- Methodology
- Open Access

# How does Poisson kriging compare to the popular BYM model for mapping disease risks?

- Pierre Goovaerts
^{1}Email author and - Samson Gebreab
^{2}

**7**:6

https://doi.org/10.1186/1476-072X-7-6

© Goovaerts and Gebreab; licensee BioMed Central Ltd. 2008

**Received: **08 January 2008

**Accepted: **04 February 2008

**Published: **04 February 2008

## Abstract

### Background

Geostatistical techniques are now available to account for spatially varying population sizes and spatial patterns in the mapping of disease rates. At first glance, Poisson kriging represents an attractive alternative to increasingly popular Bayesian spatial models in that: 1) it is easier to implement and less CPU intensive, and 2) it accounts for the size and shape of geographical units, avoiding the limitations of conditional auto-regressive (CAR) models commonly used in Bayesian algorithms while allowing for the creation of isopleth risk maps. Both approaches, however, have never been compared in simulation studies, and there is a need to better understand their merits in terms of accuracy and precision of disease risk estimates.

### Results

Besag, York and Mollie's (BYM) model and Poisson kriging (point and area-to-area implementations) were applied to age-adjusted lung and cervix cancer mortality rates recorded for white females in two contrasted county geographies: 1) state of Indiana that consists of 92 counties of fairly similar size and shape, and 2) four states in the Western US (Arizona, California, Nevada and Utah) forming a set of 118 counties that are vastly different geographical units. The spatial support (i.e. point versus area) has a much smaller impact on the results than the statistical methodology (i.e. geostatistical versus Bayesian models). Differences between methods are particularly pronounced in the Western US dataset: BYM model yields smoother risk surface and prediction variance that changes mainly as a function of the predicted risk, while the Poisson kriging variance increases in large sparsely populated counties. Simulation studies showed that the geostatistical approach yields smaller prediction errors, more precise and accurate probability intervals, and allows a better discrimination between counties with high and low mortality risks. The benefit of area-to-area Poisson kriging increases as the county geography becomes more heterogeneous and when data beyond the adjacent counties are used in the estimation. The trade-off cost for the easier implementation of point Poisson kriging is slightly larger kriging variances, which reduces the precision of the model of uncertainty.

### Conclusion

Bayesian spatial models are increasingly used by public health officials to map mortality risk from observed rates, a preliminary step towards the identification of areas of excess. More attention should however be paid to the spatial and distributional assumptions underlying the popular BYM model. Poisson kriging offers more flexibility in modeling the spatial structure of the risk and generates less smoothing, reducing the likelihood of missing areas of high risk.

## Keywords

- Geographical Unit
- Nugget Effect
- Prediction Variance
- Geostatistical Model
- Kriging Variance

## Background

Cancer mortality maps are used by public health officials to identify areas of excess and to guide surveillance and control activities. Interpretation of those maps is frequently hampered by the presence of noise caused by unreliable extreme rates computed from sparsely populated geographical units or for diseases with a low frequency of occurrence [1]. Ignoring the uncertainty attached to rate estimates can lead to misallocation of resources to investigate unreliable clusters of high risk, while areas of real concern might go undetected.

Over the years, statistical techniques of increasing complexity, usually involving Bayesian models, have been developed to improve the reliability of risk estimates by borrowing information from neighboring entities [2–6]. The implementation of these models in software such as WinBUGS [7] has fostered their adoption by both the statisticians and epidemiologists. Yet, the estimation of model parameters requires iterative procedures, such as Markov Chain Monte Carlo (MCMC) methods, that are computer intensive and require fine-tuning, which makes their application and interpretation challenging for non-statisticians [8–10]. This might explain why the majority of applications nowadays still rely on one of the early models introduced by Besag, York and Mollie in 1991 [5] which is commonly referred to as BYM model. Quoting the 2005 comparison paper by Best *et al*. [2]*"a (non exhaustive) search of the major epidemiological journals over the past decade indicates that the BYM model appears to be the only fully Bayesian spatial model to have been used in published applications of disease mapping outside of the statistical literature"*. This model is overwhelmingly used with the conditional auto-regressive (CAR) model for defining the random effect associated with spatial autocorrelation [2, 9, 11]. The arbitrary neighborhood relationship underlying the CAR model is computationally convenient but is not well-suited to situations where geographical entities have different sizes and shapes and are not arranged in a regular pattern [12]. The choice of a neighborhood weight structure (i.e. adjacency-based versus distance-based spatial weights) is also very subjective and seldom justified despite its substantial impact on the results [13]. Last, simulation studies have demonstrated the strong smoothing effect of Bayesian disease-mapping models, in particular the BYM model, which limits their ability to detect localized increases in risk [14].

Geostatistics provides an alternative, yet still little known, model-based approach to disease mapping [15]. The first initiative to tailor geostatistical tools to the analysis of disease rates must be credited to Christian Lajaunie [16] from the Center of geostatistics in Fontainebleau, France. Although it was introduced the same year as the BYM model, the geostatistical approach, called binomial cokriging, went largely unnoticed. The rare applications of the method are the study of the risk of childhood cancer in the West Midlands of England [17–19] and the mapping of lung cancer mortality across the US [20]. More recently, a similar approach, called Poisson kriging, was developed in the field of marine ecology and generalized to the analysis of cancer mortality rates and cholera incidence data [21–23]. Poisson kriging was combined with stochastic simulation to generate multiple realizations of the spatial distribution of cancer mortality risk, allowing the propagation of uncertainty through the detection of cancer clusters and outliers [24]. Point Poisson kriging, which assigns each measured rate to the geographic centroid of the unit over which it has been recorded, was later replaced by Area-to-Area (ATA) Poisson kriging that accounts for the geometry of administrative units and the spatial repartition of the population at risk in semivariogram estimation and kriging [25]. Like fully Bayesian models, Poisson kriging yields the full posterior distribution of the risk. This distribution is however assumed to be Gaussian and fully characterized by the kriging estimate and variance, while the BYM posterior distribution is modeled by the empirical distribution of simulated risk values. The trade-off cost for the simplicity of Poisson kriging is that, unlike the full Bayesian approach, the uncertainty attached to the parameters of the correlation function is ignored in the analysis, which should lead to smaller prediction variances in general [26].

Only a few comparison studies of geostatistical and Bayesian estimators have been published in the literature. In the earliest study [27], marginal and conditional generalized linear models (GLM) were compared to three implementations of universal kriging: no nugget effect, unweighted nugget effect, and a nugget effect that is weighted inversely proportional to the expected counts. Despite their simplicity, the linear kriging models provided surprisingly good predictions, in particular the third implementation that acknowledges the larger uncertainty (noise) of rates computed from small populations (fewer expected counts). Recent studies extended the comparison to semivariogram estimators and kriging algorithms that were derived specifically for count data using the properties of binomial or Poisson distributions. First, the benefit of binomial or Poisson kriging over a few simple smoothers (i.e. population-weighted estimators and empirical Bayes smoothers) was assessed under different scenarios for the disease frequency, the population size, and the spatial pattern of risk [20, 22]. Simulation studies showed that Poisson kriging outperforms other approaches for most scenarios, with a clear benefit when the risk values are spatially correlated. In a third study [26], Poisson kriging was compared to Diggle *et al*.'s "model-based kriging" [28] for mapping the relative abundance of species in the presence of spatially heterogeneous observation efforts and sparse animal sightings. Both methods gave equivalent results for 90% of the predictions; differences were observed for higher values with a smoothing for the kriging. Methods differed much more in terms of the prediction variance: the lognormal hypothesis underlying the Bayesian model induced similarities between the maps of the prediction variance and estimate (i.e. proportionality), while the Poisson kriging variance mainly reflects the observation effort (i.e. lower variance for longer observation times). Because this study was conducted on real data, the underlying risk pattern was unknown, precluding any objective assessment of prediction performances.

