For spatial point case-control data Bithell 6 introduced a spatial interpolation method based on kernel density estimation. The resulting map is called the "spatial relative risk function". Lawson and Williams 28 and Kellsall and Diggle 20 proposed modifications of this procedure.

Regional data arise from summarising individual information for administrative regions such as census tracts. The basic model for the individual data, i.e. spatial point data, is the spatial Poisson process. In case of a rare disease, aggregation of the data over distinct regions results again in Poisson distributed data. For a more common disease the regional counts may be binomially distributed.

When spatial interpolation of regional data is the objective of the disease mapping study a grid of surface interpolant co-ordinates must be provided. Then a number of techniques can be used to automatically interpolate the data by use of deterministic methods or to predict the values statistically at the grid co-ordinates. Among the possible choices are kernel smoothing, splines, loess and running medians; see 19 for a discussion of these methods. Kernel smoothing has the advantage of preserving the positivity condition implied in rate data. Brillinger 8 used kernel-type smoothing in the context of birth rate data, and Müller, Stadtmüller and Tabnak 33 applied locally weighted least squares adapted for spatial aggregation to AIDS incidence maps for the San Francisco Area.

Another approach for the interpolation of regional data onto a continuous surface is the geostatistical prediction method of kriging. Some implementations of kriging have been proposed to obtain a risk surface 102331343538. Kelsall and Wakefield 21 have proposed a more complex geostatistical approach based on generalised linear modelling (GLIM) of regional data, which is similar to the work by Diggle, Tawn and Moyeed 15. The relative merits of different interpolation approaches haves not so far been systematically investigated [24, p. 131].

The problems with the kriging method for generating isopleth maps of disease occurrence, i.e. the spatial risk function, are: (1) heterogeneous variances in the regional estimates and (2) the potential of negative interpolations. The first problem can be ameliorated by the use of empirical Bayes estimation to smooth the data prior to kriging. Appropriate geostatistical modelling can solve the second problem.

Let the study area be divided into N disjunctive sub regions indexed by i (i = 1,..., N). Define n_i as the size of the 'at-risk' population of the i-th region, and denote the number of cases by m_i. The proportion p_i = m_i/ n_i is the crude estimate for the parameter of interest θ_i.

This by now classical spatial data set on SIDS mortality rates in North Carolina from 1974 to 1984 has been analysed by many researchers. Cressie 12 gives an introduction to the problem, earlier references and results on data modelling, mapping and cluster detection. More recently Kulldorff 22 applied the spatial scan test to detect clusters of disease and Lawson and Clark 27 applied a spatial mixture modelling approach to map the standardized mortality ratio (SMR) for the period 1974 – 1978.

For the years 1974 to 1984 the number of live births ranges from 567 to 52345 over North Carolina's 100 counties. The total number of reported SIDS cases is 1503 out of 753354 live births, which results in an annual mortality rate of approximately 2 per 1000 live births. The mean of the counties boundary files coordinates in longitude and latitude were used here as the geographic coordinates for the regions centres. Figure 1 shows the shrinkage effect by using parallel box plots for the raw rates and the empirically Bayesian smoothed rates under the Poisson model for rare phenomena.

The smoothed rates are more appropriate for disease mapping than the raw rates and hence used here for choropleth mapping in Figure 2. Cut points of the grey scale shading are the 5%, 50% and 95% quantiles of the empirical distribution. These are used to highlight the upper and lower five percent of the distribution of the Bayesian smoothed mortality rates and to distinguish between higher and lower values. The smoothed rates vary from about 1.2 to 3.5 cases per 1000 live births. Visual map perception reveals some potential high and low risk areas in the north and southwest but no striking spatial trend pattern.

Geostatistical prediction, i.e. kriging of the empirical Bayesian smoothed rates, is based on appropriate modelling of the data. Here, the constant mean assumption for the spatial mean surface is chosen and justified by visual inspection of the empirical semivariogram which levels out and reaches a sill. The spatial dependence structure is modelled by an isotropic exponential semivariogram without nugget effect, which is fitted by weighted least squares estimation to the robustly estimated empirical semivariogram; see 12 for technical details. The result of this procedure is summarised in Figure 3. The close fitting of the empirical semivariogram to the model indicates appropriate model choice for the dependence structure as well as for the constant mean surface. See 1 for diagnostic methods and its applications in geostatistical modelling.

Due to the constant mean assumption ordinary kriging is the appropriate spatial prediction method and without nugget effect this leads to direct interpolation of the data at the counties centres and to smoothed values shrunken towards the global mean for the rest of the study area. The resulting isopleth map or risk surface map is given in Figure 4. Now the grey scale shading is almost continuous, i.e. the patchy nature of Figure 2 is replaced by a surface. Additional isolines are drawn for the same cut points as for the grey scale shadings in Figure 2 (i.e. at the 5%, 50% and 95% level of the empirical distribution of the empirical Bayesian estimates) and will be useful to support map interpretation and comparison. The risk surface map may be more useful to identify potential environmental risk factors than the choropleth map.

