saib et al,t j iomet iostat 2014, 5:3 journal of ......saib et al,t j iomet iostat 2014, 5:3 9...

Volume 5 • Issue 3 • 1000200J Biomet BiostatISSN: 2155-6180 JBMBS, an open access journal

Research Article Open Access

Saib et al., J Biomet Biostat 2014, 5:3 DOI: 10.472/2155-6180.1000200

Research Article Open Access

Keywords: Poisson kriging; Filtering; Cancer mortality

IntroductionOne objective of the French National Plan for Health and the

Environment (NPHE) is to prevent diseases caused by environmental factors, particularly cancer. In this context, the Cancer Inequalities Regions, Counties and Environment (CIRCE) project aims to quantify how much the socio-economic and environmental factors account for geographical inequalities, here defined mortality due to cancer.

In France, geographical health inequalities are a recent study topic. Previous studies were based on either individual-level surveys [1,2] or spatially aggregated data (administrative unit) from specific regions of France [3]. On a regional scale, data are often available at a fine level of resolution. This allows for building environmental, socioeconomic and health indicators. The following are two illustrative examples of this: (1) Rey et al. [4] built a FDep deprivation index, which was mapped using the most detailed census administrative level (French census tract IRIS); (2) Advances in computational technologies and development of widely accessible georeferenced databases are permitting the connection of information systems such as Geographic Information Systems (GIS) and risk models. Exposure indicators described in Caudeville et al. [5,6] for quantifying human exposure to chemical substances were mapped at a resolution of a 1 km² grid.

Regarding health data, because there is a protection rule for individual patient data, these data are not publicly available. Only aggregated data are available at the level for which the disclosure or reconstruction of the patient identity is impossible. These corresponding levels of these census units may be regions or counties in France. This aggregation unfortunately results in large uncertainty about rates or risks calculated for small or sparsely populated areas. This effect is known as the "small number problem" [7]. Another challenge for epidemiology is the analysis and synthesis of the relationships between spatial data collected at different spatial scales.

The geostatistical approach, in this context, presents a spatial methodology that allows for filtering the noise caused by the small number problem and enables the estimation of mortality risk and the associated uncertainty at different spatial scales.

The geostatistical analysis of disease data has received increasing attention with kriging becoming more popular. Lai performed ordinary kriging on Chinese cancer mortality data of 63 rural counties [8]. To produce a set of contour maps, the spatial structure of the cancer mortality rates was studied but other possible covariates were not incorporated. A first attempt to take into account the discrete nature of cancer data was the use of binomial cokriging which was employed to produce a map of childhood cancer risk in the West Midlands Health Authority Region (WMHAR) of England [9]. The application of this technique to Long Island (USA) data led to negative variogram estimates. To avoid this problem, binomial cokriging was extended to the case when the variance of observed rates is smaller than expected under the binomial model. One geostatistical filtering approach used is modified binomial cokriging which was applied to estimate breast cancer incidence in Long Island, New York [10]. The modified technique was shown to be more flexible and robust concerning the underlying hypothesis that all counties have the same spatial support, and the simulation studies have demonstrated its more accurate estimates [11].

Another geostatistical technique, Poisson kriging, was recently developed to filter noise from the data by accounting for spatially varying population sizes and spatial patterns. The methodology for estimating a spatial Poisson distribution was first introduced by Kaiser et al. [12]. They developed the spatial “auto-models”

*Corresponding author: Mahdi-Salim. Saib, French National Institute forIndustrial Environment and Risks; Parc Technologique Alata, BP 2, 60550 Verneuil-en-Halatte, France, Tel: +33-(0)344-618-115; Fax: +33-(0)344-556-399; E-mail:[email protected]

Received May 14, 2014; Accepted July 09, 2014; Published July 14, 2014

Citation: Saib MS, Caudeville J, Carre F, Ganry O, Trugeon A, et al. (2014) Noise Filtering Cancer Mortality Data for a Better Assessment of Health-Environment Relationships: Application to the Picardy Region. J Biomet Biostat 5: 200. doi:10.4172/2155-6180.1000200

