The methodology to account for spatial support and population density in Poisson kriging will be illustrated using directly age-adjusted mortality rates for a frequent (lung) and less frequent (cervix) cancer. These data are part of the Atlas of Cancer Mortality in the United States 24 and were downloaded from http://www3.cancer.gov/atlasplus/download.html. The analysis focuses on white female rates recorded over the 1970–1994 period and adjusted using the 1970 population pyramid. Two areas with contrasted county geographies were considered: 1) state of Indiana (Region 1), and 2) four states in the Western US (Arizona, California, Nevada and Utah) that will be referred to as Region 2. The choice of these two specific geography areas was guided by the need to assess performances in two contrasted settings: 1) all geographical units have a fairly similar size and shape, which is the "ideal" situation for methods that ignore the spatial support of the data, and 2) geographical units display a wide range of sizes and shapes, which should favour any approach that implicitly accounts for the spatial support of the data in the analysis. The West coast provided a perfect example for the second type of geography (i.e. set of 118 vastly different counties), while Indiana includes a reasonable number (i.e. 92) of counties that are geometrically fairly similar.

Figure 1 (top graphs) shows the spatial distribution of age-adjusted mortality rates per 100,000 person-years for lung cancer in Region 1 and cervix cancer in Region 2. Following the recommendations of several studies on map color schemes 2526, a double-ended color scheme with 10 equally-weighted classes (i.e. boundaries correspond to deciles of the histogram) was used: a gradient of red is used for rates higher than the median, while a gradient of blue is used for lower rates. The population-weighted average of mortality rates is 23.7 per 100,000 person-years for lung cancer and 2.85 per 100,000 person-years for cervix cancer. For Region 1, the population at risk in each county was computed as: 100,000 × the total number of "lung cancer" deaths over the 1970–1994 period divided by the corresponding age-adjusted mortality rate; both datasets are available on the website of the National Cancer Institute (NCI) of the US (URL as given above). A similar procedure was conducted in Region 2, except that the most frequent breast cancer was used to avoid zero population estimates for a few sparsely populated counties where no cervix cancer mortality was reported during that period. The population sizes are mapped in Figure 1 (middle row) using a rainbow color scheme to avoid any confusion with the rate maps.

A visual inspection of the cervix cancer mortality map conveys the impression that rates are particularly high in the centre of the study area (Nye and Lincoln Counties), as well as in a few Northern California counties. This result must, however, be interpreted with caution since the population is not uniformly distributed across the study area and rates computed from sparsely populated counties tend to be less reliable. This effect, known as "small number problem" 27, is illustrated by the bottom scattergrams in Figure 1. In particular, cervix cancer mortality in excess of 5 deaths per 100,000 person-years is observed only for counties with a 25 year cumulated population smaller than 10⁵. The use of administrative units to report the results (i.e. counties in this case) can also bias the interpretation: had the two counties with the highest rates been much smaller in size, these high values likely would have been perceived as less problematic. The higher frequency of lung cancer mortality, coupled with the narrower range of county sizes in Indiana, makes the interpretation of Region 1 cancer mortality map less hazardous. The highest rate is in fact reported for the most populated county (Marion) whose seat is Indianapolis.

To highlight the limitations of collapsing administrative units into their geographic centroids, the spatial distribution of population within each county is mapped at the top of Figure 2. These maps were produced by allocating the county-level population estimates of Figure 1 to a grid of 25 km² cells discretizing each study area, leading to 3,751 and 48,474 cells in Regions 1 and 2, respectively. The relative proportion of the county-level population within each 25 km² cell was retrieved from the readily available 2000 census block level data. These population maps illustrate the non-uniform repartition of population, in particular for the vast counties in Region 2. 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:

D i s t ( v α , v β ) = 1 ∑ s = 1 P α ∑ s ′ = 1 P β n ( u s ) n ( u s ′ ) ∑ s = 1 P α ∑ s ′ = 1 P β n ( u s ) n ( u s ′ ) ‖ u s − u s ′ ‖       ( Equation  1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGebarcqWGPbqAcqWGZbWCcqWG0baDcqGGOaakcqWG2bGDdaWgaaWcbaacciGae8xSdegabeaakiabcYcaSiabdAha2naaBaaaleaacqWFYoGyaeqaaOGaeiykaKIaeyypa0ZaaSaaaeaacqaIXaqmaeaadaaeWbqaamaaqahabaGaemOBa4MaeiikaGccbeGae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqWGUbGBcqGGOaakcqGF1bqDdaWgaaWcbaGafm4CamNbauaaaeqaaOGaeiykaKcaleaacuWGZbWCgaqbaiabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFYoGyaeqaaaqdcqGHris5aaWcbaGaem4CamNaeyypa0JaeGymaedabaGaemiuaa1aaSbaaWqaaiab=f7aHbqabaaaniabggHiLdaaaOWaaabCaeaadaaeWbqaaiabd6gaUjabcIcaOiab+vha1naaBaaaleaacqWGZbWCaeqaaOGaeiykaKIaemOBa4MaeiikaGIae4xDau3aaSbaaSqaaiqbdohaZzaafaaabeaakiabcMcaPaWcbaGafm4CamNbauaacqGH9aqpcqaIXaqmaeaacqWGqbaudaWgaaadbaGae8NSdigabeaaa0GaeyyeIuoaaSqaaiabdohaZjabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFXoqyaeqaaaqdcqGHris5aOWaauWaaeaacqGF1bqDdaWgaaWcbaGaem4CamhabeaakiabgkHiTiab+vha1naaBaaaleaacuWGZbWCgaqbaaqabaaakiaawMa7caGLkWoacaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGymaedacaGLOaGaayzkaaaaaa@920C@

where P_α and P_β are the number of points u_s and u_s' used to discretize the two units or blocks v_α and v_β, respectively. The quantity n(u_s) represents the population size within the 25 km² cell centred on the discretizing point u_s. For the examples of Figure 1, the discretizing points are identified with the nodes of the 5 km grid, yielding a total of 9 to 69 points per county in Indiana, and 11 to 2,082 discretizing points for the West Coast counties. The set of block-to-block distances (Equation (1)) are plotted against Euclidian distances between polygon centroids depicted by red dots in Figure 2 (middle maps). The scatterplots at the bottom of Figure 2 indicate a high correlation between the two sets of distances; discrepancies essentially occur for neighboring counties (i.e. for small distances). The correlation is slightly smaller for Region 2 where county shapes and sizes vary greatly. Block-to-block distances are numerically very close to the distances computed between population-weighted centroids depicted by blue triangles in Figure 2 (middle maps).

