots’ ofdisease that can aid decisions regarding intervention orprevention programs, allocation of health care resources,or provide context for further epidemiological studies[1-4]. At the same time, the utility of disease mappingoften depends on how accurately the value beingmapped estimates the spatial process of interest. Esti-mating disease risk requires a defined, closed population

and cannot be measured directly with surveillance data;accordingly, incidence rates are often used to approxi-mate disease risk [5]. However, calculating crude ratesfrom routinely collected health data indexed at a smallgeographical resolution poses specific statistical pro-blems due to the sparse nature of the data, especially forrare diseases [1-6]. In particular, crude rate maps ofsmall areas will be dominated by sampling variability.Due to variation in population size among areas, a mapdisplaying crude rates will tend to be dominated byareas with small populations since small changes to theobserved number of cases will result in large changes tothe rate [5-7]. Meanwhile, observed high rates based onsmall populations are likely to be artificially elevateddue to the high variability in the estimates. In otherwords, error due to sampling variability introducesobservational noise into the map that may obscure the

* Correspondence: [email protected] of Environmental Sciences and Engineering, University of NorthCarolina at Chapel Hill, Chapel Hill, NC, USAFull list of author information is available at the end of the article

Hampton et al. International Journal of Health Geographics 2011, 10:54http://www.ij-healthgeographics.com/content/10/1/54

INTERNATIONAL JOURNAL OF HEALTH GEOGRAPHICS

© 2011 Hampton et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

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underlying spatial process of interest, and, if notadjusted for, lead to incorrect inference about the spatialpattern of interest.One general method to improve the stability of

observed rates is to increase population sizes by upscal-ing from finer to coarser resolution levels, such as fromcensus tracts or ZIP codes to counties. Increasing theaggregation area, however, causes a loss in the resolu-tion of the data, thereby masking spatial details neededto identify, analyze, and monitor health problems at thecommunity level [5,7,8]. At the same time, analyzinghealth data at too fine a resolution, such as with caseevent or point maps, can compromise patient confiden-tiality and may be misleading in some circumstancesbecause the underlying population density is not consid-ered [8]. This paper focuses on the issue of samplingvariability in disease maps at a geographical resolutioncorresponding to an aggregation of counts over fixedsmall geographical regions - in this case, postal codes ofresidence (US postal service ‘ZIP codes’).Both deterministic and model-based approaches have

been proposed to address the issue of sampling variabil-ity, also referred to as the ‘small number problem,’ whenmapping health data at fine resolutions [1,5,9]. Theseapproaches reduce the noise in rates of small areasthrough ‘spatial smoothing,’ which allows areas to bor-row strength from neighboring regions to produce amore stable estimate of the value associated with eachregion [2-4,6,7,10,11]. Among the different methods,however, a tradeoff exists between the computationalcomplexity of each method and the ability to accountfor spatial correlation and uncertainty in the smoothedresults [12]. For example, methods such as disk smooth-ing, population-weighted averages, and empirical Bayesestimates have low computing requirements but do notaccount for spatial autocorrelation or quantify theamount of uncertainty in the smoothed rates [12,13]. Incontrast, full Bayesian hierarchical models are able toincorporate multiple covariates in the model parametersto yield the full posterior distribution of the estimatedrate, yet are computationally cumbersome due to time-consuming iterative procedures and challenging fornon-statisticians to implement and interpret due to theircomplexity [2,12,13].Geostatistical methods modified to account for the

heteroscedasticity of health data, namely that the var-iance at each location varies as a function of the popula-tion size, offer a compromise between computationalburden and quantification of the uncertainty in thesmoothed rate estimate. Methods such as Poisson kri-ging are able to account for spatial correlation and yielda full posterior distribution while being computationallyfaster than fully Bayesian hierarchical models [12-14].Furthermore, simulation studies of cancer mortality

showed that Poisson kriging outperformed simple popu-lation-weighted averages, empirical Bayes smoothers,and the Besag-York-Mollié Bayesian hierarchical modelin measures of estimation accuracy and degree ofsmoothing, particularly when background rate valueswere spatially correlated [12,13]. While spatial smooth-ing is needed to improve the stability of observed ratesbased on small populations, too much smoothingreduces the ability to identify areas of high or low ratesthat may indicate clusters or outbreaks of disease.In this paper, we examine extensions of two geostatis-

tical methods, ordinary kriging and Bayesian MaximumEntropy (BME), which have been used extensively in theanalysis of fields exhibiting spatial variability and arecomputationally efficient [15-22]. Traditional geostatis-tics do not account for the heteroscedasticity of diseaserates and must be modified to address both the numera-tor and denominator of rate data [12-14]. Poisson kri-ging as an extension of ordinary kriging was firstintroduced by Monestiez et al. [23] and applied tohealth data by Goovaerts [12]. Here, we introduce a uni-form model extension of BME (UMBME) that, like Pois-son kriging, can account for changes in variance due topopulation size and compare the performance ofUMBME to that of Poisson kriging in measures of esti-mation accuracy and smoothing strength. We firstexamine differences in underlying model assumptions byapplying each model to simulated datasets where thetrue spatial variation in the data is known. We thenapply both models to the real data example of humanimmunodeficiency virus (HIV) infection with a spatialanalysis of infection occurring in North Carolina HIVtesting data from 1994 to 2002. Attempts to describe ormonitor the distribution of HIV cases detected throughtesting have been limited by the variability in samplingthat is inherent in the testing situation.

