Exploring scale-dependent correlations between cancer mortality rates using factorial kriging and population-weighted semivariograms

Abstract

This paper presents a geostatistical methodology which accounts for spatially varying population size in the processing of cancer mortality data. The approach proceeds in two steps: (1) spatial patterns are first described and modeled using population-weighted semivariogram estimators, (2) spatial components corresponding to nested structures identified on semivariograms are then estimated and mapped using a variant of factorial kriging. The main benefit over traditional spatial smoothers is that the pattern of spatial variability (i.e. direction-dependent variability, range of correlation, presence of nested scales of variability) is directly incorporated into the computation of weights assigned to surrounding observations. Moreover, besides filtering the noise in the data the procedure allows the decomposition of the structured component into several spatial components (i.e. local versus regional variability) on the basis of semivariogram models. A simulation study demonstrates that maps of spatial components are closer to the underlying risk maps in terms of prediction errors and provide a better visualization of regional patterns than the original maps of mortality rates or the maps smoothed using weighted linear averages. The proposed approach also attenuates the underestimation of the magnitude of the correlation between various cancer rates resulting from noise attached to the data. This methodology has great potential to explore scale-dependent correlation between risks of developing cancers and to detect clusters at various spatial scales, which should lead to a more accurate representation of geographic variation in cancer risk, and ultimately to a better understanding of causative relationships.

1. INTRODUCTION

Cancer mortality maps are important tools in health research, allowing the identification of spatial patterns, clusters and disease ‘hot spots’ that often stimulate research to elucidate causative relationships (Jacquez, 1998; Rushton et al., 2000). For example, an early version of the National Cancer Institute (NCI)'s cancer atlas stimulated research that uncovered associations between snuff dipping and oral cancer (Win et al., 1981), as well as the association between shipyard asbestos exposure and lung cancers (Blot et al., 1978). Analysis of mortality maps for a series of time intervals can also contribute to a greater understanding of temporal trends and can help to pinpoint locations where health policies need to be changed. For example, comparison of maps of mortality rates of cervix uteri from 1950 through 1994 highlighted states that did not follow the national decline because poverty reduced access to health care and to early detection through the Pap smear test in particular (Friedell et al., 1992). Another use of mortality maps is the exploration of multivariate relationships in cancer mortality. A population exposed to a given carcinogen often exhibits excess risk of cancer at several different body sites, i.e. ionizing radiation exposure is associated with excess lymphomas, leukemia, and cancers of the thyroid, breast, lung and other organs (Beebe et al., 1978). Across populations the incidence of different cancers varies and is related to differences in genetics and exposure to carcinogens. Several studies have demonstrated that multivariate analysis can reveal common risk factors underlying patterns of covariation in site-specific cancer mortalities. For example, in a factor analysis of age-adjusted mortality rates for 15 cancers in 46 countries, Groves et al. (1987) found that lymphoma and cancers of the colon, rectum, lung and prostate were highly correlated with a first factor the authors associated with smoking and high-fat diets.

Detection of space-time patterns and multivariate relationships is frequently hampered by the presence of noise in mortality data, which is often caused by unreliable extreme relative risks estimated over small areas, such as United States ZIP code areas or census tracts (Mungiole et al., 1999; MacNab and Dean, 2002). Statistical smoothing algorithms have been developed to filter local small-scale variations (i.e. high-frequency variability or changes occurring over short distances) from mortality maps, enhancing larger-scale regional trends (Lawson, 2000; Talbot et al., 2000). These methods encompass simple weighted average (i.e. based on inversed squared distance), median-based head banging smoother, simple empirical Bayes and linear Bayes methods, median polish (see Kafadar, 1994, Clayton and Kaldor, 1987, and Lawson et al., 2000, for a description and performance comparison of the main types of linear and non-linear smoothers). The main shortcoming of these smoothers is that data reliability is usually ignored, leading to smoothing not only spikes of suspect origin (i.e. likely due to random noise) but also credible spikes recorded in cities with large populations. To avoid this over-smoothing Mungiole et al. (1999) developed a weighted head-banging algorithm whereby weights inversely proportional to standard errors are assigned to mortality data. Application to simulated data indicated the superiority of the technique to detect underlying spatial patterns in the data.

A limitation of both the weighted and unweighted smoothers reported in today's health science literature is that they cannot be tailored easily to the pattern of variability displayed by the data. For example, important features such as anisotropy (i.e. direction-dependent variability) or range of spatial correlation are not accounted for by the inverse squared distance method. Also the distribution of cancer mortality rates is influenced by a variety of demographic, social, economic, and environmental factors that likely operate at various spatial scales. Instead of simply filtering the noise from cancer data it would be worth decomposing the structured variability according to the corresponding spatial scales, i.e. by estimating and mapping both local and regional spatial components. Mapping such spatial components should illuminate potential factors responsible for the spatial distribution of mortality rates at different scales, and allow the scale-dependent exploration of temporal changes and correlations between different cancer rates.

Geostatistics (Goovaerts, 1997) provides a set of statistical tools for analyzing and mapping data distributed in space and time. Semivariograms allow one to detect multiple scales of spatial variability, and the observed values can then be decomposed into the corresponding spatial components using kriging, a generalized least-square regression algorithm (Goovaerts, 1992; Wackernagel, 1998). The technique, known as factorial kriging analysis, has been used in geochemical exploration to distinguish large isolated values (pointwise anomalies) from groupwise anomalies that consist of two or more neighboring values just above the chemical detection limit (Sandjivy, 1984). It has also been applied in hydrology to distinguish between local sources of contamination linked to human activities and regional changes in geological properties of the aquifer (Goovaerts et al., 1993). Multivariate applications of factorial kriging in soil science have shown repeatedly that filtering of weakly correlated local spatial components tends to enhance a relation between variables that is otherwise blurred in an approach where all different sources of variation are mixed, leading to a greater understanding of the physical underlying mechanisms controlling spatial patterns (Goovaerts, 1994; Webster et al., 1994).

There have been relatively few applications of geostatistics to cancer data, with alternative solutions to the problem of non-stationarity of the variance caused by spatially varying population sizes. In his book (p.385-402), Cressie (1993) analyzed the spatial distribution of the counts of sudden-infant-death-syndromes (SID) for 100 counties of North Carolina. He proposed a two-step transform of the data to remove first the mean-variance dependence of the data and then the heteroscedasticity. Traditional variography was then applied to the transformed residuals. In another study on the risk of childhood cancer in the West Midlands of England, Oliver et al. (1998) developed an approach that accounted for spatial heterogeneity in the population of children to estimate the semivariogram of the ‘risk of developing cancer’ from the semivariogram of observed mortality rates. Cokriging was then used to produce a map of cancer risk. An alternative is to incorporate directly the fuzziness or softness of the data into the computation of the sample semivariogram and into the kriging equations. This can be done using the BME (Bayesian Maximum Entropy) approach developed by Christakos et al. (2002). Last, in their review paper Gotway and Young (2002) showed how block kriging can account for differing supports in spatial prediction (aggregation and disaggregation approach), allowing the analysis of relationships between disease and pollution data recorded over different geographies. Filtering of spatial components, as well as study of space-time relationships between cancer rates, was not addressed in any of these studies.