The geostatistical methodology for the estimation of risk values from empirical frequencies, and its performance relative to common smoothers, is described in details in Goovaerts 15. This section provides a brief recall of the centroid-based implementation of Poisson kriging (PK) for prediction of aggregated risk values. The approach is then generalized to account for spatial supports and population density in both the case of area-to-area and area-to-point predictions.

For a given number N of geographical units v_α (e.g. counties), denote the observed mortality rates (areal data) as z(v_α) = d(v_α)/n(v_α), where d(v_α) is the number of recorded mortality cases and n(v_α) is the size of the population at risk. Let us assume for now that all units v_α have similar shapes and sizes, with a uniform population density. A straightforward approach to implement geostatistics is to assume that each unit or measurement support is a single point. This simplification amounts at assigning each measured rate to the geographic centroid of the unit over which it has been recorded. For example, let u_α = (x_α, y_α) be the vector of spatial coordinates for the centroid of the unit v_α . The disease count d(u_α) is interpreted as a realization of a random variable D(u_α) that follows a Poisson distribution with one parameter (expected number of counts) that is the product of the population size n(u_α) by the local risk R(u_α).

The risk over a given unit v_α is estimated as a linear combination of the rate observed for that unit, z(u_α), and in (K-1) neighboring units:

r ^ P K ( u α ) = ∑ i = 1 K λ i ( u α ) z ( u i )       ( Equation  2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcamaaBaaaleaacqWGqbaucqWGlbWsaeqaaOGaeiikaGccbeGae8xDau3aaSbaaSqaaGGaciab+f7aHbqabaGccqGGPaqkcqGH9aqpdaaeWbqaaiab+T7aSnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xDau3aaSbaaSqaaiab+f7aHbqabaGccqGGPaqkcqWG6bGEcqGGOaakcqWF1bqDdaWgaaWcbaGaemyAaKgabeaakiabcMcaPaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaem4saSeaniabggHiLdGccaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGOmaidacaGLOaGaayzkaaaaaa@5B17@

where λ_i(u_α) is the weight assigned to the rate z(u_i) when estimating the risk at u_α . The K weights are the solution of the following system of linear equations:

∑ j = 1 K λ j ( u α ) [ C R ( u i − u j ) + δ i j m * n ( u i ) ] + μ ( u α ) = C R ( u i − u α ) i = 1 , ... , K ∑ j = 1 K λ j ( u α ) = 1       ( Equation  3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaWaaabCaeaaiiGacqWF7oaBdaWgaaWcbaGaemOAaOgabeaakiabcIcaOGqabiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaeiykaKYaamWaaeaacqWGdbWqdaWgaaWcbaGaemOuaifabeaakiabcIcaOiab+vha1naaBaaaleaacqWGPbqAaeqaaOGaeyOeI0Iae4xDau3aaSbaaSqaaiabdQgaQbqabaGccqGGPaqkcqGHRaWkcqWF0oazdaWgaaWcbaGaemyAaKMaemOAaOgabeaakmaalaaabaGaemyBa0MaeiOkaOcabaGaemOBa4MaeiikaGIae4xDau3aaSbaaSqaaiabdMgaPbqabaGccqGGPaqkaaaacaGLBbGaayzxaaGaey4kaSIae8hVd0MaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGH9aqpcqWGdbWqdaWgaaWcbaGaemOuaifabeaakiabcIcaOiab+vha1naaBaaaleaacqWGPbqAaeqaaOGaeyOeI0Iae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aaGcbaGaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4saSeabaWaaabCaeaacqWF7oaBdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaeiykaKcaleaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGlbWsa0GaeyyeIuoakiabg2da9iabigdaXaqaaaaacaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeG4mamdacaGLOaGaayzkaaaaaa@937E@

where δ_ij = 1 if u_i = u_j and 0 otherwise, and m* is the population-weighted mean of the N rates. The "error variance" term, m*/n(u_i), leads to smaller weights for less reliable data (i.e. rates measured over smaller populations). The prediction variance associated with the estimate (2) is computed as:

σ P K 2 ( u α ) = C R ( 0 ) − ∑ i = 1 K λ i ( u α ) C R ( u i − u α ) − μ ( u α )       ( Equation  4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemiuaaLaem4saSeabaGaeGOmaidaaOGaeiikaGccbeGae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGH9aqpcqWGdbWqdaWgaaWcbaGaemOuaifabeaakiabcIcaOiabicdaWiabcMcaPiabgkHiTmaaqahabaGae83UdW2aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiabdoeadnaaBaaaleaacqWGsbGuaeqaaOGaeiikaGIae4xDau3aaSbaaSqaaiabdMgaPbqabaGccqGHsislcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiabgkHiTiab=X7aTjabcIcaOiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaeiykaKcaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGlbWsa0GaeyyeIuoakiaaxMaacaWLjaWaaeWaaeaacqqGfbqrcqqGXbqCcqqG1bqDcqqGHbqycqqG0baDcqqGPbqAcqqGVbWBcqqGUbGBcqqGGaaicqaI0aanaiaawIcacaGLPaaaaaa@6EFC@

To solve the kriging system (Equation 3), one needs a model for the covariance of the risk, C_R(h), or equivalently its semivariogram γ_R(h) = C_R(0)-C_R(h). Following Monestiez et al. 2122 the semivariogram of the risk is estimated as:

γ ^ R ( h ) = 1 2 ∑ α = 1 N ( h ) n ( u α ) n ( u α + h ) n ( u α ) + n ( u α + h ) ∑ α = 1 N ( h ) { n ( u α ) n ( u α + h ) n ( u α ) + n ( u α + h ) [ z ( u α ) − z ( u α + h ) ] 2 − m * }       ( Equation  5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqcamaaBaaaleaacqWGsbGuaeqaaOGaeiikaGccbeGae4hAaGMaeiykaKIaeyypa0ZaaSaaaeaacqaIXaqmaeaacqaIYaGmdaaeWbqaamaalaaabaGaemOBa4MaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqWGUbGBcqGGOaakcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabgUcaRiab+HgaOjabcMcaPaqaaiabd6gaUjabcIcaOiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaeiykaKIaey4kaSIaemOBa4MaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGHRaWkcqGFObaAcqGGPaqkaaaaleaacqWFXoqycqGH9aqpcqaIXaqmaeaacqWGobGtcqGGOaakcqGFObaAcqGGPaqka0GaeyyeIuoaaaGcdaaeWbqaamaacmqabaWaaSaaaeaacqWGUbGBcqGGOaakcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiabd6gaUjabcIcaOiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaey4kaSIae4hAaGMaeiykaKcabaGaemOBa4MaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGHRaWkcqWGUbGBcqGGOaakcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabgUcaRiab+HgaOjabcMcaPaaadaWadaqaaiabdQha6jabcIcaOiab+vha1naaBaaaleaacqWFXoqyaeqaaOGaeiykaKIaeyOeI0IaemOEaONaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGHRaWkcqGFObaAcqGGPaqkaiaawUfacaGLDbaadaahaaWcbeqaaiabikdaYaaakiabgkHiTiabd2gaTjabcQcaQaGaay5Eaiaaw2haaaWcbaGae8xSdeMaeyypa0JaeGymaedabaGaemOta4KaeiikaGIae4hAaGMaeiykaKcaniabggHiLdGccaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGynaudacaGLOaGaayzkaaaaaa@AF0D@

The different spatial increments [z(u_α)-z(u_α+h)]² are weighted by a function of their respective population sizes, n(u_α)n(u_α + h)/(n(u_α) + n(u_α + h)), a term which is inversely proportional to their standard deviation. More importance is thus given to the more reliable data pairs (i.e. smaller standard deviations).

In presence of geographical units with very different shapes and sizes, it is overly simplistic to assimilate each unit v_α to its geographic centroid u_α . The spatial support of each unit needs to be accounted for in both the semivariogram inference and kriging. Following the terminology in Kyriakidis 14, ATA kriging refers to the case where both the prediction and measurement supports are areas instead of points. The PK estimate (2) for the areal risk value r(v_α) thus becomes:

r ^ P K ( v α ) = ∑ i = 1 K λ i ( v α ) z ( v i )       ( Equation  6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcamaaBaaaleaacqWGqbaucqWGlbWsaeqaaOGaeiikaGIaemODay3aaSbaaSqaaGGaciab=f7aHbqabaGccqGGPaqkcqGH9aqpdaaeWbqaaiab=T7aSnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqWG6bGEcqGGOaakcqWG2bGDdaWgaaWcbaGaemyAaKgabeaakiabcMcaPaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaem4saSeaniabggHiLdGccaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGOnaydacaGLOaGaayzkaaaaaa@5B2A@

The weights λ_i(v_α) are computed by solving the following kriging system:

∑ j = 1 K λ j ( v α ) [ C ¯ R ( v i , v j ) + δ i j m * n ( ν i ) ] + μ ( v α ) = C ¯ R ( v i , v α ) i = 1 , ... , K ∑ j = 1 K λ j ( v α ) = 1.       ( Equation  7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaWaaabCaeaaiiGacqWF7oaBdaWgaaWcbaGaemOAaOgabeaakiabcIcaOiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiykaKYaamWaaeaacuWGdbWqgaqeamaaBaaaleaacqWGsbGuaeqaaOGaeiikaGIaemODay3aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqWG2bGDdaWgaaWcbaGaemOAaOgabeaakiabcMcaPiabgUcaRiab=r7aKnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOWaaSaaaeaacqWGTbqBcqGGQaGkaeaacqWGUbGBcqGGOaakcqWF9oGBdaWgaaWcbaGaemyAaKgabeaakiabcMcaPaaaaiaawUfacaGLDbaaaSqaaiabdQgaQjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aOGaey4kaSIae8hVd0MaeiikaGIaemODay3aaSbaaSqaaiab=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@94EA@

The main difference between systems (3) and (7) is that point-to-point covariance terms C_R(u_i - u_j) are replaced by area-to-area covariances C¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaqeaaaa@2DD3@_R(v_i, v_j) = Cov{Z(v_i), Z(v_j)}. Those covariances are numerically approximated by averaging the point-support covariance C(h) computed between any two locations discretizing the areas v_i and v_j:

C ¯ R ( v i , v j ) = 1 ∑ s = 1 P i ∑ s ′ = 1 P j w s s ′ ∑ s = 1 P i ∑ s ′ = 1 P j w s s ′ C ( u s , u s ′ )       ( Equation  8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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vha1naaBaaaleaacqWGZbWCaeqaaOGaeiilaWIae8xDau3aaSbaaSqaaiqbdohaZzaafaaabeaakiabcMcaPiaaxMaacaWLjaWaaeWaaeaacqqGfbqrcqqGXbqCcqqG1bqDcqqGHbqycqqG0baDcqqGPbqAcqqGVbWBcqqGUbGBcqqGGaaicqaI4aaoaiaawIcacaGLPaaaaaa@7DA5@

where P_i and P_j are the number of points used to discretize the two areas v_i and v_j, respectively. The weights w_ss' are computed as the product of population sizes within the square cells centred on the discretizing point u_s and u_s':

w s s ′ = n ( u s ) × n ( u s ′ )  with  ∑ s = 1 P i n ( u s ) = n ( v i )  and  ∑ s ′ = 1 P j n ( u s ′ ) = n ( v j )       ( Equation  9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@884B@