MethodsModel definitionsIn traditional epidemiology, the incidence rate of diseaseis defined as the number of new cases in the at-riskpopulation divided by the person-time, or summation ofeach person’s observation time, over a specified period[5,24]. With rare diseases at steady state, the loss of per-son-time per diseased person is minimal, and the totalperson-time at risk may be approximated by the totalsize of the at-risk population over the period times thelength of the period (day, month, year) [5,24]. In otherwords, the incidence rate corresponds to a proportion ofnew cases in the at-risk population divided by the timeduration considered to enumerate the new cases.Following this concept, we define in the spatial con-

text the latent incidence rate as the theoretically possi-ble rate of infection built on a hypothetical infinite


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underlying population at risk. In this paper, we considerobservation datasets aggregated to a single time periodt, resulting in space only analyses and temporally inde-pendent maps that can be used to examine the spatialvariability of the random fields. Therefore, without lossof generality, the latent incidence rate of disease in agiven region i can be defined as:

Xi = limnt→∞

Yini

, i = 1, ...I (1)

where Xi is the latent disease rate, Yi is the number ofnew cases of disease, and ni is the size of the populationat risk in area i. In other words, as the size of the popu-lation ni reaches infinity, then the observed disease rateYi/ni approaches the latent disease rate for that location.In this hypothetical situation, sampling variability isminimized because all regions being mapped are ofequally large populations and small changes in the num-ber of cases do not result in large changes to theobserved rate.In practice, however, only a finite population at risk ni

in area i may be measured, producing a finite numberof cases Yi and an observed rate Ri = Yi/ni. Note thateven when ni is an exhaustive count of the entire popu-lation at risk in area i, maps of the observed rate Ri aresubject to noise due to variation in population sizeamong regions, and areas with rates based on smallpopulations will tend to dominate the map. The differ-ence between the observed rate Ri and latent rate Xi isdirectly related to the observation that Ri can only bedefined in increments of 1/ni, while the underlyinglatent rate can take any value on the real number conti-nuum. The difference between Ri and Xi is inherently ameasurement error εi that can be expressed as:

Ri = Xi + εi (2)

where we refer to εi as the error due to samplingvariability in area i. In this paper, we contrast twoapproaches in modeling sampling variability, Poissonkriging and the UMBME method, which use unboundedand bounded distributions, respectively, for the error εi.

Poisson krigingWith Poisson kriging, the observed number of cases ismodeled as a random variable Yi that follows a Poissondistribution with one parameter. This parameter is theexpected number of cases defined as the product of thepopulation size ni by the latent rate Xi, such that:

Yi | Xi ∼ Poisson (niXi) (3)

As detailed by Goovaerts [12], the Poisson kriginglatent rate in area i is estimated as a linear combinationof K neighboring observed rates, such that:

X̂PKi =

K∑j=1

λijRj (4)

where the weights lij are derived from a Poisson krigingspecific system of linear equations, in which the samplingvariability at i is accounted for by a normally distributederror εi with variance equal to si2 = m*/ni, where m* is thepopulation-weighted mean of the observed rates [12]. Weimplemented the Poisson kriging adaptation of ordinarykriging in the MATLAB programming environment [25].Detailed descriptions of Poisson kriging may be found inGoovaerts (2005) and Goovaerts and Gebreab (2008).

Uniform model extension of Bayesian Maximum Entropy(UMBME)An underlying assumption of UMBME is that theobserved population ni is a representative sample of thepopulation at risk in area i and measured without error.In this case, the observed number of cases Yi given thelatent rate Xi can be interpreted as the product of thepopulation size and latent rate rounded to the nearestinteger, such that:

Yi | Xi = round (niXi) (5)

The discrete nature of case counts requires Yi to be awhole number value. Therefore, when given an observedcase count Yi and population ni, the latent rate Xi fol-lows a uniform distribution bounded by the roundingerror, such that:

Xi | Ri = U(Ri − 0.5

ni,Ri +

0.5ni

)(6)

where Ri is the observed rate Yi /ni. In other words, inUMBME the measurement error εi is assumed to be uni-formly distributed in an interval of size 1/ni, whichexpresses the fact that Ri is interpreted as an observationof the latent rate Xi in increments of 1/ni. To derive theUMBME latent rate estimate from observed values, theBME method treats the observed values as ‘soft data’ withmeasures of uncertainty as defined by the uniform distri-bution. As detailed by Christakos et al. [17], the BMEmethod processes information known about the rate fieldin three main stages: (i) Structural (or prior) stage, (ii) Spe-cificatory (or meta-prior) stage, and (iii) Integration stage.Programs to perform each stage were derived from func-tions found in the BMElib software package for theMATLAB programming environment [25,26]. Detaileddescriptions of the BME method may be found in Christa-kos and Li (1998) and Christakos et al. (2002), amongothers. A general introduction to information processingin BME is presented in Supplementary Materials, Appen-dix A.


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In this study, the UMBME estimate X̂UMBMEi of the

latent rate in area i is the expected value of the poster-ior probability density function (pdf) derived in the Inte-gration stage. Other possible estimators include themode or median of the posterior pdf. Furthermore, thevariance of the posterior pdf, or BME variance, providesa useful measure of the estimation uncertainty.