The objective of this paper is to present an adaptation of semivariogram and factorial kriging analysis that accounts for spatially varying population size in the processing of cancer mortality data. The performance of this new methodology to detect underlying spatial patterns of risks and estimate accurately the correlation between different cancer risks is assessed using stochastic simulation.

2. GEOSTATISTICAL ANALYSIS OF CANCER DATA: AN EXAMPLE

A geostatistical approach for the characterization of spatial patterns of cancer mortality rates and exploration of scale-dependent relationships between cancers is presented. Observed cancer rate data are used to illustrate the methodology and facilitate the understanding of a few issues associated with the analysis of aggregated data. The analysis of the spatial patterns of brain and bladder cancer in New England is however not the topic of this paper and therefore is not discussed at length.

2.1. Population-weighted Semivariogram Estimators

For a given number N of entities (e.g. counties, states, electoral ward), denote the number of recorded mortality cases by l(u_α) and the size of the population at risk n(u_α). Following most authors (Cressie, 1993; Oliver et al., 1998; Christakos and Lai, 1997; Christakos and Serre, 2000), entities are referenced geographically by their centroids (or seats) with the vector of spatial coordinates u_α=(x_α,y_α). The empirical mortality rates are then denoted as z(u_α). Figure 1 (top map) shows an example for 295 counties of 12 New England States (1950-1969 period). The directly age-adjusted mortality rates for brain cancer, as provided by the new Atlas of United States mortality (Pickles et al., 1999), are displayed for white females. Note the huge variation in the size of the population at risk (from 338 to 846,472) as well as the spatial pattern of the population data (Figure 1, middle plots).

Figure 1:

Maps of brain cancer mortality rates and the corresponding county population sizes. Bottom graphs show the semivariogram of mortality rates computed using the traditional (unweighted) or population-weighted estimator.

The common approach to characterize the spatial pattern of brain cancer rates would consist of computing the sample semivariogram defined as:

$$\widehat{\gamma }\left(\mathbf{h}\right)=\frac{1}{2N\left(\mathbf{h}\right)}\sum _{\alpha =1}^{N\left(\mathbf{h}\right)}{[\mathbf{z}\left({\mathbf{u}}_{\alpha }\right)-\mathbf{z}({\mathbf{u}}_{\alpha }+\mathbf{h})]}^2$$ (1)

where |h| corresponds to the Euclidian distance between two centroids— All the following discussion can be readily generalized to other distance measures that might be more appropriate to capture contiguity of entities of complex shape. This approach is however too simplistic since it ignores the fact that the two observations $z\left({\mathit{u}}_{\alpha }\right)$ and $z({\mathit{u}}_{\alpha }+\mathbf{h})$ are derived from different population sizes, and hence have different reliabilities. Intuitively, ignoring this local source of randomness would lead one to underestimate the actual magnitude of the spatial correlation or, in other words, to overestimate the high-frequency variance. This is seen in the sample semivariogram of mortality rates displayed at the bottom of Figure 1 (left plot) which appears as a pure nugget effect, indicating a lack of spatial correlation.

To account for the varying level of reliability of the data, we propose to weight differently the pairs $[z\left({\mathit{u}}_{\alpha }\right)-z({\mathit{u}}_{\alpha }+h)]$ in expression (1), that is to use an estimator of the type (Cressie, 1993; Rivoirard et al., 2000):

$$\widehat{\gamma }\left(\mathbf{h}\right)=\frac{1}{2\sum _{\alpha =1}^{N\left(\mathbf{h}\right)}w\left({\mathbf{u}}_{\alpha }\right)}\sum _{\alpha =1}^{N\left(\mathbf{h}\right)}w\left({\mathbf{u}}_{\alpha }\right){[\mathbf{z}\left({\mathbf{u}}_{\alpha }\right)-\mathbf{z}({\mathbf{u}}_{\alpha }+\mathbf{h})]}^2$$ (2)

The following four options were considered for the weights:

1. Use of the traditional semivariogram estimator where each data pair receives an equal weight, $w\left({\mathit{u}}_{\alpha }\right)=1\forall {\mathit{u}}_{\alpha }$.

2. Use of weights equal to the sum of population size, $w\left({\mathit{u}}_{\alpha }\right)=n\left({\mathit{u}}_{\alpha }\right)+n({\mathit{u}}_{\alpha }+\mathit{h})\forall {\mathit{u}}_{\alpha }$.

3. Use of weights equal to the sum of the square root of the population size, $w\left({\mathit{u}}_{\alpha }\right)=\sqrt{\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)}+\sqrt{\mathrm{n}({\mathbf{u}}_{\alpha }+\mathbf{h})}\forall {\mathit{u}}_{\alpha }$.

4. Use of weights equal to the sum of logarithm of the population size, $w\left({\mathit{u}}_{\alpha }\right)=\mathit{ln}\left[n\left({\mathit{u}}_{\alpha }\right)\right]+\mathit{ln}\left[n({\mathit{u}}_{\alpha }+\mathit{h})\right]\forall {\mathit{u}}_{\alpha }$.

The objective in using the last three weighting schemes is to attenuate the impact of data pairs for which mortality rates are computed from small population sizes. Population weighted estimators can be viewed as regularized semivariograms (Journel and Huijbregts, 1978), where the support of an observation is not directly based on its geographic extent but is proportional to the size of its population. Note that the population weights are equivalent to the area of the geographical units assuming that the population density is the same for all these units.

Figure 1 (right bottom plot) shows the weighted sample semivariogram of brain cancer data computed using option 3. It displays a clear spatial structure with much smaller nugget effect (i.e. discontinuity at the origin). The solid line represents a model that has been fitted by weighted least squares to the experimental values. It takes the form:

$$\gamma \left(\mathbf{h}\right)=b_0\times \delta \left(\mathbf{h}\right)+b_1\times g\left(\mathbf{h}\right)$$ (3)

where b₀ is the nugget variance, $\delta \left(\mathit{h}\right)=0$ if $\mid \mathit{h}\mid =0$ and 1 otherwise, b₁ is the sill of a so-called spherical model defined as: $g\left(\mathit{h}\right)=1.5(\mid \mathit{h}\mid ∕a)-0.5{(\mid \mathit{h}\mid ∕a)}^3$ if $\mid \mathbf{h}\mid$ does not exceed a, the range of spatial correlation equal to 205 km, and g(h)=1 otherwise.