This paper presents the first simulation-based evaluation of performance of Poisson kriging (both point and ATA implementations) and full Bayesian disease-mapping models. The benchmark for Bayesian methods is the BYM model since it is the most widely applied in the health literature. Both types of model are first illustrated using age-adjusted lung and cervix cancer mortality rates recorded for white females in two contrasted county geographies: 1) state of Indiana that consists of 92 counties of fairly similar size and shape, and 2) four states in the Western US (Arizona, California, Nevada and Utah) forming a set of 118 counties that are vastly different geographical units. Simulation studies are then conducted for the same two county geographies. Performance criteria include the magnitude of prediction errors of the underlying risk, the precision and accuracy of the model of uncertainty, and the ability to discriminate between background and high-risk areas.

## Methods

### Cancer data sets

The basic properties of Poisson kriging and Bayesian model will be illustrated using directly age-adjusted mortality rates for a frequent (lung) and less frequent (cervix) cancer. These data were described in a previous study [25] and only their most salient features are summarized here. 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 compare performances in two contrasted settings: 1) all geographical units have a fairly similar size and shape, which is the "ideal" situation for point Poisson kriging or Bayesian methods implemented under the conditional auto-regressive (CAR) model that ignores the spatial support of the data, and 2) geographical units display a wide range of sizes and shapes, which should favour Area-to-Area Poisson kriging that implicitly accounts for the spatial support of the data in the analysis. The West coast provides 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.

^{2}cells discretizing each Region; see [25] for a description of these datasets. Figures 1 and 2 (top graphs) show county maps of age-adjusted mortality rates and population at risk, for lung cancer in Region 1 and cervix cancer in Region 2. 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.

### Simulated data sets

An objective assessment of the performance of the geostatistical and Bayesian disease mapping techniques requires knowledge of the "true" underlying risk maps, which are unknown in practice. Simulation provides a way to generate multiple realizations of the spatial distribution of cancer mortality rates under specific scenarios for the underlying risk and population sizes. 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 generated in a previous study [25] using a two-step procedure: 1) county-level mortality risks were computed as population-weighted averages of continuous risk maps created by sequential Gaussian simulation, and 2) the number of death within each county was simulated by random drawing from a Poisson distribution characterized by the county-level risk and population. The present comparison study used the odd-numbered realizations (i.e. realization #1,3,5,...,99) for each set.

### Area-to-Area (ATA) Poisson kriging

The geostatistical methodology for the estimation of risk values from empirical frequencies, and the creation of isopleth maps, is described in details in Goovaerts [25]. This section provides a brief recall of the approach

*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. The noise-filtered mortality rate for a given area

*v*

_{ α }, called mortality risk, is estimated as a linear combination of the kernel rate

*z*(

*v*

_{ α }) and the rates observed in (K-1) neighboring entities

*v*

_{i}:

*v*

_{ α }) ≠

*z*(

*v*

_{ α }). The weights

*λ*

_{ i }(

*v*

_{ α }) assigned to the

*K*rates are computed by solving the following system of linear equations; known as "Poisson kriging" system:

*μ*(

*v*

_{ α }) is the Lagrange parameter,

*m** is the population-weighted mean of the

*N*rates, and

*δ*

_{ ij }= 1 if

*v*

_{i}=

*v*

_{j}and 0 otherwise. The "error variance" term,

*m**/

*n*(

*v*

_{i}), derives directly from the Poisson distribution for the disease count variable

*D*(

*v*

_{i}). This term, which corresponds to the difference between the variances of the risk and rate variables (Equation 5 in [22]), leads to smaller weights for less reliable data (i.e. rates measured over smaller populations). In addition to the population size, the kriging system accounts for the spatial correlation among geographical units through the area-to-area covariance terms ${\overline{C}}_{R}$(

*v*

_{ i },

*v*

_{ j }) = Cov{

*Z*(

*v*

_{i}),

*Z*(

*v*

_{j})} and ${\overline{C}}_{R}$(

*v*

_{ i },

*v*

_{ α }). 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}:

*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'}:

In this study the discretizing points were identified with the nodes of a 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. When the geographical units differ by several orders of magnitude like in Region 2, it is not computationally efficient to use the same discretizing level for each unit. One solution is to use flexible discretizing grids that ensure a constant number of discretizing points within each unit. For example, in TerraSeer's STIS software [29] a given number of discretization points is distributed uniformly within each polygon according to a stratified random design.

**h**), or equivalently the point-support semivariogram

*γ*(

**h**), cannot be estimated directly from the observed rates, since only areal data are available. Thus, only the regularized semivariogram of the risk can be estimated as:

where N(**h**) is the number of pairs of areas (*v*_{
α
}, *v*_{
β
}) whose population-weighted centroids are separated by the vector **h**. The different spatial increments [*z*(*v*_{
α
})-*z*(*v*_{
β
})]^{2} are weighted by a function of their respective population sizes, n(*v*_{
α
})n(*v*_{
β
})/[n(*v*_{
α
})+n(*v*_{
β
})], a term which is inversely proportional to their standard deviations. More importance is thus given to the more reliable data pairs (i.e. smaller standard deviations). Derivation of a point-support semivariogram from the model fitted to the experimental semivariogram ${\widehat{\gamma}}_{R}$(**h**) computed from areal data is called "deconvolution", an operation that has been the topic of much research [30–33]. In this paper, we adopted the iterative procedure introduced for area-to-area Poisson kriging [25] whereby one seeks the point-support model that, once regularized, is the closest to the model fitted to areal data; see [33] for a detailed presentation of the algorithm and demonstration of its performances in simulation studies.

*v*

_{ α }can be modeled using the conditional cumulative distribution function (ccdf) of the risk variable

*R*(

*v*

_{ α }). Under the assumption of normality of the prediction errors, that ccdf is defined as:

*G*(·) is the cumulative distribution function of the standard normal random variable, and

*σ*(

*v*

_{ α }) is the square root of the kriging variance estimated as:

where ${\overline{C}}_{R}$(*v*_{
α
}, *v*_{
α
}) is the within-area covariance that is computed according to Equation (3) with *v*_{i} = *v*_{j} = *v*_{
α
}. This term will increase with the size of the area, leading to larger kriging variance when the risk is estimated for large geographical units. The notation "|(*K*)" expresses conditioning to the local information, say, *K* neighboring observed rates.

### Point Poisson kriging

Assimilating each unit *v*_{
α
}to its population-weighted centroid **u**_{
α
}greatly simplifies the implementation of the geostatistical methodology since it eliminates the need to discretize geographical units or perform a semivariogram deconvolution. Besides the gain in computational speed, the centroids-based (i.e. point) Poisson kriging can be accomplished using the public-domain executable **poisson-kriging.exe** described in Goovaerts [22]. This simplified version of Poisson kriging is included in the comparison study to: 1) quantify the potential loss in accuracy and precision resulting from such an approximation, and 2) apply Poisson kriging using the same point support as the BYM model.

