Data Generation
The spatial structure used throughout the simulations is that of the 532 wards in the county of Yorkshire, England. Wards are administrative areas in the United Kingdom, with a total population of approximately 5,000 on average. We base our expected counts Ei on those calculated by Jarup et al. 2002 for prostate cancer in males 45–64 years of age over the period from 1975 to 1991. We then simulate three spatial patterns of increased risks. For each pattern, we examine three magnitudes for the elevated risks. We also examine how the inference is changed if the expected counts are multiplicatively increased by a scale factor (SF) varying from 2 to 10.
Three spatial patterns for areas of elevated risk were chosen. The choice of patterns was intended to span a spectrum ranging from a scenario with single isolated areas with elevated risks (the hardest test case for any smoothing method) to a scenario with a number of larger clusters of several contiguous areas with elevated risks (a situation with a substantial amount of heterogeneity). In all cases the elevated areas were selected in turn at random from the set of areas with the required expected counts. In the Simu 1 and Simu 3 cases, once an area was selected, a buffer of neighboring areas with background risk (excluded thereafter from the random selection) was placed around it to produce the required pattern of isolated high-risk clusters. The three generated patterns were defined as follows:
Simu 1: five isolated single wards with expected counts ranging from 0.8 to 7.3 corresponding, respectively, to the 10th, 25th, 50th, 75th, and 90th percentiles of the distribution of the expected counts
Simu 2: a group of contiguous wards representing 1% of the total expected counts. In effect, this chosen 1% cluster grouped four wards with fairly comparable expected counts ranging from 3.6 to 7.0, giving an average expected count per ward of 5.4 over the four wards
Simu 3: a situation with high heterogeneity comprising 20 such 1% clusters that are nonoverlapping.
Note that for Simu 3, the twenty 1% clusters each have a total expected count close to 17 but a large disparity in terms of numbers of constitute areas: 10 clusters had 2 or 3 areas, whereas 8 clusters had more than 8 areas, up to a maximum of 18 areas. Correspondingly, the expected counts in each of the wards in the clusters ranged from 0.3 for some wards in the 18-area cluster to 12 for the cluster with 2 areas. Simu 3 thus corresponds to a realistic situation of heterogeneity of risk where both small clusters with high expected counts, for example, typically a populated area, and large clusters each with small expected counts, for example, in rural areas, are present. This high degree of heterogeneity has to be considered when interpreting the results for the Simu 3 case where an average over all the 20 clusters is presented. Note also that contrary to the Simu 2 case, about half the background areas in Simu 3 have a neighbor that belongs to one of the 20 clusters. In each case, apart from the elevated risk areas described above, all other areas are called background areas.
For each spatial pattern in Simu 1 and Simu 2, counts Yi were generated as follows: Counts in all background areas were generated from a Poisson distribution with mean Ei. For all the other areas, an elevated relative risk with magnitude θi > 1 was used and counts were simulated as Poisson variables with mean θiEi . The simulation was repeated for three values of θi (1.5, 2, and 3) and for different SFs that multiply the expected counts Ei for all areas. Thus, results reported, for example, for an area with E = 1.92, θ = 2, and SF = 4, correspond to counts generated from a Poisson with mean 15.36 (2 × 4 × 1.92).
For Simu 3 a slightly different procedure for generating the cases was used to ensure that Σ Yi = Σ Ei (Appendix A). Note that for Simu 1 and Simu 2, the simulation procedure meant only that ΣYi ≈ ΣEi. This corresponds, for instance, to an epidemiologic situation where expected counts Ei are calculated based on an external reference rate. However, Simu 3 uses internal reference rates because otherwise ΣYi would have been much larger than Ei , which could distort the overall risk estimates. The multinomial procedure used in Simu 3 and detailed in Appendix A implies that, in effect, the multiplicative contrast between areas of elevated risk and background areas is still 3, 2, and 1.5, respectively, but the corresponding relative risks in each area (denoted * θ i ) relative to the internal (i.e., study region average) reference rates are now 2.1, 1.65, and 1.35 for the elevated areas and 0.7, 0.82, and 0.9 for the background areas.
To allow for sampling variability, each simulation case was replicated 100 times. The results presented are averaged over these 100 replications. A total of 36 simulation scenarios were investigated, corresponding to three spatial patterns (Simu 1, 2, and 3) × three different magnitudes of elevated risk (θ = 3, 2, and 1.5) × 4 SFs for the expected counts Ei (SF = 1, 2, 4, and 10).