2.2. Filtering the Noise

In general the noise in the mortality data at any location u_α can be filtered using the following weighted linear combination of s(u_α) surrounding observations:

$$Z_{Filt}\left({\mathbf{u}}_{\alpha }\right)=\sum _{i=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_i\left({\mathbf{u}}_{\alpha }\right)z\left({\mathbf{u}}_i\right)$$ (4)

Different weighting schemes can be adopted. The simplest choice is to set ${\lambda }_i\left({\mathit{u}}_{\alpha }\right)=1∕s\left({\mathit{u}}_{\alpha }\right)\forall i$ that is to use an arithmetic average. The number s(u_α) can then vary spatially to create filters of fixed geographical size or filters of constant or nearly constant population size (Talbot et al., 2000). An alternative, which accounts for data reliability, is to assign a weight that is proportional to the population size or inversely proportional to the standard error of measured rates. The proximity of observations to the data to be smoothed can also be incorporated by using weights that are a function of the distance $\mid {\mathit{u}}_{\alpha }-{\mathit{u}}_i\mid$ (e.g. inverse squared distance such as in Kafadar, 1994). In geostatistical filters, besides the distance $\mid {\mathit{u}}_{\alpha }-{\mathit{u}}_i\mid$ the weights are influenced by the model of spatial correlation fitted to the weighted semivariogram (2) and the geometric configuration of the observations (e.g. clustering). The weights are obtained by solving the following system (Goovaerts, 1997) of linear equations:

$$\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)\gamma \left({\mathbf{u}}_i\text{-}{\mathbf{u}}_j\right)+u\left({\mathbf{u}}_{\alpha }\right)=\gamma \left({\mathbf{u}}_i\text{-}{\mathbf{u}}_{\alpha }\right)+g_{0}i=1,\dots ,s\left({\mathit{u}}_{\alpha }\right)\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)=1$$ (5)

where μ(u_α) is a Lagrange multiplier that results from minimizing the estimation variance subject to the unbiasedness constraint on the estimator, $g_0=b_0\times (1-\delta ({\mathit{u}}_i-{\mathit{u}}_{\alpha }))$ where b₀ is the nugget variance in eq. (3) and $\delta ({\mathit{u}}_i-{\mathit{u}}_{\alpha })=0$ if u_i=u_α and 1 otherwise. The right-hand-side term differs from the traditional implementation of ordinary kriging by the addition of the nugget effect for zero lag distances. It leads to non-zero semivariogram values on the right-hand side of the kriging system even if the estimated location u_α coincides with a data location u_i, allowing the noise component to be filtered (non-exact interpolator). Note that in the absence of spatial correlation (i.e. pure nugget effect), the kriging estimate (4) is simply the arithmetic average of observations.

To incorporate data reliability (i.e. population size) directly into the geostatistical filter the kriging weights computed according to system (5) are rescaled as:

$${\lambda }_i^{\prime }\left({\mathbf{u}}_{\alpha }\right)=\frac{{\lambda }_i\left({\mathbf{u}}_{\alpha }\right)\times n\left({\mathbf{u}}_i\right)}{\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)\times n\left({\mathbf{u}}_j\right)}\mathrm{i}=1,\dots ,\mathrm{s}\left({\mathbf{u}}_{\alpha }\right)$$ (6)

This rescaling is applied separately to the negative and positive kriging weights, keeping constant the overall contribution of these two sets of weights; that is the sum of positive (negative) kriging weights is the same before and after rescaling, which ensures that the unbiasedness constraint in system (5) is still satisfied. Figure 2 (top) shows the maps of brain cancer mortality rates before and after noise filtering. Filtering clearly enhances spatial patterns, in particular clusters of high and low rates become much more apparent. These clusters cannot be discerned reliably in the raw rate maps.

Figure 2:

Maps of brain cancer mortality rates before and after filtering of noise by kriging analysis. Bottom maps show the decomposition of bladder mortality rates into local and regional components on the basis of nested semivariogram model.

2.3. Factorial Kriging Analysis

The geostatistical filtering described in Section 2.2 can be expressed as the decomposition of the observed mortality rates into a structured component and a noise component corresponding to the decomposition of the nested semivariogram model (3):

$$z\left({\mathbf{u}}_{\alpha }\right)=z_{Filt}\left({\mathbf{u}}_{\alpha }\right)+{\widehat{z}}_0\left({\mathbf{u}}_{\alpha }\right)$$ (7)

As the size of the study area increases, the spatial distribution of cancer mortality rates is likely influenced by a series of factors related, for example, to demography, economy and environment. If the scales at which these different factors operate are very different from one another, then they should be apparent in the semivariogram of mortality rates which would be modeled using several basic semivariogram structures, e.g. spherical models with different ranges of spatial correlation. In the simplest case, two nested structures are observed in addition to the nugget effect, leading to the following model:

$$\gamma \left(\mathbf{h}\right)=b_0\times \delta \left(\mathbf{h}\right)+b_1\times g_1\left(\mathbf{h}\right)+b_2\times g_2\left(\mathbf{h}\right)$$ (8)

where g₁(h) is a semivariogram model with a short range (local variability) while g₂(h) has a longer range (regional variability). An example is shown in Figure 2 (middle plot) where the semivariogram of bladder cancer displays two scales of variability which were modeled, using weighted least-square regression, by two spherical models with ranges of 70 and 425 kms, respectively. On the basis of the nested model (8), the filtered component in expression (7) can then be further decomposed into local and regional components:

$$z\left({\mathbf{u}}_{\alpha }\right)={\widehat{z}}_0\left({\mathbf{u}}_{\alpha }\right)+z_{Filt}\left({\mathbf{u}}_{\alpha }\right)={\widehat{z}}_0\left({\mathbf{u}}_{\alpha }\right)+{\widehat{z}}_1\left({\mathbf{u}}_{\alpha }\right)+{\widehat{z}}_2\left({\mathbf{u}}_{\alpha }\right)$$ (9)

This decomposition assumes the local component has a zero mean while the regional component incorporates the local mean of mortality rates (i.e. mean within the kriging search window). The regional component is estimated as:

$${\widehat{z}}_2\left({\mathbf{u}}_{\alpha }\right)=\sum _{i=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_i\left({\mathbf{u}}_{\alpha }\right)z\left({\mathbf{u}}_i\right)$$ (10)

where the weights are the solution of the following system:

$$\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)\gamma ({\mathbf{u}}_i-{\mathbf{u}}_j)+\mu \left({\mathbf{u}}_{\alpha }\right)=b_2{g}_2({\mathbf{u}}_i-{\mathbf{u}}_{\alpha })i=1,\dots ,s\left({\mathit{u}}_{\alpha }\right)\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)=1$$ (11)