Neighborhood selectionPoisson kriging and UMBME offer at least two advan-tages in neighborhood selection over their counterparts,particularly simple smoothers that weight all neighborsfalling within an arbitrary fixed distance of areal cen-troids. First, both methods have the ability to limit thenumber of neighbors selected to a user-defined maxi-mum, which can account for changes in the spatial den-sity of centroids, and thus the size of geographical units,across the study area [12,17]. For example, an urbanarea may have dozens of neighbors falling within a fixeddistance while a rural area may only have a handful ofneighbors falling within the same distance. Without lim-itations on the number of selected neighbors, the urbanarea would have significantly more contributors to itssmoothed value than the rural area. Second, Poisson kri-ging and UMBME are able to determine smoothingweights based upon the data’s covariance model. If theneighborhood distance selected is large enough suchthat it includes all the neighborhood information, thenincreasing the distance or number of points does notadd more information or change the model estimation.

Simulated data analysisWe applied both Poisson kriging and UMBME to simu-lated data where the true spatial variation in the latentrate field X(s) is known in order to examine the effect ofunderlying assumptions on each model’s performance inmeasures of estimation accuracy and smoothingstrength. Simulations allow us to both model the truelatent disease rate and to control variables in order toexplore how changes impact the smoothed result[12,13]. For example, we considered a study area dividedinto square regions with the spatial location s of eachregion defined as the center of the corresponding square(Figure 1), which minimized the effect of spatial supportand the modifiable areal unit problem on the smoothedestimates since regions were of uniform size and shape[12]. With real data, not knowing the true latent rate orrelative impact of error sources makes it difficult toassess the accuracy and validity of different smoothingmethods.Since sampling variability generates the greatest dis-

tortion in studies of rare diseases, a low latent rate fieldX(s) was simulated where the latent rate of disease in

each region ranged in value from 0 to 7.5 per 1,000 per-sons (Figure 1A). We modeled the latent rate field byfirst identifying one region each of relative low, medium,and high latent rates with an associated decay rate. Wethen calculated the latent rates for surrounding areasbased upon the distance from each central location. Theresulting disease field where higher rates tend to begrouped around a central location follows the core areatheory of sexually transmitted diseases such as syphilis[19,27].The sample population size n(s) was then determined

using a random number generator, resulting in valuesranging from 1 to 767 (Figure 1B). While general popu-lation density tends to be spatially correlated, it is rarefor the populations sampled by public health investiga-tions and surveillance systems to be perfectly represen-tative of the general population. Instead, factors such asoutreach efforts and concentrations of special popula-tions can cause the populations sampled in neighboringareas to vary significantly. Furthermore, population den-sity is dependent on geographic scale. Even in urbanstudy areas, the population density of, for example, cen-sus block groups may represent a random patchworkdepending on the locations of commercial and residen-tial areas.A fundamental difference in the model assumptions of

Poisson kriging and UMBME is the probability distribu-tion of the observed cases, Y(s), given the latent rate, X(s). If the observed cases, Y(s), are drawn from a Poissondistribution, then Poisson kriging should outperformUMBME in estimating the latent rate. Conversely, if Y(s)integer values are derived from the product of the latentrate, X(s), and population size, n(s), then UMBMEshould outperform Poisson kriging in estimation accu-racy. We derived two realizations of the case count, Y(s), to verify these characteristics and examine differ-ences in spatial variability between assumptions. Giventhe latent rate field, X(s), and sample population, n(s),one realization of Y(s) was sampled from a Poisson dis-tribution (Figure 2A), while the other resulted from theproduct of X(s) and n(s) (Figure 2B). In both realiza-tions, Y(s) ranged in value from 0 to 3 cases.For a given realization, we have the simulated (true) X

(s) and the observed rate, R(s), from which we calculated

both the Poisson kriging estimate, X̂PKi , and UMBME

estimate, X̂UMBMEi . We then calculated two statistics as

measures of estimation accuracy: the mean square error(MSE) and Lin’s Concordance Correlation Coefficient

(LCCC) between the estimated value ( X̂PKi or X̂UMBME

i )

and the simulated true value for X(s). Divergencebetween the estimated and true latent rate valuesdecreases as the MSE approaches zero and LCCC


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approaches one [28]. Meanwhile, smoothing strengthwas described as the degree to which each techniquemodified high or low observed rates towards the globalmean. More smoothing indicates greater movementtowards the global mean and less spatial variability inestimated values. While smoothing strength may bequalitatively assessed by examining, for example,whether regions of high observed rates are discernablein the estimated map, we also calculated the mean abso-lute difference (MAD) between estimated values and themean of observed rates. As the MAD approaches zero,the average deviation of estimated values from the glo-bal mean decreases, indicating greater smoothing ofobserved values.

Mapping HIV in North CarolinaTo examine model performance in real world situations,we mapped HIV disease rates in North Carolina usingboth Poisson kriging and UMBME. Previous studies ofsexually transmitted infections (STIs) demonstrate thatcases tend to be concentrated in residential core areaswith high rates of infection in small, definable geo-graphic areas [19,21,27,29-31]. In North Carolina,approximately 1,700 new diagnoses of HIV disease arereported to the North Carolina public health surveil-lance system each year, with about 25 per cent of NorthCarolina’s HIV disease reports consistently coming fromrural, or non-metropolitan, areas since the early 1990s[32]. Regional differences in the spread of HIV highlight

B)A)

Figure 1 Maps of the simulated A) latent rate field X(s) and B) sample population size n(s). The values of X(s) and n(s) ranged from 0 to7.5 per 1,000 persons and from 1 to 767 persons, respectively.