Copyright: © 2014 Saib MS, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Noise Filtering Cancer Mortality Data for a Better Assessment of Health-Environment Relationships: Application to the Picardy RegionMahdi-Salim Saib1,2*, Julien Caudeville1, Florence Carre1, Olivier Ganry3, Alain Trugeon4 and Andre Cicolella1 1French National Institute for Industrial Environment and Risks; Parc Technologique Alata, BP 2, 60550 Verneuil-en-Halatte, France2University of Picardie Jules Verne; 33 rue St Leu, Amiens, 80039, France 3University Hospital of Amiens; Place Victor Pauchet Amiens, 80054, France 4Regional Observatory of Health and Social Issues in Picardie (OR2S); 3, rue des Louvels, Amiens, 80036, France

AbstractCancer is one of the leading causes of mortality. However, it is necessary to analyze this disease from different

perspectives. Cancer mortality maps are used by public health officials to identify areas of excess and to guide surveillance and control activities. However, the interpretation of these maps is difficult due to the presence of extremely unreliable rates, which typically occur for sparsely populated areas and/or less frequent cancers. The analysis of the relationships between health data and risk factors is often hindered by the fact that these variables are frequently assessed at different geographical scales. Geostatistical techniques that have enabled the process of filtering noise from the maps of cancer mortality and estimating the risk at different scales were recently developed. This paper presents the application of Poisson kriging for the examination of the spatial distribution of cancer mortality in the "Picardy region, France". The aim of this study is to incorporate the size and shape of administrative units as well as the population density into the filtering of noisy mortality rates and to estimate the corresponding risk at a fine resolution.

Journal of Biometrics & Biostatistics Jour

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f Biometrics & Biostatistics

ISSN: 2155-6180


J Biomet BiostatISSN: 2155-6180 JBMBS, an open access journal

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based on the Poisson distribution to be used to incorporate spatial dependencies among the variables. However, their model is not well suited for irregularly sampled data and interpolation. Monestiez et al. [13,14] introduced Poisson kriging to model spatially heterogeneous observations. The approach applied by Monestiez is similar to binomial cokriging proposed by Oliver except that the count data are assumed to follow a Poisson distribution. Poisson riging was then generalised to estimate prostate cancer mortality risk in the United States [15], breast and cervix cancer mortality in New England States [16] and cholera and dysentery incidence risk in Bangladesh [17] by incorporating varying population sizes in the processing of cancer data. When the risk values were spatially correlated, simulation studies showed that in most cases, Poisson kriging outperformed other smoothers such as population-weighted estimators and empirical Bayes smoothers [16]. It is not practical to represent each geographic unit by its centroid, especially when geographic units vary greatly in size and shape. The geographical characteristics need to be incorporated for data analysis together with spatially varying population. The framework for Area-to-area (ATA) or Area-to-point (ATP) kriging was first introduced by Kyriakidis for interpolating point values from available areal data [18]. Goovaerts modified the ATP estimator to a Poisson estimator and applied ATP Poisson kriging to lung and cervix cancer mortality in counties of the United States [15].

The aim of this paper is to examine the spatial distribution of cancer mortality in the Picardy region using geostatistical methods, which consists of two steps: (a) filtering of the noise in the data based on Poisson kriging (Area to Area-ATA) and (b) mapping of the corresponding risk at a fine resolution (Area to Point-ATP). The approach is illustrated using age-adjusted lip, oral cavity, pharynx and lung cancer mortality rates recorded from 2000-2009.

Materials and MethodsStudy area

The region of Picardy consists of 112 counties (Figure 1), which covers an area of approximately 19,500 km² and is located between North Artois, the Ile-de-France in the south, the Bay of the Somme to the west and east Champagne. It covers the departments of Somme, Oise and Aisne. The urbanization rate in this region is far below the national average (60.4% compared to 74% for the whole country). The agricultural sector provides more than 4% of the French agricultural production. This region also has significant industrial activity. Fine and specialty chemicals account for nearly 15% of the jobs in this region and the automotive industry accounts for 40% of industrial employment (26.5% of assets employed in industry against 19.5% nationally).