Similarly, the estimate for the local spatial component is written as:

$${\widehat{z}}_1\left({\mathbf{u}}_{\alpha }\right)=\sum _{i=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_i\left({\mathbf{u}}_{\alpha }\right)z\left({\mathbf{u}}_i\right)$$ (12)

where the weights are the solution of the following system:

$$\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)\gamma ({\mathbf{u}}_i-{\mathbf{u}}_j)+\mu \left({\mathbf{u}}_{\alpha }\right)=b_1{g}_1({\mathbf{u}}_i-{\mathbf{u}}_{\alpha })i=1,\dots ,s\left({\mathit{u}}_{\alpha }\right)\sum _{j=1}^{s\left({\mathbf{u}}_{\alpha }\right)}{\lambda }_j\left({\mathbf{u}}_{\alpha }\right)=0$$ (13)

Note that decomposition (9) is not valid when the kriging weights in systems (11) and (13) are rescaled according to expression (6). Figure 2 (bottom maps) shows the maps of local and regional components of bladder cancer rates which were estimated on the basis of this nested semivariogram model. The regional map displays clear clusters of high mortality rates, while low rates are confined to the Southern states. These clusters are of larger size than the ones observed for brain cancer, which reflects the longer range identified on the weighted semivariogram (425 versus 205 km). Decomposition into local and regional components directly supports the identification of local hotspots and coldspots from the map of local components and larger-scale trends and differences from the map of regional components. This is a considerable benefit over the commonly used disease clustering methods such as the scan, LISA and related methods that,while, suited for the identification of focused, local, and global clustering, do not decompose disease rates into components that reflect the characteristic spatial scales of variation.

2.4. Scale-dependent Correlation

Consider now that the mortality rate of another type of cancer has been recorded over the same N entities, $\{y\left({\mathit{u}}_{\alpha }\right),\alpha =1,\dots ,N\}$ Assuming that the relationship between the two sets of mortality rates is linear, the common way to assess its strength would be to compute the correlation coefficient as:

$$r_{ZY}=\frac{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right)[\mathbf{z}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{Z}}]\times [\mathrm{y}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{Y}}]}{\sqrt{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right){[\mathbf{z}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{Z}}]}^2}\sqrt{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right){[\mathrm{y}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{Y}}]}^2}}$$ (14)

with w(u_α)=1/N. By analogy with the computation of the weighted sample semivariogram (2) it would be desirable to account for data reliability also in the computation of this statistic (Haining, 1991). Three weighting schemes were considered to attenuate the influence of mortality rates computed from small population sizes:

1. $\mathrm{w}\left({\mathbf{u}}_{\alpha }\right)=\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)∕{\sum }_{\alpha =1}^N\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)$

2. $\mathrm{w}\left({\mathbf{u}}_{\alpha }\right)=\sqrt{\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)}∕{\sum }_{\alpha =1}^N\sqrt{\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)}$

3. $\mathrm{w}\left({\mathbf{u}}_{\alpha }\right)=\ln \left[\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)\right]∕{\sum }_{\alpha =1}^N\ln \left[\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)\right]$

A limitation of any statistic of type (14) is that it ignores spatial information, such as the location of observations, and might therefore reflect the combined influence of several spatial processes operating at different spatial scales. For most cancers this is almost certainly the case, as cancers have multiple etiologies some of which are affected by risk factors related to geology (e.g. radon exposure), behavior (e.g. smoking), ethnicity/genetics (e.g. genetic predisposition); and these risk factors operate at different spatial scales related to local as well as regional differences in the environment, risk behavior, demography, SES and ethnicity. The key idea behind the computation of scale-dependent correlations consists of replacing the raw mortality rates z(u_α) and y(u_α) in expression (14) by their corresponding spatial components to derive either a local or regional correlation coefficient. For example, the population-weighted correlation coefficient between brain and bladder cancer mortality rates is only 0.17 while the same correlation is 0.43 once noise and local components have been filtered out. This effect is similar to the increase in correlation noticed by several authors when grouping proximal units such as census tracts (modifiable areal unit problem, see Gotway and Young (2002) for a review of the issue of spatial aggregation and combination of data measured over different spatial supports). Removing the noise prior to calculating the correlation coefficients is expected to yield a more accurate estimation of the true, underlying correlations, and this is confirmed in simulation studies (below).

3. STOCHASTIC SIMULATION OF CANCER MORTALITY RATES

The benefit of a geostatistical approach over common spatial smoothers, as well as the ability of weighted semivariograms to assess underlying spatial patterns of risks, cannot be assessed from observed data since the reality is unknown. This section shows how geostatistical simulation allows one to generate realizations of the spatial distribution of cancer mortality rates with different patterns and levels of correlation. These realizations will then be used in a comparison study.

3.1. Univariate Case

The recorded numbers of cases, l(u_α), can be viewed as realizations of Binomial variables L(u_α) with parameters p(u_α) for the probability of developing the disease (cancer risk) and n(u_α) for the sample size. Following the Binomial model, the expected value and variance of the random variable Z(u_α), corresponding to the proportion of cases z(u_α)=l(u_α)/n(u_α), are:

$$E\left[Z\left({\mathbf{u}}_{\alpha }\right)\right]=p\left({\mathbf{u}}_{\alpha }\right)\text{and}Var\left[Z\left({\mathbf{u}}_{\alpha }\right)\right]=\frac{p\left({\mathbf{u}}_{\alpha }\right)\times (1-p\left({\mathbf{u}}_{\alpha }\right))}{n\left({\mathbf{u}}_{\alpha }\right)}$$ (15)

Because of the large sample size n(u_α), it is reasonable to assume that the variable Z has approximately a normal distribution.

The objective is to generate joint realizations of the set of N random variables $\{z^{\left(l\right)}\left({\mathit{u}}_{\alpha }\right),\alpha =1,\dots ,N\}$ that reproduce a specific histogram and spatial pattern as modeled by the semivariogram. To be representative of reality, our simulations used the actual population data recorded at the locations of the 295 counties of 12 New England States displayed in Figure 1. The following general procedure was used to generate realizations of cancer mortality rates, accounting for spatial patterns of risk and uncertainty resulting from the observed spatially varying population sizes:

1. The spatial distribution of the cancer risk is first simulated using sequential Gaussian simulation (sGs) over the N locations ${\mathit{u}}_{\alpha },\{p^{\left(l\right)}\left({\mathit{u}}_{\alpha }\right),\alpha =1,\dots ,N\}$ The simulation was conditioned to the sample histogram of bladder cancer data displayed in Figure 2 and five different types of semivariogram models, which were selected arbitrarily to cover a wide range of possible situations:

   • Pure nugget effect (NE): no spatial correlation.
   • Spherical semivariogram model with a range of 200 kms and 25% NE.
   • Spherical semivariogram model with a range of 200 kms and 50% NE.
   • Spherical semivariogram model with a range of 200 kms and 75% NE.
   • Spherical semivariogram model with a range of 200 kms and no NE (highly continuous process).
2. At each simulated location u_α, a normal probability distribution for the mortality rate Z(u_α) is determined using as mean and variance the statistics defined in equation (15) with the risk simulated in step 1. Two options were considered for the population sizes:

   • The original distribution of population sizes for these counties, which are highly correlated in space., see Figure 1 (middle plots).
   • A random swapping of the set of original population sizes, which allows one to keep the same histogram while removing any spatial correlation in population size.