B)A)

Figure 2 Maps of the simulated case count Y(s) under the A) Poisson assumption and B) uniform assumption. Under the Poissonassumption, the observed cases are drawn from a Poisson distribution. Under the uniform assumption, the observed cases are derived from theproduct of the latent rate and population size. In both realizations, the values of Y(s) ranged from 0 to 3 cases.


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the necessity for accurate information on the spatial dis-tribution of the disease in order to effectively target pre-vention activities and identify possible community-levelrisk factors contributing to the epidemic.In this analysis, we calculated rates of new HIV dis-

ease reports in North Carolina’s testing population from1994 to 2002. Specifically, clients at risk of HIV infec-tion presenting to publicly-funded Voluntary Counselingand Testing (VCT) sites distributed across the state ofNorth Carolina were tested for long-term HIV infection.ZIP code of residence was recorded in a research data-base for each de-identified test result. Records for clientsreporting previous positive HIV tests were excludedfrom analysis. The remaining records were geocoded tothe residential-weighted centroid of the reported ZIPcode of residence [33]. There were 938,889 total testsand 5,677 new HIV cases reported between 1994 and2002, of which 96% to 97% matched to a location. Rea-sons for failing to geocode included having a missing,invalid, or out-of-state ZIP code. We modeled theobserved HIV infection rate using Poisson kriging andUMBME and evaluated model performance.In real world situations, the true latent rate of disease

is unknown. Therefore, in order to assess model fit, weused cross-validatory predictive checks, which have beenshown to be useful in determining whether divergentrates are due to poor model fit or to actual ‘hot-spots’of disease that warrant further investigation [34,35]. Thebasic concept of cross-validation is to remove eachobserved rate in turn, in this case observed ratesassigned to a ZIP code population centroid, then reana-lyze the original data without that observation, andassess the model’s ability to predict the removed area’sdata value. Estimation accuracy was then measured asthe MSE and LCCC between the model-predicted valueand the observed rate. Smoothing strength was assessedboth qualitatively and as the MAD between model-pre-dicted values and the mean of observed rates.

Results and discussionSimulated data analysisIn applying smoothing models to the simulated rate fieldR(s), we considered first the spatial moments of thelatent rate spatial random field X(s). In the absence ofprior information regarding the general behavior of X(s),we assigned no mean trend to the model and used theinformation provided by the rate field R(s) directly inour calculations rather than as a residual with the meantrend removed. This enabled us to model the spatialvariability of X(s) using the covariance of R(s). A usefulfeature of the covariance model is that it does notrequire prior distributional assumptions of the data andcan be applied to both the Poisson and uniform simula-tions. Other estimators, such as Monestiez et al.’s

population-weighted semivariogram and Yu et al.’s Pois-son regression model in BME, have measurable advan-tages but are dependent on the Poisson assumption[23,36]. Furthermore, covariance plots provide a quanti-tative assessment of the correlation between pairs ofpoints and are useful in assessing the strength and scaleof the disease pattern [19]. For example, the covariancerange, or distance where the curve becomes asymptoticto the x-axis, may be used to identify the neighborhoodof influence around an observation point. Observationswithin the covariance range may influence values at thecurrent location, whereas observations outside thisrange may not be influential. In addition, local diseasepatterns may be described by the behavior of the covar-iance model near the origin. Steep curves indicate rapidchange over a short distance, while long-tailed curvesindicate slower change and less variability over the samedistance.

Poisson assumptionThe experimental covariance calculated from the corre-sponding observed rates shows the change in the corre-lation between pairs of points as the distance betweenpoints increase in space. For the Poisson assumption,the experimental covariance indicated high local varia-bility with a neighborhood of influence extending lessthan 2 spatial units from each point (Figure 3). The cor-responding R(s)-covariance model (Figure 3A) wasobtained by fitting a nugget-exponential-Gaussiannested model to the experimental R(s)-covariance values,such that

cR (ρ) = c1δρ + c2 exp(−3ρ

aρ1

)+ c3 exp

(−3ρ2

aρ22

)(7)

where r is the spatial lag, c1, c2, and c3 are constantswhose sum equals the variance of R(s), δr is the Kro-necker delta function, and ar1 and ar2 represent thespatial ranges of the exponential and Gaussian compo-nents, respectively. The nugget component of this covar-iance model corresponds to the initial drop of thecovariance, while the exponential and Gaussian compo-nents correspond to the tail of the covariance curve.This covariance model is adequate if we are interestedin mapping the observed rate field R(s) over the geogra-phical region of interest using observed rate data. Onthe other hand, if we are interested in mapping the truelatent rate field X(s), we then need to model its covar-iance function. We obtain the X(s)-covariance model bysimply recognizing that the nugget component (initialdrop of covariance) of the R(s)-covariance is due to thesampling variability error term ε defined in 2. Therefore,we obtained the X(s)-covariance model (Figure 3B) byremoving the nugget component of the R(s)-covariance