### Bayesian model

Clayton and Kaldor [34] first introduced empirical Bayesian inference for relative risks. It was later extended to a fully Bayesian setting by Besag *et al*. [5]. The BYM model is one of the most popular hierarchical Bayesian models. This model incorporates random effects due to unstructured and spatially structured heterogeneity into the log-linear model for the relative risk. The inclusion of these random effects allows smoothing relative risks at global and local levels. For a detailed description of this model, the readers are referred to Lawson *et al*. [35] and Wakefield *et al*. [36]. The three stages of the BYM model, as described in Lawson *et al*. [35], are briefly reviewed in this section.

#### First-stage model

For rare diseases such as cancers, the observed number of mortality cases (*y*_{
i
}) in a geographic unit *i* is assumed to follow a Poisson distribution with mean *e*_{
i
}*θ*_{
i
}:

*y*_{
i
}~ *Poisson*(*e*_{
i
}*θ*_{
i
})

where *e*_{
i
}denotes the age-adjusted expected number of cases in the *i*^{
th
}geographic unit, and *θ*_{
i
}is the "true" but unknown relative risk in that unit. Besag *et al* [5] proposed the following model for the log relative risk:

log(*θ*_{
i
}) = *μ* + *v*_{
i
}+ *u*_{
i
}

where the intercept *μ* represents the global mean, and the terms *v*_{
i
}and *u*_{
i
}are random effects modeling the unstructured and spatially structured heterogeneity, respectively.

#### Second-stage model

*v*

_{ i }does not depend on geographic location and is assumed to follow a normal distribution with zero mean and a common variance (precision parameter)

*τ*

_{ v }

^{2}:

*u*

_{ i }, is defined according to the conditional autoregressive (CAR) Gaussian prior model [5]:

where the prior mean of each *u*_{
i
}is defined as a weighted average of the other *u*_{
j
}, j≠i. The weights *w*_{
ij
}define the relationship between the area *i* and its neighbors *j*. We adopted the common first order binary weighting scheme where *w*_{
ij
}*=* 1 if areas *i* and *j* share a common border, and *w*_{
ij
}= 0 otherwise. The precision parameter *τ*_{
u
}^{2} controls the amount of variability for the random effect.

#### Third-stage model

A fully Bayesian model specification is completed by adding hyper-prior distributions on the precision parameters *τ*_{
v
}^{2} and *τ*_{
u
}^{2}. With no prior estimation for precisions of the random effects, distributions with large variance (i.e. vague hyperpriors) are recommended. Following Kelsall and Wakefield's discussion [37] we adopted the Gamma distribution Γ(0.5, 0.0005) which yields a probability of 99% that the precision lies between 0.16 and 6635, with most of the probability concentrated towards 0 [9]. This prior choice is less informative and allows the likelihood data to dominate the prior information; hence, it will have minimum effect on the inference of relative risks.

#### Implementation of the model

As mentioned in the introduction, Bayesian inference of model parameters is not feasible through mathematically closed form. Instead, it requires computer intensive MCMC simulations, such as Gibbs sampler, which became affordable with recent increase in computational power. The BYM model for each data set (lung and cervix cancers) was fitted using WinBUGS version 1.4 [7] that implements the Markov Chain Monte Carlo (MCMC) methods [38]. The model was simulated using two independent chains starting with dispersed initial values. Convergence of the relative risks for the two chains was confirmed by graphing their traces, as well as computing the Gelman-Rubin (G-R) statistic and Monte Carlo errors (<5% of the posterior standard deviation). After a burn-in of 15,000 iterations, the following 10,000 iterations were sampled from the two chains, yielding a set of 20,000 values to approximate the posterior distribution of the relative risk for each geographical unit (i.e. county). For comparison with the Poisson kriging absolute risk estimate, each simulated relative risk value was multiplied by the population-weighted average of rates. The distributions of 20,000 sampled values were post-processed to derive summary statistics, such as mean, variance or the probability of exceeding specific risk thresholds.

## Results and Discussion

### Analysis of lung and cervix cancer data

Summary statistics for estimates of lung and cervix cancer mortality.

Estimators | Lung cancer | Cervix cancer | ||||
---|---|---|---|---|---|---|

Mean | Variance | Min-max | Mean | Variance | Min-Max | |

Observed rates | 21.19 | 18.48 | 9.071–31.79 | 2.851 | 2.446 | 0.000–8.138 |

ATA Poisson kriging (PK) estimate | 21.51 | 11.31 | 13.07–31.55 | 2.880 | 0.320 | 1.800–4.044 |

Point Poisson kriging (PK) estimate | 21.52 | 11.13 | 13.33–31.61 | 2.897 | 0.338 | 1.811–4.061 |

BYM model estimate | 21.38 | 11.35 | 13.96–31.63 | 2.889 | 0.200 | 1.915–4.074 |

ATA Poisson kriging (PK) variance | 3.122 | 2.009 | 0.238–7.582 | 0.187 | 0.040 | 0.003–0.881 |

Point Poisson kriging (PK) variance | 3.275 | 2.367 | 0.242–7.731 | 0.211 | 0.043 | 0.003–0.948 |

BYM model variance | 3.224 | 3.288 | 0.306–12.05 | 0.154 | 0.009 | 0.004–0.509 |

Correlation PK vs ATA PK estimate | 0.997 | 0.989 | ||||

Correlation BYM vs ATA PK estimate | 0.988 | 0.845 | ||||

Correlation PK vs ATA PK variance | 0.989 | 0.986 | ||||

Correlation BYM vs ATA PK variance | 0.908 | 0.549 |

#### Mortality risk estimates

#### Prediction variance

The Bayesian and geostatistical models differ much more in terms of prediction variance than estimated risk; see Figures 1 and 2 (right column). For both models the prediction variance increases as the population at risk, hence the reliability of raw rates, decreases. Because it accounts for the shape and size of counties, the ATA Poisson kriging variance also increases for larger spatial supports, i.e. counties of large extent. On the other hand, the lognormal hypothesis causes the BYM prediction variance to increase with the estimated risk. In Region 1 this latter effect is not obvious at first glance because it is masked by the impact of population size (Figure 4D). The proportional effect becomes striking when considering the ratio of the BYM variance versus the ATA Poisson kriging variance, which filters the effect of population size (Figure 4E). The correlation between this ratio and the BYM risk estimate is 0.85, indicating that the difference between the two types of prediction variance is largely controlled by the magnitude of Bayesian risk estimates. Figure 4E also indicates that conditionally to a range of BYM estimates, the BYM variance decreases faster than the ATA Poisson kriging variance as the population size, depicted by the size of the dots in the scatterplot, increases.

The smaller correlation between BYM and geostatistical estimates in Region 2, combined with the wide range of sizes and shapes of counties, enhances the differences between the prediction variance maps (Figure 2, right column). The correlation between the Bayesian and geostatistical prediction variances is only 0.549. The map of the ratio of the BYM variance versus the ATA Poisson kriging variance bears a lot of similarity with the map of risk estimates. The BYM prediction variance is smaller than the Poisson kriging variance in low risk areas, such as Utah, while the opposite trend is observed in the high risk areas along the coast (Figure 5B). The relationship between the variance ratio and the BYM risk estimate is not as strong as in Region 1: the correlation is 0.53 instead of 0.85 for Indiana. This weaker correlation reflects the impact of the size of geographical units on the kriging variance which is more pronounced for this heterogeneous county geography (Figure 5D). For example, Figure 5E illustrates the larger ATA Poisson kriging variance, hence smaller variance ratio, for counties of large size depicted by bigger dots in the scatterplot.