3. Monte-Carlo simulation is then used to draw randomly a simulated rate $z^{\left(l\right)}\left({\mathit{u}}_{\alpha }\right)$ from the local distribution modeled in step 2, yielding a realization of mortality rates $\{z^{\left(l\right)}\left({\mathit{u}}_{\alpha }\right),\alpha =1,\dots ,N\}$

Sequential Gaussian simulation used in step 1 proceeds as follows (see Goovaerts, p. 380 for more details):

1. Define a random path visiting each location u_α only once.

2. At each location u_α, determine the parameters (mean and variance) of the Gaussian probability distribution of cancer risks using simple kriging and previously simulated values as conditioning information.

3. Draw a simulated value from that distribution and add it to the data set.

4. Proceed to the next location along the random path, and repeat the two previous steps.

5. Loop until all N locations are simulated.

6. Transform the simulated normal scores so that the target histogram is reproduced.

Figure 3 (top maps) shows an example of the simulation procedure. A realization of the spatial distribution of cancer risks is generated using a semivariogram with zero nugget effect. Simulated risks are combined with the original distribution of population sizes, and the Monte Carlo procedure yields the simulated map of mortality rates displayed in the middle (left column). The bottom scatterplot illustrates how the population-related noise obscures the actual relationship between observed mortality rates and underlying risks. The right column shows the results of factorial kriging. Filtering the noise reveals the underlying spatial pattern of risks, increasing the correlation between mortality rates and risks. Note that the five counties with zero risk value have been discarded for the computation of the correlation coefficient since zero risks lead to zero variance according to expression (15) and to an artificial perfect prediction of risks by the simulated mortality rates. This simulation study demonstrates first, the ability of the technique to reconstruct the underlying disease risk from observed mortality rates, and second, that these reconstructed risks are more accurate estimates of the true, underlying risks than the raw observed rates are.

Figure 3:

Map of mortality rates obtained from a simulated map of risk and population size, and the result of noise filtering by kriging. The scatterplots illustrate the more accurate estimation of cancer risks achieved by filtering noise in observations versus using observed mortality rates.

3.2. Bivariate Case (one spatial structure)

An algorithm similar to the one used in the univariate case can be implemented to generate joint realizations of mortality rates for 2 different types of cancer, that is $\{({\mathrm{z}}^{\left(1\right)}\left({\mathbf{u}}_{\alpha }\right),{\mathrm{y}}^{\left(1\right)}\left({\mathbf{u}}_{\alpha }\right)),\alpha =1,\dots ,\mathrm{N}\}$ Correlated rates were simulated over the N=295 county locations using the following procedure:

1. The spatial distribution of the risk of developing each type of cancer is first simulated in two steps: 1) the risk of developing cancer #1, denoted p_z(u_α) is simulated using sequential Gaussian simulation (sGs), 2) the risk for cancer #2, p_y(u_α) is then simulated as a linear combination of the simulated value $p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)$ and another realization independent of p_z:

   $$p_y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)=a\times p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)+0.5\times p_z^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)\text{with}-1\le \mathrm{a}\le 1$$ (16)

   where $\mathrm{Corr}\{p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right),p_z^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)\}=0$ Both risks have the same spatial pattern (i.e. semivariograms), and their correlation ranges from moderately negative (-0.6) to moderately positive (0.6) depending on the value of the coefficient a.

2. At each simulated location u_α, normal probability distributions for the two mortality rates Z(u_α) and Y(u_α) are determined with means and variances as given in equation (15), using the risks simulated in step 1 and population sizes derived from actual data.

3. Monte-Carlo simulation is then used to draw simulated rates $z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)$ and $y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)$ randomly from the local distributions modeled in step 2.

As in the univariate case, the simulation was conditioned to the sample histogram of bladder cancer data and the five aforementioned semivariogram models.

3.3. Bivariate Case (nested spatial structures)

The simulation study is now extended to the more complex situation where the correlation between risks of developing two different cancers is scale-dependent. The concept of scale-dependent relationships is illustrated in Figure 4 where the spatial distribution of risk for two cancers is created as the sum of a local (range=100 kms) and regional (range=300 kms) components. The corresponding experimental semivariograms are shown in Figure 5. The two spatial structures are clearly apparent on the semivariograms of the risks of cancer (Figure 5, bottom plots). The local and regional components of the two types of cancer have been generated such that the correlation is negative at local scale and positive at regional scale, see scatterplots of Figure 6. A traditional analysis based on the raw observations would fail to detect such scale-dependent correlation, leading one to incorrectly conclude that the two risks are non-correlated, see bottom scatterplots.

Figure 4:

Maps of the simulated risk of developing two types of cancer (bottom maps). Each risk map is created as the sum of 2 spatial components with a short and long range. Note the negative correlation between the short-range components while the long-range components are positively correlated.

Figure 5:

Experimental semivariograms for the spatial components and risks of developing cancers #1 and #2 displayed in Figure 4.

Figure 6:

Scatterplots for the spatial components and risks of developing cancers #1 and #2 displayed in Figure 4.

The following procedure was used:

1. The spatial distribution of the risk of developing each type of cancer is first simulated at both local and regional scales.

   $$p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)=C+a\times p_y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)+0.5\times p_y^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)\text{with}-0.5\le \mathrm{a}\le 0.5$$ (17)

   where $\mathrm{Corr}\{p_y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right),p_y^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)\}=0$ and C is a constant to ensure that the simulated rate is positive even for negative values of the coefficient a. Both risks have the same spatial pattern (i.e. spherical semivariogram with range of 100), and their correlation ranges from moderately negative (-0.69) to moderately positive (0.70) depending on the value of the coefficient a.
   At regional scale, the risk p_y(u_α) is first simulated using sequential Gaussian simulation (sGs). Then, the risk p_z(u_α) is simulated as a linear combination of the simulated value p ⁽_yl) (u_α) and another realization independent of p_y:

   $$p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)=0.6\times p_y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)+0.4\times p_y^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)$$ (18)

   where $\mathrm{Corr}\{p_z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right),p_y^{\left(l^{\prime }\right)}\left({\mathbf{u}}_{\alpha }\right)\}=0$ Both risks have the same spatial pattern (i.e. spherical semivariogram with range of 300 kms), and their correlation is constant and highly positive (0.89).