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model and keeping only the exponential and Gaussiancomponents, such that

cX (ρ) = c2 exp(−3ρ

aρ1

)+ c3 exp

(−3ρ2

aρ22

)(8)

where r is the spatial lag, c2, and c3 are constants, andar1 and ar2 represent the spatial ranges of the exponen-tial and Gaussian components, respectively.The observed rates Ri, calculated from the cases simu-

lated under the Poisson kriging assumption, ranged invalue from 0 to 29.1 per 1,000 persons (Figure 4A).Using these observed rates and the corresponding covar-iance models (Figure 3), we obtained the Poisson kriging

estimate X̂PKi (Figure 4B) and the UMBME estimate

X̂UMBMEi (Figure 4C). Spatially, X̂PK

i and X̂UMBMEi exhib-

ited nearly identical spatial distributions of non-zerovalues. This result is due in part to the similar identifi-cation of neighbors and integration of spatial correlationbetween the models. However, when compared with theobserved rate map, Poisson kriging exhibited greatersmoothing strength than UMBME, particularly in areasof high observed rates, such as region (6,6) of the studydomain. Similarly, the Poisson kriging MAD was lowerthan that of UMBME (Table 1), indicating greatersmoothing towards the observed mean with Poisson kri-ging than UMBME. With regards to estimation accu-racy, scatterplots of the observed rates Ri, the Poisson

kriging estimates X̂PKi , and the UMBME estimates

X̂UMBMEi versus the true latent rates Xi (Figures 4D, E,

and 4F, respectively) show that the closer a point is tothe 45 degree best fit line, the more accurately the

observed or estimated value approximates the true X(s)-value. Artificially elevated rates fall above the best fitline, with the observed rate plot (Figure 4D) showingthe greatest deviation from the true X(s)-value. Quanti-tatively, Poisson kriging exhibited the highest estimationaccuracy in measures of both MSE and LCCC whencompared with the true X(s)-value (Table 1).

Uniform assumptionUnder the uniform assumption, the observed cases, Y(s),were derived from the product of the latent rate field, X(s), and sample population, n(s) (Figure 2B). We thenobtained experimental and model covariance values forthe corresponding observed rate field R(s) (Figure 5A)and latent rate field X(s) (Figure 5B). Similar to those ofthe Poisson assumption, the experimental covarianceindicated a neighborhood of influence extending lessthan 2 spatial units from each point. However, theexperimental covariance under the uniform assumptionhad a smaller nugget effect, indicating lower local varia-bility [5]. In other words, observations that were close inspace had more similar values under the uniformassumption than the Poisson assumption.The observed rates Ri, calculated from the cases simu-

lated under the uniform assumption, ranged in valuefrom 0 to 7.5 per 1,000 persons (Figure 6A). Using theseobserved rates and the corresponding covariance models(Figure 5), we obtained the Poisson kriging estimates

X̂PKi (Figure 6B) and the UMBME estimates X̂UMBME

i

(Figure 6C). Once again, the Poisson kriging andUMBME maps displayed nearly identical spatial distri-butions of non-zero values, while Poisson kriging exhib-ited greater smoothing strength than UMBME with

• experimental model


B)A)

Figure 3 Plots of the experimental rate covariance with covariance models under the Poisson assumption. The nugget component ofthe A) observed rate field R(s)-covariance model is due to sampling variability error and removed to obtain the B) X(s)-covariance model.


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D)A)

E)B)

F)C)

Figure 4 Maps and scatterplots of the observed, Poisson kriging estimated, and UMBME estimated rates under the Poissonassumption. Compared with the A) observed rate map, the B) Poisson kriging map displayed greater smoothing than the C) UMBME map,particularly in areas of high observed rates (cases per 1,000 persons). Meanwhile, scatterplots of the D) observed, E) Poisson kriging estimated,and F) UMBME estimated rates versus the true latent rate, X(s), combined with MSE and LCCC calculations (Table 1), demonstrated that Poissonkriging produced the highest estimation accuracy under the Poisson assumption.


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both less discernible high rate areas, such as region(6,5), and a lower MAD value (Table 1). In this case,however, the additional smoothing provided by Poissonkriging resulted in greater disparity with the true X(s)-value than UMBME, with a greater proportion ofUMBME estimated points falling along the 45 degreebest fit scatterplot line than those of Poisson kriging(Figures 6D, E, and 6F). Quantitatively, UMBME

exhibited the highest estimation accuracy in measures ofboth MSE and LCCC (Table 1).

Mapping HIV infection in North CarolinaWe applied Poisson kriging and UMBME to North Car-olina HIV disease data to examine their performance ina real-world situation. We derived experimental andmodel covariance values for the observed rate field (Fig-ure 7A) and corresponding X(s) field (Figure 7B). Theexperimental covariance indicated a spatial neighbor-hood of influence extending less than 20 km from eachlocation. While a nugget effect was present, without apoint of comparison, it was difficult to ascertain whetherthe amount of spatial correlation was large or small forthe given disease and geographical region. We thenobtained maps of the HIV observed rate, the Poisson

kriging estimates X̂PKi , and the UMBME estimates

X̂UMBMEi (Figures 8A, B, and 8C, respectively). Similar to

the simulated data analyses, Poisson kriging exhibitedgreater smoothing strength than UMBME with both lessdistinction between areas of high and low rates, such asin the northeast corner of the state, and a lower MADvalue (Table 1). In cross-validation, UMBME performedonly slightly better than Poisson kriging in MSE andLCCC values, with both models performing better thanthe observed data model with no smoothing (Table 1).

Model comparisonUnder both the Poisson and uniform assumptions of thesimulated data, Poisson kriging exhibited greatersmoothing strength than UMBME. The two methodsutilize the same values for a number of parameters that

Table 1 Measures of model estimation accuracy,smoothing strength, and variance quality for thesimulated Poisson assumption data, simulated uniformassumption data, and real HIV data.