The kriging variance for the ATA kriging estimator is computed as:

σ P K 2 ( v α ) = C ¯ R ( v α , v α ) − ∑ i = 1 K λ i ( v α ) C ¯ R ( v i , v α ) − μ ( v α )       ( Equation  10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemiuaaLaem4saSeabaGaeGOmaidaaOGaeiikaGIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGH9aqpcuWGdbWqgaqeamaaBaaaleaacqWGsbGuaeqaaOGaeiikaGIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGSaalcqWG2bGDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiabgkHiTmaaqahabaGae83UdW2aaSbaaSqaaiabdMgaPbqabaaabaGaemyAaKMaeyypa0JaeGymaedabaGaem4saSeaniabggHiLdGccqGGOaakcqWG2bGDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiqbdoeadzaaraWaaSbaaSqaaiabdkfasbqabaGccqGGOaakcqWG2bGDdaWgaaWcbaGaemyAaKgabeaakiabcYcaSiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiykaKIaeyOeI0Iae8hVd0MaeiikaGIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGymaeJaeGimaadacaGLOaGaayzkaaaaaa@7687@

where C¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaqeaaaa@2DD3@_R(v_α, v_α) is the within-area covariance that is computed according to Equation (8) with v_i = v_j = v_α.

A particular case of ATA kriging is when the prediction support is so small that it can be assimilated to a point u_s, leading to the following area-to-point kriging estimator and kriging variance:

r ^ P K ( u s ) = ∑ i = 1 K λ i ( u s ) z ( v i )       ( Equation  11 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcamaaBaaaleaacqWGqbaucqWGlbWsaeqaaOGaeiikaGccbeGae8xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGH9aqpdaaeWbqaaGGaciab+T7aSnaaBaaaleaacqWGPbqAaeqaaOGaeiikaGIae8xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqWG6bGEcqGGOaakcqWG2bGDdaWgaaWcbaGaemyAaKgabeaakiabcMcaPaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaem4saSeaniabggHiLdGccaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGymaeJaeGymaedacaGLOaGaayzkaaaaaa@5BB7@

σ P K 2 ( u s ) = C R ( 0 ) − ∑ i = 1 K λ i ( u s ) C ¯ R ( v i , u s ) − μ ( u s )       ( Equation  12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6F39@

The kriging weights and the Lagrange parameter μ(u_s) are computed by solving the following system of linear equations:

∑ j = 1 K λ j ( u s ) [ C ¯ R ( v i , v j ) + δ i j m ∗ n ( v i ) ] + μ ( u s ) = C ¯ R ( v i , u s ) i = 1 , ... , K ∑ j = 1 K λ j ( u s ) = 1.       ( Equation  13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaWaaabCaeaaiiGacqWF7oaBdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGlbWsa0GaeyyeIuoakiabcIcaOGqabiab+vha1naaBaaaleaacqWGZbWCaeqaaOGaeiykaKYaamWaaeaacuWGdbWqgaqeamaaBaaaleaacqWGsbGuaeqaaOGaeiikaGIaemODay3aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqWG2bGDdaWgaaWcbaGaemOAaOgabeaakiabcMcaPiabgUcaRiab=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@94D8@

The ATP kriging system is similar to the ATA kriging system (Equation 7), except for the right-hand-side term where the area-to-area covariances C¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaqeaaaa@2DD3@_R(v_i, v_α) are replaced by area-to-point covariances C¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaqeaaaa@2DD3@_R (v_i, u_s) that are approximated as:

C ¯ R ( v i , u s ) = 1 ∑ s ′ = 1 P i w s ′ s ∑ s ′ = 1 P i w s ′ s C ( u s ′ , u s )       ( Equation  14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGdbWqgaqeamaaBaaaleaacqWGsbGuaeqaaOGaeiikaGIaemODay3aaSbaaSqaaiabdMgaPbqabaGccqGGSaalieqacqWF1bqDdaWgaaWcbaGaem4CamhabeaakiabcMcaPiabg2da9maalaaabaGaeGymaedabaWaaabCaeaacqWG3bWDdaWgaaWcbaGafm4CamNbauaacqWGZbWCaeqaaaqaaiqbdohaZzaafaGaeyypa0JaeGymaedabaGaemiuaa1aaSbaaWqaaiabdMgaPbqabaaaniabggHiLdaaaOWaaabCaeaacqWG3bWDdaWgaaWcbaGafm4CamNbauaacqWGZbWCaeqaaOGaem4qamKaeiikaGIae8xDau3aaSbaaSqaaiqbdohaZzaafaaabeaakiabcYcaSiab=vha1naaBaaaleaacqWGZbWCaeqaaOGaeiykaKcaleaacuWGZbWCgaqbaiabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWGPbqAaeqaaaqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabigdaXiabisda0aGaayjkaiaawMcaaaaa@6DFA@

where P_i is the number of points used to discretize the area v_i and the weights w_s's are computed according to expression (9).

ATP kriging can be conducted at each node of a grid covering the study area, resulting in a continuous (isopleth) map of mortality risk and reducing the visual bias that is typically associated with the interpretation of choropleth maps. Another interesting property of the ATP kriging estimator is its coherence: the population-weighted average of the risk values estimated at the P_α points u_s discretizing a given entity v_α yields the ATA risk estimate for this entity:

r ^ P K ( v α ) = 1 n ( v α ) ∑ s = 1 P α n ( u s ) r ^ P K ( u s )       ( Equation  15 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcamaaBaaaleaacqWGqbaucqWGlbWsaeqaaOGaeiikaGIaemODay3aaSbaaSqaaGGaciab=f7aHbqabaGccqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabd6gaUjabcIcaOiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiykaKcaamaaqahabaGaemOBa4MaeiikaGccbeGae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcuWGYbGCgaqcamaaBaaaleaacqWGqbaucqWGlbWsaeqaaOGaeiikaGIae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkaSqaaiabdohaZjabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFXoqyaeqaaaqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabigdaXiabiwda1aGaayjkaiaawMcaaaaa@65E1@