Data sources

The health data came from the Regional Health Observatory of Picardy [19], where the age-adjusted mortality rates are calculated for each county from 2000 to 2009. Ten years is likely to be more representative in this case than a simple year in order to reduce temporal rate fluctuation. The average population of counties was computed annually by sex and age group for the years 2000 to 2009. These estimates were based on the census population conducted in 1999 and 2009, infant deaths recorded from 2000 to 2009 and the national mortality rates (metropolitan France). The relative proportion of the population in each cell of 1 km² was derived from the INSEE population data and was downloaded from the INSEE (National Institute of Statistics and Economic Studies) website.

Table 1 shows the cumulative, maximum and minimum number

of mortality and age-adjusted rates/per 100 000 person-years by county from 2000 to 2009.

Geostatistical approach

Spatial prediction (Area-to-area (ATA) and Area-to-point (ATP) Poisson kriging): The cancer count d(να) is interpreted as a realization of a random variable D(να) that is Poisson distributed with a parameter (expected number of counts), which is the product of the population size n(να), by the local risk R(να). The local risk R(να) can be thought of as a noise-filtered mortality rate for area να, which we also refer to as the mortality risk. It is estimated by using a variant of kriging with nonsystematic errors, known as Poisson kriging [13]. The aggregation of data into areal units of different shapes and sizes can cause a visual bias. A particular case of ATA kriging is when the prediction support is so small that it can be assimilated to a single point, in which case ATP kriging [15,18] is used to create high-resolution maps of the estimated mortality risk to reduce this visual bias. To account for the shape of geographical units and their heterogeneous population density, the distance between any two counties is here estimated as a population-weighted average of Euclidian distances between points discretizing the pair of counties [20].

The mortality risk and the associated kriging variance for a unit x are estimated as:

1 ( ) ( ) ˆ

K

i ii

r x z vλ=

= ∑ (1)

Kriging variance is computed as follows:

1

2 ( ) ( , ) ( , ) ( )K

R i Ri

ix C x x C v x xσ λ µ=

= − −∑ (2)

where x represents either an area (να) (ATA kriging) or a point us within that area (ATP kriging). The kriging weights (λi) and the Lagrange parameter μ(x) are computed by solving the Poisson kriging system of equations:

*

,1

[ ( , ) ] ( ) ( ) 1, ( )

K

j R i j ij R iij

mC v v x C v ix Kn v

λ δ µ=

+ + = = …∑ (3)

1

1K

jj

λ=

=∑where δij=1 if i=j and 0 otherwise. The “error variance” term, m*/n(vi), leads to smaller weights for rates measured over smaller populations. The ATA covariances ( , )

isjiR vvC and ATP covariances CR(vi,x= us) are

approximated as the population-weighted average of the point-support covariance CR(h) computed between any two locations discretizing the areas vi and vj, or vi and us. An important property of the ATP kriging estimator is its coherence: the population-weighted average of the risk values estimated at the Pα points us discretizing a given entity να yields the ATA risk estimates for this entity:

1

1( ) ( ) ( )(

ˆ ˆ)

p

ss

sr v n u r un v

α

α∝ =

= ∑ (4)

where us∈να with s=1,...,Pα, and n(us) is the population count assigned to the interpolation grid node us. Constraint (4) is satisfied if the same K areal data are used for the ATA kriging of ˆ( )r vα and the ATP kriging of the Pα risk values.

Deconvolution of the semivariogram of the risk: An important step in the application of the kriging techniques is the inference of

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0201 Anizy-le-Château 0202 Aubenton 0203 Bohain-en-Vermandois 0204 Braine 0205 La Capelle 0206 Le Catelet 0207 Charly 0208 Château Thierry 0209 Chauny 0210 Condé-en-Brie 0211Coucy-le-Château Auffrique 0212 Craonne 0213 Crécy-sr-Serre 0214 La Fère 0215 Fère-en-Tardenois 0216 Guise 0217 Hirson 0218 Laon Nord 0219 Marle 0220 Moy-de-l’Aisne 0221 Neufchatel-sur-Aisne 0222 Neuilly-Saint-Front 0223 Le Nouvion-en-Thiérache 0224 Oulchy-le-Château 0225 Ribemont 0226 Rozoy-sur-Serre 0227 Sains-Richaumont 0229 Saint-Simon 0230 Sissonne 0231 Soissons-Nord 0232 Vailly-sur-Aisne 0233 Vermand 0234 Vervins 0235 Vic-sur-Aisne 0236 Villers-Cotterets 0237 Wassigny 0238 Laon-Sud 0239 Saint-Quentin-Nord 0240 Saint-Quentin-Sud 0241 Soissons-Sud 0242 Tergnier 0297 Laon 0298 Saint-Quentin 0299 Soisson