Method MSE LCCC MAD G

Poisson Simulation

Observed 1.00E-05 0.214

Poisson kriging 1.47E-06 0.550 0.938 0.669

UMBME 4.83E-06 0.396 1.33 0.798

Uniform Simulation

Observed 1.52E-06 0.618

Poisson kriging 1.15E-06 0.555 0.510 0.670

UMBME 6.43E-07 0.794 0.701 0.781

HIV Data

Observed 1.87E-04 0.013

Poisson kriging 1.80E-04 0.039 0.00297

UMBME 1.79E-04 0.040 0.00369

The MSE and LCCC for the simulated data describe divergence between themodel estimated and true latent rate values. For the HIV data, the MSE andLCCC values describe divergence between the model cross-validation resultsand observed rate values. Bolded values represent the model with thegreatest estimation accuracy (MSE, LCCC), greatest smoothing strength (MAD),or best fit variance (G) in each dataset.



B)A)

Figure 5 Plots of the experimental rate covariance with covariance models under the uniform assumption. The nugget component ofthe A) observed rate field R(s)-covariance model was removed to obtain the B) X(s)-covariance model.


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D)A)

E)B)

F)C)

Figure 6 Maps and scatterplots of the observed, Poisson kriging estimated, and UMBME estimated rates under the uniformassumption. Compared with the A) observed rate map, the B) Poisson kriging map displayed greater smoothing than the C) UMBME map(cases per 1,000 persons). However, scatterplots of the D) observed, E) Poisson kriging estimated, and F) UMBME estimated rates versus the truelatent rate, X(s), combined with MSE and LCCC calculations (Table 1), demonstrated that UMBME produced the highest estimation accuracyunder the uniform assumption.


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affect smoothing strength, such as the maximum num-ber of neighbors used in the estimation, the populationsize, and the covariance model [13]. The two methodsdiffer, however, in their measurement error variances ofthe observed values and in their distributions of the pos-terior pdf. The error variance term is m*/ni for Poissonkriging, where m* is the population-weighted mean ofthe observed rates and ni is the population in area i,while the UMBME error term is distributed in an inter-val of size 1/ni. Furthermore, Poisson kriging yields aGaussian posterior pdf characterized by the kriging esti-mate and variance, thereby allowing extreme values dueto the tail of the Gaussian distribution [13]. UMBME,on the other hand, yields a non-Gaussian posterior pdfthat is truncated by the interval soft data. PossibleUMBME estimated values are therefore limited by thebounds of the truncated posterior pdf.The distributional form of the posterior pdf may also

contribute to differences in the estimated variance ofeach method. Under both the Poisson and uniformassumptions of the simulated data, the Poisson krigingand UMBME estimated variances were generally inver-sely proportional to the areal population size, while themagnitude of the Poisson kriging variance was consis-tently higher than that of UMBME. However, of greaterinterest is how well the estimated variance captures thetrue latent rate. Assuming normality of the predictionerrors, we calculated each method’s statistical coverage,or probability that the true latent rate in each area fallswithin the p-probability interval characterized by theestimated mean and variance. For example, for a p valueof 0.95, the ideal model coverage probability would also

equal 0.95. We then calculated Deutsch’s “goodness”statistic [12,37] to examine divergence between themodel-estimated and theoretical probability intervals,such that

G = 1 − 1L

L∑l=1

w(pl) | ς(pl) − pl |with 0 ≤ G ≤ 1 (9)

where L is the discretization level of the computation(i.e. L = 95 when pl = 0.95), ς(pl) is the model-estimatedcoverage for probability interval pl, and w(pl )= 1 if ς(pl)>pl, and 2 otherwise. Double weight is given to devia-tions where the model-estimated coverage is less thanexpected [12]. The goodness of fit of the estimated var-iance increases as the value of G approaches one. WithL = 100, UMBME outperformed Poisson kriging in Gvalues for both the Poisson and uniform simulations(Table 1).While a drawback of Poisson kriging’s greater smooth-

ing strength is that it becomes harder to distinguishareas of high or low estimated rates, in measures of esti-mation accuracy Poisson kriging performed better thanUMBME under the Poisson assumption, UMBME per-formed better than Poisson kriging under the uniformassumption, and both models performed better than theobserved rate with no smoothing. The Poisson assump-tion experimental covariance had a greater nugget effectthan that of the uniform assumption, indicating less spa-tial correlation among the Poisson-derived observedrates than the uniform-derived rates, as shown in Fig-ures 3 and 5. These results support those of previoussimulation studies, which show that methods with



B)A)

Figure 7 Plots of the experimental rate covariance with covariance models for the North Carolina HIV data. The experimental covarianceindicated a spatial neighborhood of influence extending less than 20 kilometers from each location. The nugget component of the A) observedrate field R(s)-covariance model was removed to obtain the B) X(s)-covariance model.


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0 5 10 15 20 25 30

A)

B)

C)

Figure 8 Maps of the North Carolina HIV A) observed, B) Poisson kriging estimated, and C) UMBME estimated rates. While the Poissonkriging and UMBME maps displayed nearly identical spatial distributions of non-zero values, Poisson kriging exhibited greater smoothingstrength than UMBME, thereby providing less distinction between areas of high and low rates (cases per 1,000 tests).