Constraint (15) is satisfied if the same K areal data are used for the ATP kriging of the P_α risk values. Indeed, in this case the population-weighted average of the right-hand-side covariance terms of the K ATP kriging systems is equal to the right-hand-side covariance of the single ATA kriging system:

1 n ( v α ) ∑ s = 1 P α n ( u s ) C ¯ R ( v i , u s ) = 1 n ( v α ) ∑ s = 1 P α n ( u s ) [ 1 n ( v i ) ∑ s ′ = 1 P i n ( u s ′ ) C ( u s ′ , u s ) ] = C ¯ R ( v i , v α ) ,       ( Equation  16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A458@

per relations (8), (9) and (14). Therefore, the following relationship exists between the two sets of ATA and ATP kriging weights:

λ i ( v α ) = 1 n ( v α ) ∑ s = 1 P α n ( u s ) λ i ( u s ) i = 1 , ... , K       ( Equation  17 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqacaaabaacciGae83UdW2aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqWG2bGDdaWgaaWcbaGae8xSdegabeaakiabcMcaPiabg2da9maalaaabaGaeGymaedabaGaemOBa4MaeiikaGIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkaaWaaabCaeaacqWGUbGBcqGGOaakieqacqGF1bqDdaWgaaWcbaGaem4CamhabeaakiabcMcaPiabeU7aSnaaBaaaleaacqWGPbqAaeqaaaqaaiabdohaZjabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFXoqyaeqaaaqdcqGHris5aOGaeiikaGIae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkaeaacqWGPbqAcqGH9aqpcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGlbWsaaGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabigdaXiabiEda3aGaayjkaiaawMcaaaaa@6D48@

To solve both ATA and ATP kriging systems, one needs to know the point-support covariance of the risk C(h), or equivalently the point-support semivariogram γ(h). This function cannot be estimated directly from the observed rates, since only aggregated data are available. Derivation of a point-support semivariogram from the "regularized" experimental semivariogram computed from areal data is called "deconvolution" 1428. Unlike the mining applications that have led to the initial development of geostatistical deconvolution 8, the size and shape of geographical units in health science applications can vary greatly, which makes the deconvolution much more challenging. Goovaerts 10 developed an innovative approach to conduct the deconvolution in presence of irregular units and heterogeneous population distributions. Like most inverse problems, the deconvolution is best tackled using an iterative procedure. The basic idea is to seek the point-support model that, once regularized, is the closest to the model fitted to areal data. Only the most salient features of the deconvolution procedure are described hereafter, and the reader is referred to Goovaerts 10 for a detailed presentation of the algorithm and demonstration of its performances in simulation studies.

The deconvolution is based on the following relationship between the regularized semivariogram model γ_v(h) fitted to areal data and area-to-area semivariogram values that are a function of the unknown point-support model γ(h):

γ_v(h) = γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@(v, v_h) - γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@_h(v, v)     (Equation 18)

The area-to-area semivariogram value γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@(v, v_h) represents the mean value of the point-support semivariogram between an arbitrary point in the support v and another in the translated support v_h. Because each area (e.g. county) has potentially a different size and shape, one needs to consider any two areas that can be encountered across the study area and average the semivariogram values over all N(h) pairs of areas separated by the distance h. More precisely, the area-to-area semivariogram value is estimated as:

γ ¯ ( v , v h ) = 1 N ( h ) ∑ α = 1 N ( h ) γ ¯ ( v α , v α + h )       ( Equation  19 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaiabcIcaOiabdAha2jabcYcaSiabdAha2naaBaaaleaacqWGObaAaeqaaOGaeiykaKIaeyypa0ZaaSaaaeaacqaIXaqmaeaacqWGobGtcqGGOaakcqWGObaAcqGGPaqkaaWaaabCaeaacuWFZoWzgaqeaiabcIcaOiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiilaWIaemODay3aaSbaaSqaaiab=f7aHjabgUcaRiabdIgaObqabaGccqGGPaqkaSqaaiab=f7aHjabg2da9iabigdaXaqaaiabd6eaojabcIcaOiabdIgaOjabcMcaPaqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabigdaXiabiMda5aGaayjkaiaawMcaaaaa@6318@

The semivariogram value between any two areas, v_α and v_α+h, separated by the population-based distance h (Equation 1) is computed as:

γ ¯ ( v α , v α + h ) = 1 P α P α + h ∑ s = 1 P α ∑ s ′ = 1 P α + h γ ( u s , u s ′ )       ( Equation  20 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaiabcIcaOiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiilaWIaemODay3aaSbaaSqaaiab=f7aHjabgUcaRiabdIgaObqabaGccqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabdcfaqnaaBaaaleaacqWFXoqyaeqaaOGaemiuaa1aaSbaaSqaaiab=f7aHjabgUcaRiabdIgaObqabaaaaOWaaabCaeaadaaeWbqaaiab=n7aNbWcbaGafm4CamNbauaacqGH9aqpcqaIXaqmaeaacqWGqbaudaWgaaadbaGae8xSdeMaey4kaSIaemiAaGgabeaaa0GaeyyeIuoaaSqaaiabdohaZjabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFXoqyaeqaaaqdcqGHris5aOGaeiikaGccbeGae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGSaalcqGF1bqDdaWgaaWcbaGafm4CamNbauaaaeqaaOGaeiykaKIaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabikdaYiabicdaWaGaayjkaiaawMcaaaaa@7223@

where P_α and P_α+h are the number of points used to discretize the two areas v_α and v_α+h, respectively.