The spatial support (i.e. point versus area) has a much smaller impact on the prediction variance than the statistical methodology (i.e. geostatistical versus Bayesian models); see Table 1 (last two rows). For both regions the correlation between ATA and point kriging variances is strong: 0.986–0.989. Ignoring the spatial support of county-level data however leads, on average, to larger kriging variances: 3.275 vs 3.122 for Region 1 and 0.211 vs 0.187 for Region 2. This result is consistent with the "change-of-support" theory that predicts larger error variances for point versus areal estimation. The relative magnitude of the overestimation by point kriging is the largest for Region 2 that includes very large counties and has a wide range of county sizes and shapes.

#### Decision-making

The local Moran test to evaluate local clustering or spatial autocorrelation [39] was applied to the map of Bayesian and geostatistical risk estimates. The use of PK instead of BYM estimates increases the size of the central cluster of high mortality risk and the Utah cluster of low risk, compare Figure 6A and 6B. This difference results directly from the geostatistical prediction of lower risks in low-valued areas (i.e. Utah) and higher risks in high-valued areas (i.e. Nevada), recall Figure 5A. Beware that the use of smoothed rates in local cluster analysis (LCA) ignores the uncertainty attached to estimated risk and tends to create artificially large clusters. A geostatistical-based simulation procedure that propagates the rate uncertainty through LCA, as described in [24], would be more appropriate.

To depict the uncertainty attached to risk maps, several authors recommend mapping the 95^{th} percentile range of the posterior distribution of risk values or the probability that the risk in each entity exceeds a specific threshold of interest [9, 40]. This probability is readily derived from the PK estimate and variance using the analytical expression (6) or it can be computed from the empirical BYM posterior distribution. As an example, the population-weighted average of cervix cancer mortality rates (i.e. area-wide rate) was used as threshold of interest. The corresponding probability maps displayed in Figures 6C &6D capture the features of both the risk and prediction variance maps. In particular, note the larger probability of exceedence computed for Nevada using the kriging results. Richardson *et al*. [14] proposed to use this type of probability to decide whether an area should be classified as having an excess risk of cancer. Following their decision rule that the probability must exceed 0.75 in order for an area to be classified as having increased risk yields the classification maps at the bottom of Figure 6. The geostatistical model leads to declaring substantially more counties as having elevated risk: 28 instead of the subset of 18 for the BYM model. Four of these additional ten counties are located in Nevada, a result in agreement with the local cluster analysis. Note that the decision rule considered in this example ignores the multiple testing (or multiple comparison) problem caused by the repeated use of statistical tests; see [41] for a discussion on correction for multiple testing.

### Simulation studies

Figures 1, 2, 3, 4, 5, 6 supported an empirical comparison of results obtained using the BYM model and Poisson kriging. The smoothing of BYM risk estimates and smaller proportion of significant raised-risk areas agree with previous studies that showed Bayesian disease-mapping models to be conservative [14]. However, since the "true" mortality risk is unknown, one cannot conclude that one approach outperforms the other. To investigate the performance of Bayesian and geostatistical disease-mapping models for recovering the "true" risk surfaces, in particular the ability to detect risk-raised areas, a similar analysis was conducted on 50 simulated rate maps. Predicted risks were then compared to the risk maps used in the simulation.

^{th}simulated rate map in Region 2 (Figure 7B). To be more specific, the number of cases for each county was simulated by random sampling of a Poisson distribution whose mean parameter is the product of the mortality risk by the population. The division of the simulated death counts by the county population leads the set of simulated mortality rates; see [25] for more details. The noise caused by the small number problem is particularly apparent for sparsely populated counties. For example, a couple of counties in the North central part of the simulated map display high mortality rates, while the underlying risk value is low.

Performance comparison of Bayesian and geostatistical estimators: mean error of prediction.

Estimators | Lung cancer | Cervix cancer | ||
---|---|---|---|---|

| Average | % best result | Average | % best result |

Global Empirical Bayes | -0.031 | 22 | 0.053 | 18 |

Local Empirical Bayes | 0.012 | 14 |
| 28 |

BYM Model | 0.010 | 12 |
| 16 |

Point Poisson kriging (adjacent counties) | 0.009 | 14 | 0.019 | 22 |

ATA Poisson kriging (adjacent counties) | 0.014 | 20 | 0.023 | 8 |

ATA Poisson kriging (32 neighbors) |
| 18 | 0.040 | 8 |

| ||||

Global Empirical Bayes | -0.026 | 26 | 0.012 | 32 |

Local Empirical Bayes | -0.011 | 22 | 0.001 | 32 |

BYM Model |
| 18 |
| 12 |

Point Poisson kriging (adjacent counties) | -0.011 | 16 |
| 10 |

ATA Poisson kriging (adjacent counties) | -0.011 | 4 |
| 8 |

ATA Poisson kriging (32 neighbors) | -0.016 | 14 | 0.002 | 6 |

Performance comparison of Bayesian and geostatistical estimators: mean absolute error of prediction.

Estimators | Lung cancer (mean = 21.25) | Cervix cancer (mean = 2.993) | ||
---|---|---|---|---|

Arithmetical average | Average | % best result | Average | % best result |

Global Empirical Bayes | 1.396 | 2 | 0.517 | 0 |

Local Empirical Bayes | 1.380 | 0 | 0.458 | 0 |

BYM model | 1.280 | 6 | 0.426 | 0 |

Point Poisson kriging (adjacent counties) |
| 30 | 0.400 | 16 |

ATA Poisson kriging (adjacent counties) | 1.246 | 32 | 0.397 | 14 |

ATA Poisson kriging (32 neighbors) | 1.250 | 30 |
| 70 |

| ||||

Global Empirical Bayes | 0.972 | 2 | 0.152 | 0 |

Local Empirical Bayes | 0.980 | 2 | 0.145 | 0 |

BYM model | 0.918 | 12 | 0.141 | 4 |

Point Poisson kriging (adjacent counties) |
| 32 | 0.133 | 18 |

ATA Poisson kriging (adjacent counties) | 0.909 | 24 |
| 28 |

ATA Poisson kriging (32 neighbors) | 0.911 | 28 |
| 50 |

Performance comparison of Bayesian and geostatistical estimators: Smoothing effect and prediction variance.

Estimators | Lung cancer | Cervix cancer | ||
---|---|---|---|---|

Dispersion variance | Prediction variance | Dispersion variance | Prediction variance | |

True risk values | 9.817 | - | 1.153 | - |

Global Empirical Bayes | 6.059 | - | 0.537 | - |

Local Empirical Bayes | 7.905 | - | 1.063 | - |

BYM model | 7.034 | 2.791 | 0.757 | 0.347 |

Point Poisson kriging (adjacent counties) | 7.835 | 2.542 | 1.006 | 0.338 |

ATA Poisson kriging (adjacent counties) | 7.563 | 2.363 | 0.977 | 0.287 |

ATA Poisson kriging (32 neighbors) | 7.400 | 2.297 | 0.830 | 0.237 |

Performance comparison of geostatistical and Bayesian estimators: goodness and precision of models of uncertainty.