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greater smoothing strength are more accurate estimatorswhen the spatial autocorrelation of the observed data islow, while methods that smooth less have greater esti-mation accuracy with data that is more spatially corre-lated [12].In a real-world situation where the true latent rate of

disease is unknown, the Poisson kriging and UMBMEestimated HIV rate maps compared with the observedHIV rate map (Figure 8) show how the noise generatedby artificially high rates due to sampling variability maybe reduced through smoothing. Smoothing out samplingvariability is particularly important in the study of HIVinfection in North Carolina because it facilitates correc-tion of the rates observed in rural areas with small test-ing populations by borrowing strength from ratesobserved in neighboring areas with larger testing popu-lations. By exhibiting greater smoothing strength, thePoisson kriging map (Figure 8B) also eliminated muchof the distinction between high and low rates needed toidentify outbreaks or clusters of disease. The UMBMEmap (Figure 8C), on the other hand, provided a similarspatial representation of HIV that smoothed outextreme rates based on small populations while main-taining areas of high observed rates that may be usefulin monitoring the HIV epidemic. For example, the num-ber of areas with rates above 10 cases per 1,000 testswas reduced by 59% with Poisson kriging, but only 12%with UMBME as compared with the observed rates(Table 2). Both Poisson kriging and UMBME, however,produced nearly identical cross-validation MSE andLCCC values. Without knowing the true latent rate ofinfection, additional information, such as the distribu-tional form of the dataset, is needed to assess model fit.Similarly, we could not calculate Deutsch’s goodness sta-tistic, G, for the real data to assess model uncertaintybecause the true latent rate of infection was unknown.However, as with the simulated data, the model-esti-mated variances (Figure 9A for UMBME) were generallyinversely proportional to the underlying population size(Figure 9B).

ConclusionsGeostatistics modified to account for the specific natureof health data provides a rich class of methods able to

incorporate and output more information about the dis-ease field than simple smoothers while being computa-tionally faster and easier to implement than fullBayesian hierarchical methods. This facilitates the inves-tigation of both internal and external factors that affectmodel performance. As shown in this paper, both Pois-son kriging and UMBME smoothing models more accu-rately predict the latent rate than the observed data withno smoothing. Meanwhile, Poisson kriging yieldedsmoother results than UMBME due to internal modelassumptions.Choosing the most appropriate smoothing method

depends heavily on the characteristics of the diseasebeing studied and the geographical space. As shownwith the simulation data, accuracy of each estimationmodel was associated with the observed spatial correla-tion of the disease field. Methods that smoothed lessperformed better as the spatial correlation of the diseasefield increased. However, the observed spatial correla-tion, in turn, depends on the assumptions and charac-teristics of the latent and observed rate fields. For thelatent rate, spatial characteristics of the disease, such asgeographic risk factors and disease frequency, must beconsidered. For example, STIs tend to be concentratedin small, definable geographic areas where cases reside[19,21,27,29-31]. Therefore, maps of the residential loca-tions of STI cases would typically exhibit greater spatialcorrelation than maps of the residential locations of anon-transmissible disease, such as leukemia. As a result,estimation methods with limited smoothing, such asUMBME, would be expected to produce more accuratepredictions of the latent rate than methods that smoothmore. However, the robustness of the observed datamust also be considered. Factors that would increase ordecrease the observed spatial correlation include thespatial support of the study area, such as the size andshape of geographical units, and the temporal resolutionof the study. Observations measured over small timedurations, such as monthly for STIs in North Carolina,tend to be spatially less correlated than observationsidentified over longer time periods, such as quarterly orannually.Finally, in real world situations, the relative degree of

spatial correlation in the observed data is difficult toascertain when no point of reference exists. Furtherresearch is needed to compare the advantages and dis-advantages of different smoothing models, examinemodel performance in different data environments, andexamine model performance using different estimators.Additional research is also needed to examine whetherfitting real data by a distributional form improves modelperformance and to identify other information about thedisease field that would aid investigators in choosing themost appropriate smoothing model. Better disease map

Table 2 Number of ZIP codes with values above the 90th

percentile of observed rates.

Method No. ZIPs > 10 cases/1,000 tests %Δ fromObserved

Observed 74 –

UMBME 65 -12%

Poissonkriging

30 -59%


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construction would improve the ability of public healthofficials to monitor spatial and temporal trends in dis-ease rates, creating new opportunities for the definitionof at-risk populations, identification of outbreaks, andallocation of resources toward areas and populationsmost affected by disease.

Supplementary MaterialsAPPENDIX A: The BME methodIn the spatial application of BME, the distribution of adisease rate field is represented in terms of a Spatial

Random Field (SRF), X(s), where s is the spatial location.Each region i is defined in terms of a point spatial loca-tion, such as the latitude and longitude of the areal cen-troid, where si = (s1i , s2i) and i = 1,...I. The spatialmoments of X(s) can be used to describe the behaviorof the SRF. For example, the mean function

mx(s) = X(s) (A1)

(the overbar denotes stochastic expectation), charac-terizes trends and systematic structures in space, while

x 10

A)

B)

Figure 9 Maps of the A) UMBME estimated variance and B) testing population for the North Carolina HIV dataset. The estimatedvariance was generally inversely proportional to the underlying population size.