The second term in the regularization expression (Equation 18) is the within-area semivariogram value γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@_h(v, v). It fluctuates as a function of the separation distance h since the size of the areas used in semivariogram computation varies as a function of the distance between them: smaller areas tend to be paired at shorter distances, while larger areas can only be paired for distances that exceed half their minimum dimension. This quantity is estimated as the arithmetical average of within-area semivariogram values for any pair of areas separated by a given distance h:

γ ¯ h ( v , v ) = 1 2 N ( h ) ∑ α = 1 N ( h ) [ γ ¯ ( v α , v α ) + γ ¯ ( v α + h , v α + h ) ]       ( Equation  21 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeamaaBaaaleaacqWGObaAaeqaaOGaeiikaGIaemODayNaeiilaWIaemODayNaeiykaKIaeyypa0ZaaSaaaeaacqaIXaqmaeaacqaIYaGmcqWGobGtcqGGOaakcqWGObaAcqGGPaqkaaWaaabCaeaacqGGBbWwcuWFZoWzgaqeaiabcIcaOiabdAha2naaBaaaleaacqWFXoqyaeqaaOGaeiilaWIaemODay3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGHRaWkcuWFZoWzgaqeaiabcIcaOiabdAha2naaBaaaleaacqWFXoqycqGHRaWkcqWGObaAaeqaaOGaeiilaWIaemODay3aaSbaaSqaaiab=f7aHjabgUcaRiabdIgaObqabaGccqGGPaqkcqGGDbqxaSqaaiab=f7aHjabg2da9iabigdaXaqaaiabd6eaojabcIcaOiabdIgaOjabcMcaPaqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabikdaYiabigdaXaGaayjkaiaawMcaaaaa@746F@

where γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@(v_α, v_α) and γ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqeaaaa@2E72@(v_α+h, v_α+h) are computed using Equation (20) with h = 0.

The deconvolution procedure starts with the choice of an initial point-support model γ⁽⁰⁾(h); for example the model γ_R(h) fitted to the areal data. This model is regularized according to expression (18), and the theoretically regularized model, γv(0) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaqhaaWcbaGaemODayhabaGaeiikaGIaeGimaaJaeiykaKcaaaaa@329C@(h), is compared to the model inferred from areal data, γ_R(h). The optimization criterion is the relative difference between the two curves, denoted D. A new candidate point-support semivariogram γ⁽¹⁾(h) is derived by rescaling of the initial point-support model γ⁽⁰⁾(h), and then regularized according to expression (18). Model γ⁽¹⁾(h) becomes the new optimum if it lowers the deviation between the theoretically regularized semivariogram model γv(1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaqhaaWcbaGaemODayhabaGaeiikaGIaeGymaeJaeiykaKcaaaaa@329E@(h) and the model fitted to areal data, that is if D⁽¹⁾ <D⁽⁰⁾. Rescaling coefficients are then updated to account for the difference between γv(1) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaqhaaWcbaGaemODayhabaGaeiikaGIaeGymaeJaeiykaKcaaaaa@329E@(h) and γ_R(h), leading to a new candidate model γ⁽²⁾(h) for the next iteration. The procedure stops when the maximum number of allowed iterations has been tried (e.g. 35 in this paper) or the decrease in the D statistic becomes negligible from one iteration to the next. The use of lag-specific rescaling coefficients provides enough flexibility to modify the initial shape of the point-support semivariogram and makes the deconvolution insensitive to the initial solution adopted.

A couple of geostatistical approaches have been proposed in the health science literature to derive continuous maps of health outcomes from areal data. The most straightforward method is to interpolate the raw rates to the nodes of a regular grid using point ordinary kriging 2:

z ^ O K ( u s ) = ∑ i = 1 K λ i ( u s ) z ( u i )       ( Equation  22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG6bGEgaqcamaaBaaaleaacqWGpbWtcqWGlbWsaeqaaOGaeiikaGccbeGae8xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGH9aqpdaaeWbqaaGGaciab+T7aSnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aOGaeiikaGIae8xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqWG6bGEcqGGOaakcqWF1bqDdaWgaaWcbaGaemyAaKgabeaakiabcMcaPiaaxMaacaWLjaWaaeWaaeaacqqGfbqrcqqGXbqCcqqG1bqDcqqGHbqycqqG0baDcqqGPbqAcqqGVbWBcqqGUbGBcqqGGaaicqaIYaGmcqaIYaGmaiaawIcacaGLPaaaaaa@5BAE@

The corresponding kriging variance is computed as:

σ O K 2 ( u s ) = C ( 0 ) − ∑ i = 1 K λ i ( u s ) C ( u i − u s ) − μ ( u s )       ( Equation  23 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaem4ta8Kaem4saSeabaGaeGOmaidaaOGaeiikaGccbeGae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqGH9aqpcqWGdbWqcqGGOaakcqaIWaamcqGGPaqkcqGHsisldaaeWbqaaiab=T7aSnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabdUealbqdcqGHris5aOGaeiikaGIae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcqWGdbWqcqGGOaakcqGF1bqDdaWgaaWcbaGaemyAaKgabeaakiabgkHiTiab+vha1naaBaaaleaacqWGZbWCaeqaaOGaeiykaKIaeyOeI0Iae8hVd0MaeiikaGIae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGOmaiJaeG4mamdacaGLOaGaayzkaaaaaa@6C63@

The weights λ_i(u_s) are the solution of the following ordinary kriging system:

∑ j = 1 K λ j ( u s ) C ( u i − u j ) + μ ( u s ) = C ( u i − u s ) i = 1 , ... , K ∑ j = 1 K λ j ( u s ) = 1       ( Equation  24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGacaaabaWaaabCaeaaiiGacqWF7oaBdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGlbWsa0GaeyyeIuoakiabcIcaOGqabiab+vha1naaBaaaleaacqWGZbWCaeqaaOGaeiykaKIaem4qamKaeiikaGIae4xDau3aaSbaaSqaaiabdMgaPbqabaGccqGHsislcqGF1bqDdaWgaaWcbaGaemOAaOgabeaakiabcMcaPiabgUcaRiab=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@810D@

The covariance C(h) is derived from the model fitted to the experimental semivariogram of raw rates that is calculated as:

γ ^ ( h ) = 1 2 N ( h ) ∑ α = 1 N ( h ) [ z ( u α ) − z ( u α + h ) ] 2       ( Equation  25 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFZoWzgaqcaiabcIcaOGqabiab+HgaOjabcMcaPiabg2da9maalaaabaGaeGymaedabaGaeGOmaiJaemOta4KaeiikaGIae4hAaGMaeiykaKcaamaaqahabaGaei4waSLaemOEaONaeiikaGIae4xDau3aaSbaaSqaaiab=f7aHbqabaGccqGGPaqkcqGHsislcqWG6bGEcqGGOaakcqGF1bqDdaWgaaWcbaGae8xSdegabeaakiabgUcaRiab+HgaOjabcMcaPiabc2faDnaaCaaaleqabaGaeGOmaidaaaqaaiab=f7aHjabg2da9iabigdaXaqaaiabd6eaojabcIcaOiab+HgaOjabcMcaPaqdcqGHris5aOGaaCzcaiaaxMaadaqadaqaaiabbweafjabbghaXjabbwha1jabbggaHjabbsha0jabbMgaPjabb+gaVjabb6gaUjabbccaGiabikdaYiabiwda1aGaayjkaiaawMcaaaaa@6677@

This approach overlooks two critical facts: 1) rates can be very unreliable if recorded over sparsely populated geographical units, and 2) the spatial support of the data (i.e. county) can not be simply equalled to the prediction support (i.e. grid node). In other words, it is unrealistic to collapse vastly different administrative units into their geographic centroids. Another limitation of this approach is its lack of coherence: there is no guarantee that the average of kriging estimates for all P_α points discretizing the county v_α yields the areal data z(v_α):

z ( v α ) ≠ 1 P α ∑ s = 1 P α z ^ O K ( u s )       ( Equation  26 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG6bGEcqGGOaakcqWG2bGDdaWgaaWcbaacciGae8xSdegabeaakiabcMcaPiabgcMi5oaalaaabaGaeGymaedabaGaemiuaa1aaSbaaSqaaiab=f7aHbqabaaaaOWaaabCaeaacuWG6bGEgaqcamaaBaaaleaacqWGpbWtcqWGlbWsaeqaaaqaaiabdohaZjabg2da9iabigdaXaqaaiabdcfaqnaaBaaameaacqWFXoqyaeqaaaqdcqGHris5aOGaeiikaGccbeGae4xDau3aaSbaaSqaaiabdohaZbqabaGccqGGPaqkcaWLjaGaaCzcamaabmaabaGaeeyrauKaeeyCaeNaeeyDauNaeeyyaeMaeeiDaqNaeeyAaKMaee4Ba8MaeeOBa4MaeeiiaaIaeGOmaiJaeGOnaydacaGLOaGaayzkaaaaaa@5A94@

To correct for the "small number problem", Berke 3 proposed to replace the raw rates z(u_i) in Equations (22) and (25) by rates r^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcaaaa@2E29@_GBS(u_i) obtained using global empirical Bayes smoothers. Each smoothed rate is easily computed as a linear combination of the raw rate z(u_i) and the global mean m*:

r^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcaaaa@2E29@_GBS(u_i) = λ(u_i)z(u_i) + [1 - λ(u_i)]m*     i = 1,...,N     (Equation 27)

The Bayes shrinkage factor λ(u_i) is computed as:

λ ( u i ) = { s 2 − m ∗ / n ¯ s 2 − m ∗ / n ¯ + m ∗ / n ( u i ) if  s 2 ≥ m ∗ / n ¯ 0 otherwise       ( Equation  28 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7B1B@

where m* and s² are the population-weighted sample mean and variance of rates, and n¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGUbGBgaqeaaaa@2E29@ is the average population size across the study area. Whenever the rate z(u_i) is based on small population sizes n(u_i) relative to the average size n¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGUbGBgaqeaaaa@2E29@, the factor λ(u_i) is small and the Bayesian estimate (Equation 27) is close to the global mean m*.

The kriging of empirical Bayesian smoothed rates suffers from the same shortcomings as the kriging of raw rates: lack of coherence and ignorance of the spatial support of the data. To attenuate the smoothing effect caused by the use of a global mean in the Bayes smoother (Equation 27), in this paper kriging was also applied to local Bayes estimates. Local Bayes smoothers are calculated similarly except that all the statistics (i.e. the population-weighted sample mean and variance, population size) are computed within local search windows 29; for example using the K neighboring observed rates. The estimator thus becomes:

r^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGYbGCgaqcaaaa@2E29@_LBS(u_i) = λ(u_i)z(u_i) + [1 - λ(u_i)]m* (u_i)     (Equation 29)

where m*(u_i) is the population-weighted average of the rates within the search window. The Bayes shrinkage factor λ(u_i) is now computed as:

λ ( u i ) = { s 2 ( u i ) − m ∗ ( u i ) / n ¯ ( u i ) s 2 ( u i ) − m ∗ ( u i ) / n ¯ ( u i ) + m ∗ ( u i ) / n ( u i ) if  s 2 ( u i ) ≥ m ∗ ( u i ) / n ¯ ( u i ) 0 otherwise MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@99B4@

As for global Bayes smoothers, the relative weight λ(u_i) assigned to the observed rate z(u_i) is smaller for less densely populated counties. For counties with similar population sizes, the factor λ(u_i) is also smaller in regions of greater homogeneity, as measured by a lower local variance s²(u_i).

Figures 3 through 8 illustrated the major differences between alternative approaches for creating isopleth risk maps from aggregated rates. An objective assessment of the prediction performances of the various techniques requires, however, the availability of the underlying risk maps, which are unknown in practice. Simulation provides a way to generate multiple realizations of the spatial distribution of cancer mortality rates that can be analyzed using alternative approaches. Predicted risks can then be compared to the risk maps used in the simulation.