Cross-validation can be used to explore the predictive performance of the kriging model. The cross-validation residuals (weighted with respect to the spatial dependence structure given by the semivariogram) should be approximately Gaussian distributed. Normal probability plots can be used to disclose grossly model inadequacies [12, p. 498] as well as for outlier identification. Figure 5 shows the normal probability plot of the cross-validation residuals based on the refitted model with the outliers removed.

Aside from some positive and negative extreme values the general appearance of the residual process is Gaussian. The potential outliers are regions with steep gradients in risk and hence part of disease clusters (or their surroundings) which were previously identified 22 and are located in the north and south-west of North Carolina. Of course, disease clusters are of interest and the proposed modelling approach reveals their existence and points to the respective high risk areas. This is in line with the gross aims of exploratory disease mapping.

Echinococcus multilocularis (E.m.) is a tapeworm occurring in the northern hemisphere, including endemic regions in central Europe, most of northern and central Eurasia and parts of North America. In central Europe the red fox is the main definitive host with rodents such as mice or muskrats serving as intermediate hosts 16. The parasite E.m. causes the zoonosis alveolar echinococcosis (A.E.), which has a potentially high fatality rate. Recent studies reflect an alarmingly wide geographic range of the parasite in foxes, with average prevalences varying up to 60% for central Europe. However, the spatial distribution of E.m. in foxes is complex and insufficiently known. There are indications of emerging risk factors for human A.E. such as increasing parasite prevalences in red foxes, growing fox population and progressive spread of foxes to cities.

The federal state of Lower Saxony is part of northern Germany and contains the federal city-state of Bremen as an enclave. During the period from 1991 to 1997, 5365 red foxes were sampled in Lower Saxony and examined for infections with E.m.. The data are given in Table 13.

Figure 6 is a choropleth map of the empirical Bayes smoothed period prevalences from 43 regions in the study area. The cut points of the grey scale shading are again the 5%, 50% and 95% quantiles of the empirical distribution or the smoothed data. The period prevalences range from 3% to 51% with the median at 9%.

The map indicates high period prevalences in the south and north of Lower Saxony. Extraordinarily high prevalences were observed in the southern regions, which indicate the presence of a positive disease cluster that is identified by use of the spatial scan statistic 4.

Figure 7 shows the corresponding isopleth map resulting from universal kriging, with overlaid isolines. The isopleth map gives an impression of gradual changes instead of jumps at the regional borders. Furthermore, the missing region of Bremen in the central north of Lower Saxony has also been supplied with predicted values.

Figure 6 points out the presence of heterogeneity in the spatial mean. Here, universal kriging is based on the assumption that the spatial mean function of the data could be represented by an incomplete quadratic polynomial in the spatial co-ordinates. Other potential explanatory variables are not at hand. Let s = (x, y)' denote an arbitrary location in the study region, where x and y are the co-ordinates to the east and north. By inspection of the variogram cloud 12 based on trend residuals, two observations (Sites 3 and 13) were identified as outliers and removed from the structure analysis. In the second step of an iterated modelling approach then the trend polynomial fitted by ordinary least squares (OLSE) is given by μ(s) = β₀ + β₁y + β₂y², with , , . Instead of OLSE one could use iterated WLSE, but iteration may lead to biased estimates. The residuals of the trend surface fit to the empirical Bayes smoothed regional period prevalences were then used to model a spherical semivariogram by weighted least squares estimation of the robustly estimated empirical semivariogram. Figure 8 shows the empirical semivariogram as well as the fitted model for both, the detrended and trend contaminated data (sill = 0.0039, range = 5.29 and sill = 0.0047, range = 7.83 respectively), to indicate the benefit from the trend model as measured by the 20% decrease of the sill value.

The normal probability plot of the kriging cross-validation residuals in Figure 9 draws attention to the presence of three or four outliers (Observations 3, 8 and 13 and possibly 15). Observation 3 and 15 were previously identified as extremes and excluded from the analysis. Regions 8 and 13 are the nearest neighbours of Region 15. All three are part of a positive spatial cluster 4, a spatial structure that could not be captured by the spatial linear model used for universal kriging. However, the general appearance of the data is Gaussian. This in turn justifies the appropriateness of the modelling and prediction approach based on the empirical Bayesian smoothed regional data.

Geostatistical modelling of the empirical Bayesian smoothed regional prevalences allows the calculation of an error map representing the kriging standard errors. The darker the error map in Figure 10 the larger are the kriging standard errors, which range up from 0 to 0.06.