6001 Attichy 6002 Auneuil 6004 Beauvais Sud-Ouest 6005 Betz 6006 Breteuil 6007 Chaumont-en-Vexin 6008 Clermont 6009 Compiègne Nord 6010 Le Coudray-Saint-Germer 6011 Creil-Nogent-sur-Oise 6012 Crépy-en-Valois 6013 Crèvecoeur-le-Grand 6014 Estrées-Saint-Denis 6015 Formerie 6016 Froissy 6017 Grandvilliers 6018 Guiscard 6019 Lassigny 6020 Liancourt 6021 Maignelay-Montigny 6022 Marseille-en-Beauvaisis 6023 Méru 6024 Mouy 6025 Nanteuil-le-Haudoin 2026 Neuilly-en-Thelle 2027 Nivillers 6028 Noailles 6029 Noyon 6030 Pont-Sainte-Maxence 6031 Ressons-sur-Matz 6032 Ribécourt-Dreslincourt 6033 Saint-Just-en-Chaussée 6034 Senlis 6035 Songeons 6036 Chantilly 6037 Compiègne-Sud-Est 6039 Montataire 6040 Beauvais-Nord-Ouest 6041 Compiègne Sud-Oest 6097 Compiègne 6098 Creil 6099 Beauvais

8001 Abbeville Nord 8002 Abbeville Sud 8003 Acheux-en-Amiénois 8004 Ailly-le-Haut-Clocher 8005 Ailly-ser-Noye 8006 Albert 8007 Amiens Ouest 8008 Amiens Nord-Ouest 8009 Amiens-Nord-Est 8010 Amiens-Est 8011 Ault 8012 Bernaville 8013 Boves 8014 Bray-sur-Somme 8015 Chaulnes 8016 Combles 8017 Conty 8018 Corbie 8019 Crécy-en-Ponthieu 8020 Domart-en-Ponthieu 8021 Doullens 8022 Gamaches 8023 Hallencourt 8024 Ham 8025 Homoy-le-Bourg 8026 Molliens-Dreuil 8027 Montdidier 8028 Moreuil 8029 Moyenneville 8030 Nesle 8031 Nouvion 8032 Oisemont 8033 Péronne 8034 Picquigny 8035 Poix-de-Picardie 8036 Roisel 8037 Rosières-en-Santerre 8038 Roye 8039 Rue 8040Saint-Valery-sur-Somme 8041 Villers-Bocage 8042 Amiens Sud-Est 8043 Amiens Sud-Ouest 8044 Amiens Nord 8046 Friville-Escarbotin 8098 Abbeville 8099 Amiens

Figure 1: Map of the study area.



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the point-support variogram γR(h) or, equivalently, the point-support covariance CR(h) defined as CR(0)–γR(h). This function cannot be estimated directly from the experimental variogram because the latter is computed from areal rate data. The regularized semivariogram of the risk can be estimated as:

( )2 *

( ),

,

( ) ( )1( ) { [ ( ) ( )]( ) ( ) ( ) ( )

2 ( ) ( )

α βα β

α β α βα β

α βα β

γ = − −+

+

∑∑

N h

R N h

n v n vh z v z v m

n v n v n v n vn v n v

(5)

where, N(h) is the number of pairs of areas (να,νβ), the population-weighted centroids of which are separated by the vector h. The usual

(b) (c)

(d) (e)

1413121110

40

30

20

10

0

Log population

Ag

e a

just

ed

mo

rta

lity

1413121110

150

125

100

75

50

Log population

Ag

e a

just

ed

mo

rta

lity

0 10 20km

128.94 to 138.63

119.255 to 128.94

109.56 to 119.255

99.87 to 109.5690.18 to 99.87

80.49 to 90.18

70.81 to 80.49

61.11 to 70.81

51.42 to 61.11

41.73 to 51.42

33.94 to 37.430.49 to 33.9427.03 to 30.4923.57 to 27.0320.11 to 23.57

16.65 to 20.1113.19 to 16.659.73 to 13.196.27 to 9.732.81 to 6.27

13.5 to 14

13 to 13.5

12.5 to 1312 to 12.5

11.5 to 12

11 to 11.510.5 to 11

10 to 10.59.5 to 10

9 to 9.5

0 10 20km

0 10 20km

(a)