Estimators | Lung cancer | Cervix cancer | ||
---|---|---|---|---|

GOODNESS STATISTIC | Average | % best result | Average | % best result |

BYM model |
| 48 | 0.950 | 48 |

Point Poisson kriging (adjacent counties) | 0.941 | 28 |
| 24 |

ATA Poisson kriging (adjacent counties) | 0.922 | 14 | 0.939 | 16 |

ATA Poisson kriging (32 neighbors) | 0.914 | 10 | 0.927 | 12 |

AVERAGE WIDTH OF PI | ||||

BYM model | 2.544 | 14 | 0.816 | 4 |

Point Poisson kriging (adjacent counties) | 2.439 | 20 | 0.813 | 0 |

ATA Poisson kriging (adjacent counties) | 2.348 | 12 | 0.745 | 0 |

ATA Poisson kriging (32 neighbors) |
| 54 |
| 96 |

% ACCURATE AND PRECISE PI | ||||

BYM model vs Point PK (adjacent counties) | 9.35 | 6 | 12.73 | 8 |

BYM model vs ATA PK (adjacent counties) | 7.53 | 4 | 6.22 | 2 |

BYM model vs ATA PK (32 neighbors) | 7.35 | 4 | 4.30 | 4 |

Point PK (adjacent counties) vs BYM model |
| 36 | 28.02 | 32 |

ATA PK (adjacent counties) vs BYM model | 28.12 | 24 | 36.02 | 20 |

ATA PK (32 neighbors) vs BYM model | 27.66 | 26 |
| 34 |

Performance comparison of geostatistical and Bayesian estimators: discriminatory power of models of uncertainty.

Estimators | Lung cancer | Cervix cancer | ||
---|---|---|---|---|

LOW RISK THRESHOLD (RR = 1) | Average | % best result | Average | % best result |

BYM model | 4.009 | 14 | 3.532 | 16 |

Point Poisson kriging (adjacent counties) | 4.112 | 20 | 3.751 | 4 |

ATA Poisson kriging (adjacent counties) |
| 36 | 3.915 | 38 |

ATA Poisson kriging (32 neighbors) | 4.132 | 30 |
| 42 |

HIGH RISK THRESHOLD | RR = 1.1 | RR = 1.25 | ||

BYM model | 7.037 | 6 | 5.859 | 2 |

Point Poisson kriging (adjacent counties) | 7.842 | 12 | 6.300 | 0 |

ATA Poisson kriging (adjacent counties) | 8.059 | 36 | 6.721 | 6 |

ATA Poisson kriging (32 neighbors) |
| 46 |
| 92 |

Performance comparison of geostatistical and Bayesian estimators: proportion of false positives.

Estimators | Lung cancer | Cervix cancer | ||
---|---|---|---|---|

LOW RISK THRESHOLD (RR = 1) | Average | % best result | Average | % best result |

BYM model | 0.071 | 30 | 0.077 | 12 |

Point Poisson kriging (adjacent counties) |
| 20 | 0.066 | 14 |

ATA Poisson kriging (adjacent counties) | 0.070 | 30 | 0.063 | 24 |

ATA Poisson kriging (32 neighbors) | 0.072 | 20 |
| 50 |

HIGH RISK THRESHOLD | RR = 1.1 | RR = 1.25 | ||

BYM model | 0.069 | 20 | 0.080 | 4 |

Point Poisson kriging (adjacent counties) | 0.066 | 14 | 0.065 | 12 |

ATA Poisson kriging (adjacent counties) |
| 36 | 0.064 | 32 |

ATA Poisson kriging (32 neighbors) |
| 30 |
| 52 |

#### Bias and accuracy of prediction

where *r*(*v*_{
α
}) is the "true" simulated risk and ${r}_{P}^{\ast}$(*v*_{
α
}) is the risk estimated using Bayesian or geostatistical smoothers. The prediction error for each of the N counties is either equally weighted (*ω*_{
α
}= 1) or weighted according to the population size (*ω*_{
α
}= n(*v*_{
α
})), in order to penalize more the errors that affect a larger population.

Table 2 indicates that, on average over 50 realizations, all prediction methods are unbiased regardless of the weighting scheme or the cancer frequency. Methods differ more in terms of the mean absolute error of prediction, in particular for the least frequent cervix cancer (Table 3). The empirical Bayesian smoothers are consistently outperformed by other methods, although the average magnitude of the error never exceeds 5% of the mean rate. The benefit of Poisson kriging over BYM model is fairly systematic since it leads to smaller MAE values 100% of the time for cervix and 92% for lung cancer (see Figures 8A &8B for an illustration). Differences between methods are smaller for population-weighted statistics, yet Poisson kriging still yields the smallest prediction errors 96% of the time for cervix and 84% for lung cancer. Point Poisson kriging performs as well as ATA Poisson kriging, in particular in Region 1 where all counties have similar size and shape.

#### Smoothing effect and prediction variance

The comparison of risk maps in Figures 1, 2 and 7 indicated the smaller dispersion variance (i.e. larger smoothing) of estimates calculated using the BYM model. Results averaged over all 50 realizations and summarized in Table 4 support this finding. The variance of the reference risk values, averaged over the five scenarios for the risk map, is 9.817 and 1.153 for lung and cervix cancers, respectively. As expected, the global empirical Bayes estimator generates the largest smoothing effect: the variance represents 47–62% of the reference risk variance. The BYM model is the second worse (66–77%). The variance of ATA Poisson kriging estimates is closer to the variance of the true risk values, in particular when fewer observations (i.e. adjacent counties versus 32 nearest counties) are used in the estimation (77–85%). The smoothing effect is smaller for point Poisson kriging, which is expected since point estimates typically vary more than areal estimates.

In addition to a risk estimate, the BYM model and Poisson kriging provide a measure of the uncertainty attached to this estimate in the form of a prediction variance. For each interpolation method, the prediction variance was averaged over all counties, and the mean variances over all 50 realizations are listed in Table 4. As expected, the Poisson kriging variance increases when the estimation is based on fewer observations (i.e. adjacent counties versus 32 nearest counties) or ignores the support of the data (i.e. point versus ATA kriging). Yet, it remains on average 10–17% smaller than the prediction variance calculated for the BYM model. According to this criterion, the geostatistical model of uncertainty is more precise than the Bayesian one.

#### Quality of the model of uncertainty

The prediction variance does not fully capture the uncertainty attached to risk estimates, in particular for the BYM model where the entire probability distribution in the form of a set of simulated risk values is available for each geographic unit. From the distribution of 20,000 BYM simulated values we computed, for each of the counties, a series of symmetric *p*-probability intervals (PI) bounded by the (1-*p*)/2 and (1+*p*)/2 quantiles of that distribution. For example, the 0.5-PI is bounded by the lower and upper quartiles. Similar PIs were computed analytically from the Poisson kriging estimate and variance, under the assumption of normality of the prediction errors (Equation 6).

*p*

_{ k }, is quantified using the following "goodness" statistic:

where *w*_{
k
}= 1 if ${p}_{k}^{\ast}$ > *p*_{
k
}, and 2 otherwise. *K*' represents the discretization level of the computation. Twice more importance is given to deviations when ${p}_{k}^{\ast}$ <*p*_{
k
}(inaccurate case). The weights penalize less the accurate case, which is the case where the fraction of true values falling into the *p*-probability interval is larger than expected. The goodness statistic is completed by the so-called "accuracy plot" that allows one to visualize departures between observed and expected fractions as a function of the probability *p*.

The accuracy plots in Figure 8C indicate that the geostatistical (both point and ATA Poisson kriging) and Bayesian approaches yield fairly similar results for realization #50. Although the BYM curve is the closest to the 45° line of perfect agreement, it also lies more frequently below that line (inaccurate case). Because this case is penalized more heavily in the computation of the goodness statistics, the three models end up with similar value for this criterion. Table 5 confirms the tendency for the BYM model to yield, on average over all 50 simulations, larger goodness statistic than the geostatistical model of uncertainty.