For both lung and cervix cancers, L = 100 maps of county-level mortality rates {z^(l)(v_α), α = 1,..., N; l = 1,..., L} were simulated using the following procedure:

1. Continuous maps of mortality risk values, {r^(l)(u_s), s = 1,..., S}, are first simulated using non-conditional sequential Gaussian simulation (see 169, p. 380 for a description of the algorithm). The simulation grid, which has a 5 km spacing, consists of S = 3,751 and S = 48,474 nodes for Regions 1 and 2, respectively. Five different risk maps (i.e. L = 5) were generated for each cancer, and two realizations for each region are displayed at the top of Figures 10 and 11. Each realization reproduces the histogram of raw cancer mortality rates, while the deconvoluted semivariogram models of Figure 3 were used in the simulation algorithm: exponential model with a range of 75 km for Region 1 and a spherical model with range of 425 km for Region 2. Both models have a zero nugget effect and a unit sill.

2. Each simulated risk map was combined with the population maps of Figure 2 to compute a number of deaths for all grid nodes. Both simulated death counts and population data were then aggregated within each county, and their ratio defines the simulated county-level mortality risk:

r ( l ) ( v α ) = 1 n ( v α ) ∑ s = 1 P α n ( u s ) r ( l ) ( u s ) with n ( v α ) = ∑ s = 1 P α n ( u s )       ( Equation  30 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaGaemOCai3aaWbaaSqabeaacqGGOaakcqWGSbaBcqGGPaqkaaGccqGGOaakcqWG2bGDdaWgaaWcbaacciGae8xSdegabeaakiabcMcaPiabg2da9maalaaabaGaeGymaedabaGaemOBa4MaeiikaGIaemODay3aaSbaaSqaaiab=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@830E@

The maps of aggregated risk values, {r^(l)(v_α), α = 1,..., N}, are displayed in the middle of Figures 10 and 11.

3. For each cancer and each of the five risk maps, 20 realizations of the number of deaths recorded over each county v_α were generated by random drawing of a Poisson distribution whose mean parameter is r^(l)(v_α) × n(v_α). The division of the simulated death counts by the county population leads to 5 × 20 = 100 sets of simulated mortality rates for each cancer; see bottom of Figures 10 and 11 for the first realization generated for the risk maps displayed in the same figures. The noise caused by the small number problem is particularly apparent for cervix cancer. For example, a couple of counties in the North central part of realization #1 display high mortality rates, while the underlying risk value is low.

Each map of simulated rates {z^(l)(v_α), α = 1,..., N} underwent a (geo)statistical analysis in order to estimate point risk values using four alternate approaches: point kriging of raw rates, point kriging of global and local empirical Bayesian smoothed rates (EBS), and area-to-point Poisson kriging (ATP). In all four cases, the estimation at each grid node was based on K = 32 closest areal data (local search window). For each realization, the semivariogram needed for each type of kriging was estimated before being automatically modelled and deconvoluted in the case of ATP kriging.

Figure 12 shows the different semivariogram models for the two simulated maps of lung and cervix cancer mortality displayed at the bottom of Figures 10 and 11. The reference point (black curve) and areal (yellow curve) support models are the semivariograms fitted directly to the original simulated risk maps before and after aggregation to the county level; see Figure 12 (left column). Models resulting from the deconvolution procedure are plotted as green curves. Comparison of the green and black solid curves indicates that the deconvolution yields point-support models that are reasonably close to the underlying ones in terms of range values. Each deconvoluted model was regularized according to Equation (18), and the resulting model (blue curve) agrees fairly well with the risk semivariogram model fitted to areal data (red curve), which demonstrates the convergence of the iterative deconvolution procedure. Note also how this risk semivariogram model is close to the reference areal support model (yellow curve). Because of the good agreement between theoretically regularized and the risk semivariogram models inferred from the county-level rates, discrepancies at the point-support level are essentially caused by the use of the regularization formula (Equation 18). An important factor that influences the deconvolution results is the behavior at the origin of the regularized and point-support semivariogram models; for example the fitting of a nugget effect or the use of a spherical (linear behavior) versus cubic (parabolic behavior) model. Unfortunately, in absence of point data this part of the semivariogram model can not be validated. In this paper, no nugget effect was systematically fitted to the point-support model to account for the characteristics of the model used in the simulation.

The semivariogram models used by the point kriging approaches are plotted in the right column of Figure 12. In particular for cervix cancer, the noise caused by the small number problem leads to semivariograms for raw rates that have a high sill and a large relative nugget effect (blue curves). Rate smoothing using global or local empirical Bayes approaches substantially reduces the sample variability, resulting in smaller sills and nuggets effects, as well as parabolic behaviour for most of the semivariograms at the origin (green and black curves). The risk semivariogram estimator (Equation 5), which accounts for population size in the estimation, yields models with intermediate sills (red curves).

Figures 13 and 14 show, for the simulated rate maps #1 of Figures 10 and 11, the risk values estimated at the nodes of a 5 km grid using the four alternate methods. The analysis of simulated maps confirms the conclusions drawn from the analysis of real cancer mortality rates in Figures 5 and 6. The location of county centroids is the most apparent on the kriged map of raw rates which is also the most variable because it is based on the interpolation of unstable mortality rates, in particular for cervix cancer. Kriging of global and local EBS estimates leads to very smooth maps free of the influence of unreliable rates. Poisson kriging, through its coherence constraint, generates maps of risk estimates that are locally more detailed (i.e. less smoothing) and reveal features of the true risk maps that were blurred by empirical Bayes smoothing, such as the low risk area extending up to the eastern border of Region 2.

For all 100 realizations of each type of cancer, predicted risks {rP(l) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCdaqhaaWcbaGaemiuaafabaGaeiikaGIaemiBaWMaeiykaKcaaaaa@3282@(u_s), s = 1,..., S} and the corresponding prediction variance {[σP(l) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaqhaaWcbaGaemiuaafabaGaeiikaGIaemiBaWMaeiykaKcaaaaa@32DF@(u_s)]², s = 1,..., S} were compared to the underlying risk map {r(u_s), s = 1,..., S}. As in a previous study on the performances of alternative smoothing techniques 15, various criteria were computed and averaged over all realizations to attenuate the impact of statistical fluctuations.