Figure 2: (a) Map of log population density. Geographic distribution of age-adjusted mortality rates per 100,000 person-years recorded over the period 2000–2009 for: (b) lip, oral cavity and pharynx; (c) lung cancer mortality. The bottom scatter plots illustrate: (d) the age-adjusted mortality rates for lip, oral cavity and pharynx cancers plotted against population density and (e) the age-adjusted mortality rates of pleura cancers plotted against population density.

Cancer mortality

Numbers of cases

Age-adjusted rates/per 100 000 person-years

Lip, oral cavity and pharynx cancer mortalityCumulative 1327 16.26Minimum 1 2.81Maximum 128 37.4

lung cancer mortalityCumulative 7338 97.89 Minimum 11 41.37Maximum 534 138.63

Table 1: Cumulative, maximum and minimum number of mortality and age-adjusted rates/per 100 000 person-years by county, 2000-2009.



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squared differences 2[ ( ) ( )]z v z vα β− are weighted by a function of

their respective population sizes, ( ) ( ) / [ ( ) ( )]n v n v n v n vα β α β+ ,

which are inversely proportional to their standard deviations.

Results and DiscussionFigure 2 shows the spatial distribution of mortality due to the cancer

of lip, oral cavity and pharynx as well as lung cancer, age-adjusted per 100 000 person-years. It should be noted that the population is not evenly distributed throughout the study area (Figure 2a), and the rate calculated for a less populated county tends to be less reliable. This implies that the interpretation of the map must be carried out with caution. The scatter plot at the bottom of Figure 2 illustrates this effect, commonly known as the "small number problem," that translates into the larger spread of mortality rates for smaller populations.

The highest age-adjusted mortality rates per 100 000 person-years recorded from 2000–2009 for lip, oral cavity and pharynx cancers were more concentrated in the north of the region, but they are generally spread throughout the area, whereas the highest rates of lung cancers were located in the eastern part of the area.

The spatial distribution of population used to avoid the constraints of county geographical boundaries in the estimation is mapped in Figure 3a. This map shows a large variability of population concentration within each county. This variability was taken into account; the geographic centroids are replaced by population-weighted centroids (Figure 3b).

Figure 4 shows the omnidirectional semivariogram of the lip, oral cavity and pharynx, and lung cancer mortality risk computed from county-level rates using an estimator (5). The semivariogram model (see theoretical regularized model in the Figure) is used to estimate the

(a)

(b)

6.61 to 11.476.018 to 6.615.67 to 6.0185.39 to 5.675.15 to 5.394.9 to 5.154.62 to 4.94.28 to 4.623.53 to 4.280 to 3.53 0 10 20km

Centroids:GeographicPopulation-weighted

0 10 20km

Figure 3: Distribution of the Log population per 1 km² grid cell allocated from the INSEE data (a) and geographic and population-weighted centroids (b).



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Deconvolution model

Experimental model

Theoretical regularized model

Distance (m)

Deconvolution model

Experimental model

Theoretical regularized model

Distance (m)0.0 200000.0

0.0 90000.0

γγ

Figure 4: Semivariogram and the associated deconvolution of the lip, oral cavity and pharynx mortality risk (a) as well as the lung cancer mortality risk (b) computed from county-level rates using an estimator (5).