Not only should the true risk value fall into the PI according to the expected probability p, but this interval should be as narrow as possible to reduce the uncertainty about that value. In other words, among two probabilistic models with similar goodness statistic one would privilege the one with the smallest spread (more precise). Different measures of ccdf spread can be used: variance, interquartile range, and entropy. Following Goovaerts [43] the average width of the PIs that include the true value are plotted for a series of probabilities *p*. Once again, the geostatistical and Bayesian approaches yield fairly similar results for realization #50 (Figure 8D). Yet, the BYM-based probability intervals are slightly wider than the intervals based on the ATA kriging variance, a result that is consistent across most of the 50 simulations (Table 5). The larger precision of the geostatistical model of uncertainty agrees with the previous observation that the PK variance is smaller than the prediction variance calculated for the BYM model. Since the kriging variance decreases as the number of observations increases, the narrowest probability intervals are computed using Poisson kriging with 32 neighbors. The widest intervals are obtained using point Poisson kriging, which is expected since the point kriging variance is larger than the ATA kriging variance (recall Table 4).

The geostatistical model of uncertainty for realization #50 presents the advantage of yielding PIs that are slightly narrower than the Bayesian ones, while being more likely to include the true value (i.e. larger observed fractions). This property however is not captured by the separate computation of the goodness statistic and average PI width. A more informative criterion is the proportion of probabilities *p* for which the geostatistical PI is narrower than the Bayesian PI (PK width < BYM width) while including a larger fraction of true values (PK fraction > BYM fraction). This situation corresponds to the case where both blue and red ratio curves in Figures 8E–F exceed a value of one depicted by the horizontal dashed line. For realization #50, the point kriging model is jointly more accurate and precise than the BYM model 40% of the time, while the reverse is true only 7% of the time. The benefit of the geostatistical approach increases for ATA kriging: the smaller kriging variance results in narrower probability intervals hence increased precision (see red curve in Figure 8F). The ATA kriging model is jointly more accurate and precise than the BYM model 71% of the time, while the reverse is true only 1% of the time. The same conclusions hold on average across all 50 realizations. Table 5 shows that, according to this criterion, Poisson kriging outperform the Bayesian model across more than 80% of realizations, with the largest benefit displayed by ATA kriging for the heterogeneous Region 2.

#### Classification accuracy

By analogy with the classification conducted on observed cancer mortality rates in Figures 6E &6F, each simulated map underwent a classification into low and high-risk areas based on the probability of exceeding the area-wide rate. Figures 9A &9B present the results for the 50^{th} realization of Region 2 and a 0.75 probability threshold. Like in Figures 6E &6F, the geostatistical model leads to declaring substantially more counties as having elevated risk: 40 instead of 30 for the BYM model. The availability of true mortality risks here allows the computation of the number of false positives (declaring a county as having elevated risk when in fact its underlying true risk is smaller than the area-wide rate) and negatives (declaring a county to be in the background when in fact its underlying true risk exceeds the area-wide rate). The BYM model generates fewer false positives (2 instead of 6), which is expected since much less counties are flagged as having elevated risk. This result is however achieved at the expense of many more false negatives: 17 instead of 11 for ATA Poisson kriging. The geostatistical model thus leads to a smaller total number of misclassified counties. The same trend is observed when the risk threshold is up to 75% larger than the area-wide rate (RR = 1.75, Figures 9C &9D). Point and ATA Poisson kriging yield very similar proportions of errors.

Another way to compare the performances of Bayesian and geostatistical approaches, independently of the subjective choice of a probability threshold, is to compute the Receiver Operating Characteristic (ROC) curves. ROC curves plot the probability of false positive versus the probability of detection [44]. The probability of detection corresponds here to the proportion of true high-risk counties (i.e. counties with a mortality risk larger than a threshold) that are detected as the probability threshold decreases. In practice, the probability thresholds are identified with the probabilities of exceedence computed for these high-risk counties. For each of them the probability of false positive is calculated as the proportion of background counties that are wrongly declared as having elevated risks. Figure 9E &9F shows the ROC curve for the 50^{th} realization using a low (Relative Risk, RR = 1) and high risk threshold (RR = 1.25).

The most efficient approach is the one that allows the detection of a larger fraction of high-risk counties at the expense of fewer false positives; that is the ROC curve should be as close as possible to the vertical axis. For example, Figure 9E indicates that for both BYM and geostatistical models 60% of high-risk counties (i.e. counties with underlying true risk larger than the area-wide rate, RR = 1) are detected without causing any false positive. Differences between the two models increase when the risk threshold is raised to 125% the area-wide rate (RR = 1.25, Figure 9F). A quantitative measure of the detection error is the relative area above the ROC curve, which represents the average proportion of false positives (FP). According to this criterion, the geostatistical model outperforms the BYM model for higher risk thresholds: the average percentage of false positives is 5.10% for Poisson kriging and 7.81% for the BYM model. Similar results are observed across all 50 realizations, with a clear benefit of Poisson kriging for the least frequent cervix cancer. Once again, point Poisson kriging yields slightly more false positives than ATA kriging, in particular for Region 2.

## Conclusion

Since its early development for the assessment of mineral deposits, geostatistics has been used in a growing number of disciplines dealing with the analysis of data distributed in space and/or time. Its application to health data is fairly recent and, compared to the well established Bayesian modeling approach, the field of health geostatistics is still in its infancy. Nevertheless, the current geostatistical methodology offers a flexible framework to model the spatial variability of disease rates and estimate the underlying risk, taking into account not only the shape and size of geographical units, but also the spatial repartition of population within those units [15]. The approach described in this paper has been fully automated in TerraSeer's STIS software [29]: the "behind-the-scene" estimation, modeling, and deconvolution of risk semivariogram, as well as the automatic discretization of geographical units, makes Poisson kriging estimators as straightforward as empirical Bayesian smoothers while providing a measure of uncertainty in the form of the kriging variance. Furthermore, the comparison of point and area-to-area (ATA) Poisson kriging demonstrated the benefits of the geostatistical approach even under a simplified implementation whereby each geographical unit is assimilated with its population-weighted centroid.

Over the years, statisticians have developed models of increasing complexity, combining fixed effects with both uncorrelated and spatially structured random effects, leading to mixed effects or hierarchical models. The rich class of Bayes models yields the full posterior distribution of the risk while accounting for the uncertainty in the parameters of the model. Implementation of these sophisticated methods however is still cumbersome and relies on strong distributional assumptions (e.g. lognormal hypothesis), while parameter estimation requires time-consuming iterative procedures. Another drawback is that the full Bayesian modeling approach is overwhelmingly used with the conditional auto-regressive (CAR) model for defining the random effect associated with spatial autocorrelation. This computationally convenient choice is reasonable if all geographical entities are of similar size and arranged in a regular pattern but it is not particularly attractive otherwise [12]. In particular, it prohibits any change of support and creation of isopleth maps of risk, an operation easily conducted within the framework of area-to-point kriging [25].

The objectives of this paper were two-fold: 1) to illustrate the major differences between Poisson kriging and the Bayesian disease-mapping approach in two contrasted county geographies, and 2) to present the first simulation-based comparison of performance of Poisson kriging and full Bayesian models. The comparison study is certainly not exhaustive and only the BYM model, among the many alternative formulations available for disease mapping [2, 14], was considered. Yet, studies have demonstrated the robustness of the BYM model which remains the favorite among practitioners [2, 45].