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the covariance function

cx(ρ) = [X(s) − X(s)][X(s’) − X(s’)] (A2)

expresses relevant correlations and dependenciesbetween pairs of points in X(s), where ρ =| s’ − s |denotes the spatial lag.The BME approach categorizes all prior information

known about the rate field into two major knowledgebases (KB): the general KB G and the specificatory (orsite-specific) KB S , where the total knowledge baseK = G ∪ S . As discussed in Choi et al. (2003), theG − KB is considered ‘general’ in the sense that it char-acterizes global characteristics of the rate field, such asits mean trend, spatial moments, relevant epidemiologiclaws or theories, and other assumptions about the beha-vior of the field that may apply. On the other hand, theS − KB refers to information that is ‘specific’ to eachmapping situation, such as observed rate values mea-sured at specific data points. In general, the vector ofrandom variables xmap representing the rate field X(s)consists of the vector of hard data random variablesxhard representing the latent disease rate at all locationswhere exact measurement values could be obtained, thevector of soft data random variables xsoft representingthe latent rate at all locations with uncertain measuredvalues (expressed in terms of confidence intervals orprobabilistic functions), and the unknown latent ratevalue xk to be estimated at some estimation point, suchthat xmap = (xhard, xsoft, xk).Processing the information known about the rate field

consists of three main stages (Gesink Law et al., 2006),as follows (Christakos and Li, 1998; Choi et al., 2003):(i) Structural (or prior) stage: The general G − KB of

the rate field is considered at all mapping points corre-sponding to xmap. The structural probabilistic densityfunction (pdf) of the random variables xmap,fG(χmap) ,where the vector of values cmap is a realization of therandom variables xmap, is derived by selecting thefG(χmap) that maximizes entropy for the given generalknowledge base. Using the Shannon measure of infor-mation we have

Info[xmap] = log(1/fX(xmap))= − log fX(xmap)

, (A3)

and the entropy is defined in terms of the correspond-ing expected information, i.e.

H(xmap) = E[Info[xmap]]

= −∞∫

−∞dχmapfG(χmap) log fG(χmap)

(A4)

where H(xmap) is the Entropy function, and E[.] is thestochastic expectation operator. Information is pro-cessed by selecting the fG(χmap) that maximizes H

(xmap) for the given G − KB .(ii) Specificatory (or meta-prior) stage: The specifica-

tory S-KB is identified and expressed in terms of hardand soft data, and the estimation point at which thefield is to be estimated is defined. Here, we consider thespecific framework in which all observed rates in themap are soft data csoft with measures of uncertainty dueto sampling variability. In processing the data, 6 may berewritten as

χ soft : Pr[R(s) − 0.5

N(s)≤ X(s) ≤ R(s) +

0.5N(s)

]= 1 (A5)

where Pr[.] is the probability operator, R(s) is theobserved rate, N(s) is the population size, and X(s) is thelatent rate, and s represents each region i where thesesoft data are available. Furthermore, the aim is to pro-vide a more accurate estimate of the latent rate fieldonly at points s where soft data is available. In otherwords, a soft datum expressed by A5 is available foreach estimation point.(iii) Integration stage: The integration (posterior) pdf,

fK , is derived by means of an operational Bayesian con-ditionalization rule that considers the total KB,K = G ∪ S , such that

fK (χk) = A−1

∞∫−∞

dχ softfS(χkχ soft

)fG

(χmap

)(A6)

where A =∞∫

−∞dχk

∞∫−∞

dχ softfS(χkχ soft

)fG

(χmap

)is the

normalization parameter, and fS(χkχ soft

)is the multi-

variate uniform pdf defined by A5 jointly for the estima-tion point and the soft data points in its neighborhood..The integration pdf provides a full stochastic assessmentof the value xk of the latent rate field at any estimationpoint, from which one may derive a variety of estimators.In this work, the expected value of the integration pdf wasused as an appropriate estimatorX̂k of the latent rate, i.e.

X̂k = E [xk] =

∞∫−∞

dχkfK (χk) , (A7)

which we refer to as the BME mean estimator. Otherestimators include the mode or the median of the inte-gration pdf. Additionally, the variance of the integrationpdf, or BME variance, provides a useful measure of esti-mation uncertainty.


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List of AbbreviationsBME: Bayesian Maximum Entropy; HIV: Human Immunodeficiency Virus;LCCC: Lin’s Concordance Correlation Coefficient; MAD: Mean AbsoluteDifference; MSE: Mean Square Error; Pdf: probability density function; STI:Sexually Transmitted Infection; UMBME: Uniform Model extension ofBayesian Maximum Entropy; VCT: Voluntary Counseling and Testing.

AcknowledgementsThe authors thank Peter A. Leone, Del Williams, and the North CarolinaDepartment of Health and Human Services, HIV/STD Prevention and CareBranch. This work was supported in part by the National Institute of MentalHealth, R01-MH068686 (CDP) and by the National Institute of Allergies andInfectious Diseases, R01-AI067913 (WCM).

Author details1Department of Epidemiology, University of North Carolina at Chapel Hill,Chapel Hill, NC, USA. 2Department of Environmental Sciences andEngineering, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA.3Dalla Lana School of Public Health, University of Toronto, Toronto, Ontario,Canada. 4HIV/AIDS Division, San Francisco General Hospital, University ofCalifornia-San Francisco, San Francisco, CA, USA. 5Department of Medicine,University of North Carolina at Chapel Hill, Chapel Hill, NC, USA.

Authors’ contributionsKHH contributed to the study design, completed the analyses, and draftedthe manuscript. MLS contributed to the study design, execution of theanalyses, and interpretation of results. DCG, CDP, and WCM contributed tothe study design and interpretation of results. All authors read and approvedthe final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 24 May 2011 Accepted: 6 October 2011Published: 6 October 2011

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doi:10.1186/1476-072X-10-54Cite this article as: Hampton et al.: Adjusting for sampling variability insparse data: geostatistical approaches to disease mapping. InternationalJournal of Health Geographics 2011 10:54.

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