Lip, oral cavity and pharynx cancer mortality

(a) (b)

(c) (d)

22.6 to 24.1521.07 to 22.619.53 to 21.0718 to 19.5316.46 to 1814.93 to 16.4613.39 to 14.9311.86 to 13.3910.30 to 11.868.79 to 10.32

22.67 to 13.87

11.47 to 22.67

10.26 to 11.47

9.06 to 10.26

7.86 to 9.06

6.65 to 7.86

5.45 to 6.65

4.25 to 5.45

3.04 to 4.25

1.84 to 3.04

(a) (b)

(d)(c)

0 10 20 cm

0 10 20 cm 0 10 20 cm

0 10 20 cm

39.07 to 42.28

35.86 to 39.07

32.66 to 35.86

29.45 to 32.66

26.24 to 29.45

23.03 to 26.24

19.83 to 23.03

16.62 to 19.83

13.41 to 16.62

10.21 to 13.41

23.94 to 25.8222.06 to 23.9420.18 to 22.0618.3 to 20.1816.42 to 18.314.54 to 16.4212.66 to 14.5410.78 to 12.668.91 to 10.787.03 to 8.91

Figure 5: Maps of the lip, oral cavity and pharynx cancer risk estimated by using ATA Poisson kriging (a), and ATP Poisson kriging (c) with the corresponding prediction variance for ATA (b) and ATP (d).



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lip, oral cavity and pharynx, and lung cancer mortality risk and the associated prediction variance at the county-level (ATA kriging) or at the nodes of a 1 km spacing grid (ATP kriging).

The experimental variograms were fit using a spherical model with a range of 12.5 km for lip, oral cavity and pharynx cancer mortality and an exponential model with a range of 26.7 km for lung cancer mortality. Each model was deconvoluted using the iterative method [15]. The deconvoluted variogram model was then used to compute aggregated risk values at the county-level using ATA and ATP kriging, see Figure 3. The kriging estimate is based on the K=32 closest observations selected based on population-weighted distance between the counties. The noise due to the small population size was filtered; the original rate map is less smooth than all the other maps.

The lip, oral cavity and pharynx cancer mortality rate varies between 2.81 and 37.40 per 100 000 inhabitants. After the application of Poisson kriging, the minimum rate increased from 2.81 to 8.79 deaths/100 000 inhabitants, and the maximum rate decreased from 37.40 to 24.46 deaths per 100 000 inhabitants. Notably, the high rates recorded in sparsely populated counties, such as Sains-Richaumont county, (37.40 deaths/100 000 person-years), north of the Aisne department, are strongly smoothed (24.15 deaths/100 000 person-years). The highest

rates recorded in densely populated counties, such as Abbeville North county, (26.60 deaths/100 000 person-years), remain almost the same after smoothing (24.90 deaths/100 000 person-years). The map shows that the situation is favorable in the south of the region, and it is rather unfavorable in the northeast and northwest (Figure 5).

The lung cancer mortality rate varied from 41.70 to 138.63 per 100 000 inhabitants. The rate, after application of Poisson kriging ranged from 79.53 to 104.3 per 100 000 inhabitants. The highest rates recorded in densely populated counties remained the same after smoothing, such as Abbeville North county. Conversely, the highest rates recorded in the least populated counties were highly smoothed, for example, Aubenton county (Figure 6).

Compared to Figure 5, the map in Figure 6 shows a rather unfavorable situation in the northwestern region, specifically the in the Aisne department, after application of Poisson kriging in terms of lung cancer rates.

The ATP kriging risk maps are viewed as the products of the disaggregation of the ATA kriging risk maps because the ATP risk estimates are non-negative and their sum is equal to the original areal county ATA risk (Table 2). The ATP kriging map shows that high risks

Lung cancer mortality

(a) (b)

(c) (d)


46.34 to 50.4442.24 to 46.3438.14 to 42.2464.04 to 38.1429.94 to 34.0425.84 to 29.9421.74 to 25.8417.64 to 21.7413.54 to 17.649.44 to 13.540 10 20 cm 0 10 20 cm

0 10 20 cm0 10 20 cm



(a)

(c) (d)

(b)

Figure 6: Maps of the lung cancer risk estimated by using ATA Poisson kriging (a), and ATP Poisson kriging (c) with the corresponding prediction variance for ATA (b) and ATP (d).



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are not confined to a single county but can potentially spread to areas around the county with extreme risk, (i.e., the high cancer mortality risk found in Guise county, spread to the nearby Ribemont county, (Figure 6c), which is why designing prevention strategies should not be performed at the level of a single county without taking into account the associated neighboring areas.