   For each type of cancer, the final value for the simulated risk is obtained by adding the values simulated at local and regional scales.

2. At each simulated location u_α, normal probability distributions for the two mortality rates Z(u_α) and Y(u_α) are determined with means and variances as given in equation (15), using the risks simulated in step 1 and population sizes derived from actual data.

3. Monte-Carlo simulation is then used to draw simulated rates $z^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)$ and $y^{\left(l\right)}\left({\mathbf{u}}_{\alpha }\right)$ randomly from the local distributions modeled in step 2.

4. RESULTS

4.1. Semivariogram Inference

The objective is to investigate which of the weighted semivariogram estimators of type (2) provides the most accurate description of the underlying spatial pattern of risk from the experimental cancer mortality rates. Using the procedure outlined in section 3.1, 200 realizations were generated for each of the ten combinations of five semivariogram models and two population spatial distributions (original=non-random and random swapping). For each realization, the weighted semivariograms were modeled using least-square regression (Pardo-Iguzquiza, 1999) as the sum of a nugget effect and spherical model.

The closeness between inferred and underlying models could be evaluated using any statistic measuring the difference between the semivariogram of risk γ_R (h) fitted to the realization of risk and the model fitted to the estimator of the semivariogram γ(h) of mortality rates. The following measure was used here:

$$\mathit{Diff}=\frac{1}{K}\sum _{k=1}^K\mid \widehat{\gamma }\left({\mathbf{h}}_{\mathbf{k}}\right)-{\gamma }_R\left({\mathbf{h}}_{\mathbf{k}}\right)\mid$$ (19)

Average values of the statistics for the five different types of semivariogram of risk are listed in Table 1. Bold numbers indicate smallest discrepancies for each combination. There is no most accurate estimator for all situations but the square root population weighting scheme provides the best results in 60% of cases. Assigning too much importance to population size (e.g. population weighting scheme) in fact tends to create spurious spatial structures in the semivariogram estimate, which reflects more the spatial pattern of population size than the underlying pattern of cancer risks. This artifact has been confirmed by the fact that even when the risk is spatially unstructured (pure NE scenario) the weighted semivariograms computed for non-random population sizes show a structure.

Table 1.

Average value of the difference statistics (19) for all combinations of semivariogram estimators, spatial pattern of population sizes and risk of cancer. Bold number indicates best results (i.e. smallest differences) for each combination.

NE of α R(h)	No weight	Population weight	√population weight	Ln(population) weight
Non-random
0%	0.0824	0.0868	0.0611	0.0758
25%	0.0568	0.0133	0.0322	0.0534
50%	0.0469	0.0501	0.0181	0.0399
75%	0.0528	0.0436	0.0158	0.0456
100%	0.0999	0.1102	0.0983	0.0161
Random
0%	0.0802	0.1586	0.1031	0.0844
25%	0.0425	0.1574	0.0686	0.0459
50%	0.0331	0.0256	0.0152	0.0262
75%	0.0529	0.0374	0.0358	0.0488
100%	0.0597	0.0714	0.0359	0.0562

4.2. Prediction Errors

Semivariogram inference is a preliminary step towards the estimation of underlying cancer risks by factorial kriging. Hence, the different semivariogram estimators should be compared in terms of prediction performances. For the 200 realizations and each combination listed in Table 1, the risk of cancer p(u_α) has been estimated using the kriging estimator (4) and the 32 closest experimental mortality rates. The kriging system is solved using the semivariogram model fitted to each of the four alternative semivariogram estimators.

For each combination of semivariogram and kriging estimators, the mean absolute prediction error (MAPE) was computed as:

$$\mathit{MAPE}=\frac{1}{N}\sum _{\alpha =1}^N\mid \widehat{p}\left({\mathbf{u}}_{\alpha }\right)-p\left({\mathbf{u}}_{\alpha }\right)\mid$$ (20)

The reference prediction error was computed by replacing p(u_α) by the cancer mortality rates z(u_α), which amounts to ignoring the variability induced by spatially varying population sizes and differences between risks of developing the disease and mortality rates. As explained before, the five zero risk values were discarded for the computation of the MAPE to avoid an artificial underestimation of the reference prediction error.

Table 2 shows the average prediction errors obtained for all possible combinations of semivariogram estimators and spatial patterns of risk and population size. Kriging weights, rescaled according to expression (6) to account for data reliability, were found to give smaller prediction errors than classical kriging weights, hence only these results are displayed here. Comparison of reference error with other columns clearly demonstrates the benefit of geostatistical filtering of noise: the reduction in prediction errors increases as the risk values are more spatially structured (i.e. smaller nugget effect). The $\sqrt{\mathit{popul}}$. weighted semivariogram provides the smallest prediction errors, which confirms results obtained in section 4.1 for semivariogram inference. The last check was to assess whether kriging estimators are more accurate than simple weighted averages of surrounding data. Looking at Table 3 the benefit of kriging is indisputable since all weighted averages actually yield prediction errors larger than the reference values.

Table 2.

Average prediction error for all combinations of semivariogram estimators, spatial pattern of population sizes and risk of cancer (Rescaled kriging weights (6)). Bold number indicates best results (i.e. smallest errors) for each combination.

NE of α R(h )	Reference	No weight	Population weight	√population weight	Ln(population) weight
Non-random
0%	0.6251	0.4883	0.4187	0.3837	0.4444
25%	0.6093	0.4463	0.4778	0.4199	0.4327
50%	0.6058	0.4903	0.4629	0.4492	0.4772
75%	0.6156	0.5309	0.4934	0.5148	0.5231
100%	0.6282	0.5128	0.4887	0.4709	0.4956
Random
0%	0.6423	0.4319	0.4473	0.3741	0.4070
25%	0.5968	0.4472	0.5243	0.3992	0.4283
50%	0.6164	0.4970	0.4927	0.4438	0.4775
75%	0.6253	0.5380	0.5251	0.4885	0.5213
100%	0.6154	0.4997	0.5047	0.4611	0.4886

Table 3.

Average prediction error for all combinations of weights, spatial pattern of population sizes and risk of cancer (estimator = arithmetical average). Bold number indicates best results (i.e. smallest errors) for each combination.