The analysis of lung and cervix cancer mortality rates highlighted the stronger smoothing effect of the BYM model: geostatistical risk estimates tend to be lower in low-valued areas, while they exceed BYM predictions in high-valued areas. The use of BYM results to guide surveillance and control activities will thus lead to declaring substantially fewer counties as having elevated risk. These results confirm previous studies that showed Bayesian disease-mapping models to be conservative [14]. The impact of the modeling procedure was the largest for the prediction variances that differ in both magnitude and spatial pattern. As expected, the prediction variance increases as the population at risk, hence the reliability of raw rates, decreases. Because it accounts for the shape and size of counties, the Poisson kriging variance also increases for larger spatial supports, i.e. counties of large extent. On the other hand, the lognormal hypothesis causes the BYM prediction variance to increase with the estimated risk.

According to our simulation studies, the geostatistical approach yields smaller prediction errors, more precise and accurate probability intervals, and allows a better discrimination between counties with high and low mortality risks. The BYM model also generates smoother risk surfaces, leading to a much larger proportion of false negatives than the geostatistical model in particular as the risk threshold raises. The benefit of Poisson kriging is the largest for cervix cancer measured in Region 2 because of the combined effect of the low reliability of mortality rates and the wide range of county shapes and sizes. In general, the performance of Poisson kriging improves when data beyond the adjacent counties are used in the estimation.

The smoothing effect of Poisson kriging is controlled by the number of neighbors used in the estimation (more neighbors induce more smoothing), the population size (unreliable rates computed from small populations are smoothed more), and the variogram model (larger nugget effect induces more smoothing). On the other hand, the smoothing effect of the Bayesian model depends on the choice of a neighborhood weight structure (i.e. adjacency-based versus distance-based spatial weights [13]), the population size (unreliable rates computed from small populations are smoothed more), the type of distribution for the relative risk (mean-based smoothing using the Gaussian distribution versus median-based smoothing for double-exponential distribution), and the type of spatial model for the risk surface (i.e. continuous surface for the BYM model versus discontinuous surface for the partition and spatial mixture models). Both Bayesian and kriging estimators in this paper are mean-based smoothers and use the same adjacent neighbors. Estimating the nugget effect of a semivariogram is always somewhat subjective, in particular when only areal data are available which precludes any sample information on the short-range variability. The nugget effect of the point-support semivariogram model was here set to zero to facilitate the convergence of the deconvolution procedure [33]. Relaxing this constraint increases the smoothing effect of Poisson kriging that becomes comparable to the BYM model in Region 1 (results now shown). It has however no impact on the results for Region 2, where discrepancies between the dispersion variances of kriging and BYM model estimates were the largest. An alternative would be to adopt semi-parametric Bayesian spatial models that allow discontinuities in the risk surface and typically produce less smoothing than the BYM model [2]. Yet, these models are less robust [45] and are unlikely to benefit the simulation results since all realizations are based on continuous risk surfaces generated using semivariograms with no nugget effect.

The spatial support (i.e. point versus area) has a much smaller impact on the results than the statistical methodology (i.e. geostatistical versus Bayesian models). The trade-off cost for the easier implementation of point Poisson kriging is slightly larger kriging variances, which reduces the precision of the model of uncertainty and the ability to discriminate between background and risk-raised areas. The major drawback still remains the inability of point kriging to conduct any change of support and creation of isopleth maps of risk, an operation easily conducted within the framework of area-to-point kriging.

A critical element of any simulation-based comparison study is the assumptions used to generate the reference risk maps. In the present study, the mortality rates (ratio of number of deaths over population size) were not directly simulated using a geostatistical procedure, but rather were generated according to a Poisson distribution, which is similar to the approach used in other comparison studies of Bayesian models [2, 14]. Because the BYM model is not designed to pick up discontinuities in the risk surface, all simulations were generated using smooth risk surfaces. The ability of Poisson kriging to uncover more complex risk structures should be investigated in future simulation studies which will include other classes of Bayesian hierarchical spatial models that generate less smoothing.

## Declarations

### Acknowledgements

This research was funded by grants R44-CA105819-02 and 2R44CA110281-02 from the National Cancer Institute. The views stated in this publication are those of the authors and do not necessarily represent the official views of the NCI.