For a county with a large population, the ATA kriging variance map primarily reflects the highest degree of confidence in the estimated mortality risk. However, the distribution of the population can be highly heterogeneous in large counties with contrasted urban and rural areas. This information is taken into account by the kriging process. The ATP kriging variance maps highlight the location of urban centers, such as Amiens county, which are densely populated with low uncertainty in the risk assessment. Incorporating information from the high-resolution population map strengthens the impact of low or high rates in the vicinity of urban areas and helps in reducing the prediction variance around these areas. The variance of the risk estimates decreases as the area of geographical units increases: the grid-level to the county-level. The risk variance estimated for lung cancer at the county-level varies from 9.44 to 50.44 (Figure 6b), and the variance estimated at grid-level varies from 29.88 to 115.78 (Figure 6d). This uncertainty attached to the risk estimate can be incorporated in the analysis of relations along with socioeconomic and environmental factors, such as exposure indicators described in Caudeville et al. [5,6], modeled at a resolution of a 1 km² grid by weighting each estimation according to the inverse of its kriging variance. Thus, rates with a large variance will have a low weight in the analysis [21].

Several authors have already addressed the spatial relationships between health data and environmental data. One of the issues faced by spatial epidemiologists and for exposure assessment is the combination of data measured for very different spatial scales and with different levels of reliability. In reality, the analysis of cancer mortality maps is often hindered by the presence of noise caused by unreliable extreme rates computed from sparsely populated geographic units. A number of approaches have been developed to improve the reliability of risk estimates [22,23]. The most commonly used are Bayesian methods [24], which are commonly referred to as the BYM model. Bayesian methods prohibit any change of scales, an operation that is easily conducted within the framework of kriging. Goovaerts and Gebreab [25] conducted a simulation-based evaluation of the performance of geostatistical and full Bayesian disease-mapping models, and they found that the geostatistical approach yielded smaller prediction errors and more precise and accurate probability intervals and that it allowed for better discrimination between counties with high and low mortality risks.

The analysis of age-adjusted lip, oral cavity, pharynx, and lung cancer mortality rates illustrated the benefits of Poisson kriging: the

incorporation of the high-resolution population map for filtering the noise caused by small, sparsely populated areas and the estimation of the risk and associated uncertainty at fine spatial scales. The approach should facilitate the analysis of relationships between health data and putative covariates (i.e. environmental, socio-economic, or demographic factors) that are typically measured over different spatial scales [26]. These covariates could also be used directly as secondary information in area-to-point kriging, leading to more detailed risk maps at finer scale [27]. An important consideration in the interpretation of this study is that ATP kriging cannot actually create higher resolution data from areas (ATP kriging cannot realistically be a replacement for data collected at different scales). Whilst such kriging methods can provide another useful visualisation and analysis technique, they are not a substitute for higher resolution data. The original data is subject to the MAUP "modifiable area unit problem" [28,29], and therefore, the results of any analysis using this data will also have this limitation.

ConclusionCharacterizing spatial disparities in cancer mortality is a

requirement for the reduction of diseases that are leading causes of death. The analysis of cancer mortality maps is often hindered by the presence of noise in mortality data, which is caused by low population densities with drastic variations in cancer rates. The methodology that we applied was based on geostatistics. It allows for both filtering noise caused by the "small number problem" and estimating the mortality risk at a fine resolution, while also taking into account the size and shape of county as well as the distribution of the population in each county. This methodology is more reliable for characterizing spatial disparities in cancer mortality, allowing for an estimation of the risk and the associated uncertainty on different scales. This form of Poisson kriging will facilitate the analysis of the relationships of cancer mortality rates with environmental and socio-economic data measured on very different supports.

Acknowledgment

The authors wish to acknowledge the financial support by the French Environment and Energy Management Agency ADEME and the French Picardy Region provided within the framework of the CIRCE project.

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Mean Min MaxAmiens county

Observed rates 21.66ATA Poisson kriging 20.85ATP Poisson kriging 20.39 19.89 20.82

Sains-Richaumont countyObserved rates 37.40

ATA Poisson kriging 24.15ATP Poisson kriging 24.28 22.52 25.23

Table 2: Summary of Kriging Estimates for Lip, oral cavity and pharynx cancer mortality by county.

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