NE of α R (h)	Reference	No weight	Population weight	√population weight	Ln(population) weight
Non-random
0%	0.6251	0.6547	0.7229	0.6731	0.6554
25%	0.6093	0.6138	0.6773	0.6183	0.6116
50%	0.6058	0.6603	0.6886	0.6649	0.6588
75%	0.6156	0.7045	0.7346	0.7068	0.7031
100%	0.6282	0.6431	0.6508	0.6406	0.6411
Random
0%	0.6423	0.6539	0.6812	0.6445	0.6487
25%	0.5968	0.6128	0.6861	0.6177	0.6105
50%	0.6164	0.6602	0.7224	0.6567	0.6551
75%	0.6253	0.7043	0.7294	0.7032	0.7019
100%	0.6154	0.6429	0.6611	0.6373	0.6393

4.3. Bivariate Simulation (one spatial structure)

For each of the five types of semivariogram models of risk, 20 sets (i.e. realizations) of 295 simulated rates were generated for cancer 1 and for each of these sets 20 realizations for cancer 2 were generated, leading to 20×20=400 possible pairs of sets of simulated rates. For each pair, the correlation was estimated using the weighted correlation coefficient between mortality rates defined in expression (14).

To evaluate the benefit of factorial kriging for assessing correlation between cancer risks, the weighted correlation coefficient is also computed on kriging estimates:

$$r_{ZY}=\frac{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right)[{\widehat{\mathrm{p}}}_{\mathrm{z}}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{z}}]\times [{\widehat{\mathrm{p}}}_{\mathrm{y}}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{y}}]}{\sqrt{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right){[{\widehat{\mathrm{p}}}_{\mathrm{z}}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{z}}]}^2}\sqrt{\sum _{\alpha =1}^{N}w\left({\mathbf{u}}_{\alpha }\right){[{\widehat{\mathrm{p}}}_{\mathrm{y}}\left({\mathbf{u}}_{\alpha }\right)-{\mathrm{m}}_{\mathrm{y}}]}^2}}$$ (21)

Following results of the previous section, the kriging weights were rescaled according to expression (6) to account for changes in data reliability linked to differences in population size. The four different types of weighted semivariogram estimators were used. Experiments have shown that the best results (i.e. smallest deviations between actual and estimated correlation coefficients) were obtained when the second type of weights $w\left({\mathit{u}}_{\alpha }\right)=n\left({\mathit{u}}_{\alpha }\right)∕{\sum }_{\alpha =1}^N\mathrm{n}\left({\mathbf{u}}_{\alpha }\right)$ were used,regardless of the type of weighted semivariogram estimator. Hence, all simulation runs were performed using this option.

Figure 7 shows the plots of estimated versus actual correlation coefficients for two types of spatial patterns for the risk of developing cancer (0 and 50% NE). The top plots show results obtained by computing the weighted correlation coefficient (14) from the observed mortality rates, while the bottom plots show the correlation obtained after filtering of the noise using kriging. For each of the four approaches, three statistics of the distribution of 400 estimated correlation coefficients are plotted: mean (thick line), and 0.05 and 0.95 percentiles (thin lines). The top plots indicate that, regardless of the weighting scheme, using observed mortality rates leads one to underestimate the magnitude of the correlation, either positive or negative. In this case the most accurate predictions are obtained when the observations are weighted by the population size in expression (14), which substantiates the use of this weighting scheme for computing the correlation coefficient between kriging results. The comparison of top and bottom plots clearly indicates that the filtering of the noise using factorial kriging leads to more accurate prediction of correlation between risks of cancers, with best results obtained for the $\sqrt{\mathit{popul}}$. weighting scheme (long dash curves). In this case the 95% confidence interval includes most of the actual correlation coefficients (45° line).

Figure 7:

Plots of actual correlation coefficients versus coefficients estimated from raw or filtered (i.e. kriging) mortality rates using four different weighted estimators. The simulated values of cancer risk were generated using the zero and 50% nugget effect scenario.

4.4. Bivariate Simulation (nested spatial structures)

For each of the five types of semivariogram of risks, 20 sets (i.e. realizations) of 295 simulated rates were generated for cancers 1 and 2, leading to 20×20=400 possible pairs of sets of simulated rates, using the procedure described in Section 3.3. As for the case of one structure, the weighted correlation coefficient was computed before and after filtering the observed rates using factorial kriging and the nested model fitted to the four different types of semivariogram estimators.

Figure 8 shows the plots of estimated versus actual correlation coefficients for the observed rates (top) and the filtered rates (middle). In each case, three statistics of the distribution of 400 estimated correlation coefficients are plotted: mean (thick line), and 0.05 and 0.95 percentiles (thin lines). Since the use of the observed rates does not allow one to detect scale-dependent correlation, only one global correlation coefficient can be computed. The top plots indicate that, regardless of the weighting scheme, using observed mortality rates underestimates the magnitude of the correlation, either positive or negative. For the short-range component, none of the correlation coefficients is ever negative, even though the actual correlation is negative. Filtering by kriging yields more accurate predictions (middle plots), in particular for the long-range component. This is illustrated by the lower average difference between predicted and actual correlation coefficients as displayed in the bottom plots. Unfortunately no weighting scheme is systematically better than the others, but any one of them is better than no weighting scheme at all.

Figure 8:

Plots of actual correlation coefficients versus coefficients estimated from raw or filtered (i.e. kriging) mortality rates using four different weighted estimators. Bottom plot shows the absolute prediction error of correlation coefficient obtained from mortality rates before (thin line) and after (thick line) noise filtering by kriging. The simulated values of cancer risk were generated using the 0 % nugget effect scenario.

The results indicate that despite more accurate predictions achieved by kriging, the negative correlation at local scale is hardly detected. One reason might be that random fluctuations introduced by the sampling of probability distributions mask too much the actual relationship between risks of developing cancer. To check this conjecture the simulation and subsequent analysis was repeated using a standard deviation one order of magnitude smaller than the theoretical one given in expression (15). It almost amounts to using cancer risks instead of mortality rates in the analysis. The corresponding results (Figure 9) indicate that, as expected, the confidence intervals are much narrower. While the smaller variability does not improve results obtained from the raw rates, results are substantially improved for kriging. The negative correlation at a local scale is now clearly apparent and kriging still outperforms the use of raw rates as the local correlation becomes strongly positive, compare bottom plots of Figures 8 and 9. This more accurate prediction of negative correlation at local scale seems, however, to be balanced by less accurate predictions of positive correlation at regional scale.

Figure 9:

Plots of actual correlation coefficients versus coefficients estimated from raw or filtered (i.e. kriging) mortality rates using four different weighted estimators. Bottom plot shows the absolute prediction error of correlation coefficient obtained from mortality rates before (thin line) and after (thick line) noise filtering by kriging. The simulated values of cancer risk were generated using the 0% nugget effect scenario with small uncertainty (i.e. variance of Binomial distribution).