## Authors’ Affiliations

## References

- Mungiole M, Pickle LW, Hansen Simonson K: Application of a weighted head-banging algorithm to mortality data maps. Statistics in Medicine. 1999, 18: 3201-3209.PubMedView ArticleGoogle Scholar
- Best N, Richardson S, Thomson A: A comparison of Bayesian spatial models for disease mapping. Statistical Methods in Medical Research. 2005, 14: 35-59.PubMedView ArticleGoogle Scholar
- Pickle LW: Exploring spatio-temporal patterns of mortality using mixed effects models. Statistics in Medicine. 2000, 19: 2251-2263.PubMedView ArticleGoogle Scholar
- Christensen OF, Waagepetersen R: Bayesian prediction of spatial count data using generalized linear mixed models. Biometrics. 2002, 58: 280-286.PubMedView ArticleGoogle Scholar
- Besag J, York J, Mollie A: Bayesian image restoration with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics. 1991, 43: 1-59.View ArticleGoogle Scholar
- Waller LA, Gotway CA: Applied Spatial Statistics for Public Health Data. 2004, New Jersey: John Wiley and SonsView ArticleGoogle Scholar
- Spiegelhalter DJ, Thomas A, Best NG, Lunn DJ: WinBUGS: Bayesian Inference Using Gibbs Sampling Manual, Version 1.4. 2002, London: Imperial College; Cambridge, UK:MRC Biostatistics UnitGoogle Scholar
- Leyland AH, Davies CA: Empirical Bayes methods for disease mapping. Statistical Methods in Medical Research. 2005, 14: 17-34.PubMedView ArticleGoogle Scholar
- Johnson GD: Small area mapping of prostate cancer incidence in New York State (USA) using fully Bayesian hierarchical modelling. International Journal of Health Geographics. 2004, 3: 29-PubMedPubMed CentralView ArticleGoogle Scholar
- Woodward P: BugsXLA: Bayes for the common man. Journal of Statistical Software. 2005, 14: 5-View ArticleGoogle Scholar
- Best NG, Arnold RA, Thomas A, Waller LA, Conlon EM: Bayesian models for spatially correlated disease and exposure data. Bayesian Statistics 6. Edited by: Bernardo JM, Berger JO, Dawid AP, Smith AFM. 1999, Oxford, UK, Oxford University Press, 131-156.Google Scholar
- Kelsall J, Wakefield J: Modeling spatial variation in disease risk: a geostatistical approach. Journal of the American Statistical Association. 2002, 97 (459): 692-701.View ArticleGoogle Scholar
- Earnest A, Morgan G, Mengersen K, Ryan L, Summerhayes R, Beard J: Evaluating the effect of neighbourhood weight matrices on smoothing properties of Conditional Autoregressive (CAR) models. International Journal of Health Geographics. 2007, 6: 54-PubMedPubMed CentralView ArticleGoogle Scholar
- Richardson S, Thomson A, Best N, Elliot P: Interpreting posterior relative risk estimates in disease-mapping studies. Environmental Health Perspectives. 2004, 112: 1016-1025.PubMedPubMed CentralView ArticleGoogle Scholar
- Goovaerts P: Geostatistical analysis of health data: state-of-the-art and perspective. geoENV VI – Geostatistics for Environmental Applications. Edited by: Soares A et al. 2008, The Netherlands, Springer-Verlag, 3-22.http://goovaerts.pierre.googlepages.com/GEOENV_VI.pdfView ArticleGoogle Scholar
- Lajaunie C: Local risk estimation for a rare noncontagious disease based on observed frequencies. Note N-36/91/G. Centre de Geostatistique, Fontainebleau, Ecole des Mines de Paris. 1991Google Scholar
- Oliver MA, Lajaunie C, Webster R, Muir KR, Mann JR: Estimating the risk of childhood cancer. Geostatistics Troia 2004. Edited by: Soares A. 1993, Dordrecht, The Netherlands, Kluwer Academic Publishers, 2: 899-910.Google Scholar
- Webster R, Oliver MA, Muir KR, Mann JR: Kriging the local risk of a rare disease from a register of diagnoses. Geographical Analysis. 1994, 26: 168-185.View ArticleGoogle Scholar
- Oliver MA, Webster R, Lajaunie C, Muir KR, Parkes SE, Cameron AH, Stevens MCG, Mann JR: Binomial cokriging for estimating and mapping the risk of childhood cancer. IMA Journal of Mathematics Applied in Medicine and Biology. 1998, 15: 279-297.PubMedView ArticleGoogle Scholar
- Goovaerts P: Simulation-based assessment of a geostatistical approach for estimation and mapping of the risk of cancer. Geostatistics Banff 2004. Edited by: Leuangthong O, Deutsch CV. 2005, Dordrecht, The Netherlands, Kluwer Academic Publishers, 2: 787-796.View ArticleGoogle Scholar
- Monestiez P, Dubroca L, Bonnin E, Durbec JP, Guinet C: Geostatistical modelling of spatial distribution of Balenoptera physalus in the northwestern Mediterranean Sea from sparse count data and heterogeneous observation efforts. Ecological Modelling. 2006, 193: 615-628.View ArticleGoogle Scholar
- Goovaerts P: Geostatistical analysis of disease data: estimation of cancer mortality risk from empirical frequencies using Poisson kriging. International Journal of Health Geographics. 2005, 4: 31-PubMedPubMed CentralView ArticleGoogle Scholar
- Ali M, Goovaerts P, Nazia N, Haq MZ, Yunus M, Emch M: Application of Poisson kriging to the mapping of cholera and dysentery incidence in an endemic area of Bangladesh. International Journal of Health Geographics. 2006, 5: 45-PubMedPubMed CentralView ArticleGoogle Scholar
- Goovaerts P: Geostatistical analysis of disease data: visualization and propagation of spatial uncertainty in cancer mortality risk using Poisson kriging and p-field simulation. International Journal of Health Geographics. 2006, 5: 7-PubMedPubMed CentralView ArticleGoogle Scholar
- Goovaerts P: Geostatistical analysis of disease data: accounting for spatial support and population density in the isopleth mapping of cancer mortality risk using area-to-point Poisson kriging. International Journal of Health Geographics. 2006, 5: 52-PubMedPubMed CentralView ArticleGoogle Scholar
- Monestiez P, Dubroca L, Bonnin E, Durbec JP, Guinet C: Comparison of model based geostatistical methods in ecology: application to fin whale spatial distribution in northwestern Mediterranean Sea. Geostatistics Banff 2004. Edited by: Leuangthong O, Deutsch CV. 2005, Dordrecht, The Netherlands, Kluwer Academic Publishers, 2: 777-786.View ArticleGoogle Scholar
- Gotway CA, Wolfinger RD: Spatial predictions of counts and rates. Statistics in Medicine. 2003, 22: 1415-1432.PubMedView ArticleGoogle Scholar
- Diggle PJ, Tawn JA, Moyeed RA: Model-based geostatistics. Applied Statistics. 1998, 47: 229-350.Google Scholar
- Avruskin GA, Jacquez GM, Meliker JR, Slotnick MJ, Kaufmann AM, Nriagu JO: Visualization and exploratory analysis of epidemiologic data using a novel space time information system. International Journal of Health Geographics. 2004, 3: 26-PubMedPubMed CentralView ArticleGoogle Scholar
- Mockus A: Estimating dependencies from spatial averages. Journal of Computational and Graphical Statistics. 1998, 7 (4): 501-513.Google Scholar
- Kyriakidis P: A geostatistical framework for area-to-point spatial interpolation. Geographical Analysis. 2004, 36 (3): 259-289.View ArticleGoogle Scholar
- Gotway CA, Young LJ: A geostatistical approach to linking geographically-aggregated data from different sources. Journal of Computational and Graphical Statistics. 2007, 16 (1): 115-135.View ArticleGoogle Scholar
- Goovaerts P: Kriging and semivariogram deconvolution in presence of irregular geographical units. Mathematical Geology. 2008, 40 (1): 101-128.PubMedPubMed CentralGoogle Scholar
- Clayton DG, Kaldor J: Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics. 1987, 43: 671-682.PubMedView ArticleGoogle Scholar
- Lawson AB, Browne W, Rodeiro CV: Disease mapping in WinBUGS and MLwiN. 2003, New York: WileyView ArticleGoogle Scholar
- Wakefield JC, Best NG, Waller LA: Bayesian approaches to disease mapping. Spatial epidemiology: Methods and Applications. Edited by: Elliott P, Wakefield JC, Best NG, Briggs D. 2000, New York, Oxford University Press, 105-126.Google Scholar
- Kelsall JE, Wakefield JC: Discussion on Bayesian models for spatially correlated disease and exposure data. Bayesian Statistics 6. Edited by: Bernardo JM, Berger JO, Dawid AP, Smith AFM. 1999, Oxford, UK, Oxford University Press, 151-Google Scholar
- Gilks WR, Richardson S, Spiegelhalter DJ: Markov Chain Monte Carlo in Practice. 1996, Chapman and Hall, LondonGoogle Scholar
- Anselin L: Local indicators of spatial association – LISA. Geographical Analysis. 1995, 27: 93-115.View ArticleGoogle Scholar
- Clayton DG, Bernardinelli L: Bayesian methods for mapping disease risk. Geographical and Environmental Epidemiology: Methods for Small-area Studies. Edited by: Elliot P, Cuzick J, English D, Stern R. 1992, Oxford, UK, Oxford University Press, 205-220.Google Scholar
- Goovaerts P, Meliker JR, Jacquez GM: A comparative analysis of aspatial statistics for detecting racial disparities in cancer mortality rates. International Journal of Health Geographics. 2007, 6: 32-PubMedPubMed CentralView ArticleGoogle Scholar
- Deutsch CV: Direct assessment of local accuracy and precision. Geostatistics Wollongong '96. Edited by: Baafi EY, Schofield NA. 1997, Dordrecht, The Netherlands, Kluwer Academic Publishers, 1: 115-125.Google Scholar
- Goovaerts P: Geostatistical modelling of uncertainty in soil science. Geoderma. 2001, 103: 3-26.View ArticleGoogle Scholar
- Swets JA: Measuring the accuracy of diagnostic systems. Science. 1998, 240: 1285-1293.View ArticleGoogle Scholar
- Lawson AB, Biggeri AB, Boehning D, Lesaffre E, Viel J-F, Clark A, Schlattmann P, Divino F: Disease map models: an empirical evaluation. Statistics in Medicine. 2000, 19: 2217-2241.PubMedView ArticleGoogle Scholar

## Copyright

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