The last check was to investigate whether the technique would perform better if the correlation at both scales were of the same sign. Figure 10 shows similar plots in the case the local correlation ranges from 0.02 to 1.0 while the regional correlation is constant at 0.89. Again, filtering the raw mortality rates yields more accurate estimates of the correlation between risks except when the local correlation exceeds 0.6. In the latter case the positive correlation is underestimated.

Figure 10.

:Plots of actual correlation coefficients versus coefficients estimated from raw or filtered (i.e. kriging) mortality rates using four different weighted estimators. Bottom plot shows the absolute prediction error of correlation coefficient obtained from mortality rates before (thin line) and after (thick line) noise filtering by kriging. The simulated values of cancer risk were generated using the 0% nugget effect.

5. DISCUSSION

This paper presents a simulation-based demonstration of some key features of the proposed geostatistical methodology to analyze cancer mortality rates:1. Weighted semivariogram estimators are able to detect underlying spatial patterns of risks, with the best results generally obtained for the $\sqrt{\mathit{popul}}$. weighting scheme.2. Factorial kriging allows one to filter the noise attached to mortality rates, leading to more accurate predictions of the underlying risk of developing cancer. Since the magnitude of this noise is expected to increase as the population size decreases, it is important to account for data reliability in kriging. The proposed rescaling of kriging weights systematically lowers the average prediction error for the square root and logarithm weights. It is also reassuring to notice that kriging systematically outperforms the use of simple weighted averages of surrounding rates.3. The more accurate prediction of cancer risks achieved by kriging translates into more accurate prediction of the correlation coefficient between these risks. More precisely, the kriging-based approach attenuates the underestimation of the magnitude of the correlation that is associated with the use of observed mortality rates. Best results are obtained when the $\sqrt{\mathit{popul}}$. weighting scheme is used for semivariogram estimator and the correlation coefficient is weighted directly by the population size.4. Estimation of scale-dependent correlation is more difficult to achieve. Moderate performances are however partially due to the automatic modeling of semivariograms and a likely overestimation of the noise attached to mortality rates, which does not facilitate the detection of nested structures on the experimental semivariograms. At local scale all weighting schemes yield similar results, while at a regional scale the weighting by the population size gives the largest prediction errors in all three case scenarios considered.

In summary, it is worth incorporating spatial patterns, as modeled using population-weighted semivariograms, in the geostatistical filtering of mortality rates. This approach yields maps of spatial components that are closer to the underlying risk maps in terms of prediction errors and provide a more accurate visualization of regional patterns. These maps allow one to explore scale-dependent correlation between risks of developing cancers and could be used to detect clusters at different spatial scales, which would reflect the influence of factors (i.e. demography, economy, environment) acting at different scales.

Validity of Assumptions: The approach is founded on the assumption of additive and independent spatial processes. This assumption makes possible the decomposition into local and regional components. That factors contributing to cancer risk might occur at different spatial scales is obvious. Almost all cancers have more than one risk factor that may include age, socio-economic status, genetics, and behavior, as well as environmental and occupational exposures. Further, it seems highly reasonable for these risk factors to be structured at spatial scales related to local as well as regional differences in the environment, risk behavior, demography, and so on. That spatial risk processes operating at local and regional scales might be correctly assumed to always be additive and independent is less clear. Consider melanoma, which exhibits strong north-south (regional) variation in risk caused by differences in solar radiation with latitude. People living in rural areas may have additional risk caused by differences in occupation, such as farming, that are associated with more time spent outside. For this example, additivity appears to be a reasonable assumption. But exposure-risk relationships often are non-linear, and it thus may not always be the case that local and regional risks will be strictly additive. When processes at different spatial scales are not independent, one needs to account for cross covariance in risk in order to decompose risk into local and regional components. This quickly results in more unknowns than information, making the modeling of the dependence between spatial processes a difficult prospect in practice. Moreover, a specific example of a cancer with interactions between regional- and local-scale risk processes is not easily conjectured. In the absence of a specific alternative, additivity and independence seems to these authors to be a parsimonious and reasonable assumption. When the form of the dependence between spatial processes is known, this information can be incorporated into the semivariogram model.

Spatial prediction using kriging is based on the assumption of stationarity, which implies that the spatial variability is only a function of the distance between observations, thereby ignoring the non-stationarity of the variance caused by spatially varying population sizes. Weighted semivariograms used in this study indirectly account for this effect. Yet, a more rigorous approach would be the one developed by Oliver et al. (1998) that accounted for spatial heterogeneity in the population of children to retrieve the semivariogram of the ‘risk of developing cancer’ from the semivariogram of observed mortality rates. This approach would also allow one to estimate separately the variability arising from data reliability (spatially varying population size) and the potential nugget variability of the underlying risk. In this manuscript both sources of variability are lumped together in the nugget effect of the weighted semivariogram, leading one to filter them indiscriminately in the factorial kriging analysis.

Implications for cancer surveillance and control: Cancer mortality maps are used by public health officials to identify areas of excess and to guide surveillance and control activities. Maps of incidence as well as mortality are used as input to disease clustering procedures whose purpose is to identify local areas of excess. While some controversy revolves around the utility of these techniques, it is indisputable that the finding of a confirmed cancer cluster is often of considerable concern. The accurate quantification of local excesses, as well as regional trends and differences in cancer incidence and mortality, is therefore a problem of considerable practical importance. The advent of a technique that has been demonstrated in realistic,population-based simulations to effectively and reliably filter noise while revealing underlying local- and regional-scale risk patterns has several implications for cancer surveillance and control. First, it makes possible the accurate identification of hotspots that may be caused by environmental exposures or by spatial excesses of risk factors such as genetic predisposition and risk-related behaviors. Second, it directly supports the quantification of risk trends that may reflect regional differences in exposure (such as solar radiation for melanoma), risk behavior (such as the predilection for high-fat diets, increased alcohol consumption and smoking in some northern states), and other measures such as socio-economic status, population density and ethnic origin. Finally, it provides more accurate estimates of underlying risk as well as the correlations between different cancer risks. These more accurate estimates should be used in cancer surveillance and control instead of raw rates that are less reliable and less accurate.

Future research: While this technique is highly promising additional simulation studies are needed to assure the method is robust under different underlying risk functions using population distributions representative of those in the United States. Additional plausible risk functions should be explored with different amounts of noise, hotspots, and regional trends. The sensitivity of the method needs to be further assessed - how small do regional and local differences in risk have to be in order for this method to be able to distinguish them from noise? It is clear from this preliminary study that this geostatistical approach has considerable potential for increasing the accuracy of risk estimates calculated from observed mortality rates, and ultimately for improving our understanding of geographic variation in cancer mortality.