How Smooth Are the Posterior Means?
The results of a Bayesian disease-mapping analysis are typically presented in the form of a map displaying a point estimate (usually the mean or median of the posterior distribution) of the relative risk for each area. To interpret such maps, one needs to understand the extent to which the statistical model is able to smooth the risk estimates to eliminate random noise while at the same time avoiding oversmoothing that might flatten any true variations in risk. To address this issue, we consider the two aspects separately: a) do the Bayesian methods provide adequate smoothing of the background rates, and b) to what extent is the posterior mean estimate different from the background risk in the small number of areas simulated with a true elevated risk?
In all the cases simulated, we found substantial shrinkage of the relative risk estimates for the background rates. This is well illustrated in Figure 1, which displays raw and smooth estimates for all the background areas of Simu 2 and an SF of 1 or 4. Note that when SF = 1, the histogram of the raw standardized mortality or morbidity ratio (SMR) estimates is very dispersed (Figure 1A), with a range of 0–11, and shows a skewed distribution. Clearly, mapping the raw SMRs would present a misleading picture of the risk pattern, whereas any of the three Bayesian models give posterior mean relative risk estimates for the background areas that are well centered on 1 (Figure 1B–D), with just a few areas having estimates outside the 0.9–1.1 range. When the expected counts are higher (SF = 4), the histogram of the raw SMRs is less spread but still substantially overdispersed, whereas those corresponding to the three models are even more concentrated on 1 than when SF = 1 (Figure 1F–H). Thus the false patterns created by the Poisson noise are adequately smoothed out by all the disease-mapping models.
Details of the performance of the BYM model in estimating the relative risk of the high-risk areas are presented in Table 1, with findings for L1-BYM and MIX shown in Tables 2 and 3, respectively. Overall, for the BYM model, a great deal of smoothing of the relative risks is apparent. For the isolated areas in Simu 1, one can see that relative risks of 1.5 in any single area are smoothed away, even in the most favorable case of an area with expected counts of 70 (90% area SF = 10). When the simulated relative risk is 2, the posterior mean risk estimate is above 1.2 only when the expected count is around 50 or more (e.g., 75% area with SF = 10). Relative risks of 3 are smoothed to about half their values when the expected counts are around 10 (e.g., 25% area with SF = 10 or 75% area with SF = 2). Comparison of Simu 2 with Simu 1 (75% area) shows that having a cluster of high-risk areas rather than a single area with elevated risk slightly decreases the amount of smoothing for the same average expected count. Again, this is apparent in the many-cluster situation of Simu 3, where even though the true θ*i are smaller, the relative risk estimates are higher than those for Simu 2.
Overall, the performance of the L1-BYM model (Table 2) is similar to that of the BYM model. However, as expected, the L1-BYM model effects a little less smoothing in cases of large expected counts or high relative risk estimates. For Simu 3 the estimates are nearly identical to those of the BYM model. Thus, simply changing the distributional assumptions in the autoregressive specifications results in only a small modification in the estimates.
The results for the MIX model given in Table 3 show a different pattern than those for the BYM or L1-BYM. For Simu 1 and an elevated relative risk of 1.5, strong smoothing toward 1 is apparent as for BYM. However, for Simu 2, posterior mean relative risks become higher than 1.2 for the largest SF. At the other end of the spectrum, relative risks of 3 are well estimated with posterior means above 2.5 as soon as the expected count is above 10 either for single areas (e.g., 50% area with SF = 4) or for the 1% clustered areas with SF = 2. These results are in accordance with the nature of the MIX model. When there is sufficient evidence in the data to create a group of areas with higher risk, the posterior mean risks for the areas in this group are well estimated and close to the simulated values. Otherwise, all areas are allocated to the background category and smoothed toward 1.
Having many heterogeneous clusters as in Simu 3 does not improve the MIX performance as much as that of BYM. Because of the more diffuse nature of some of the clusters, more areas in the background are randomly included in the group of areas with higher risk. Thus, the MIX model still has a mode close to the true relative risk, but the histogram of the mean posterior risks for all the high-risk areas has a longer left-hand tail than in the Simu 2 scenario (Figure 2).
The difference in performance of the three models is further illustrated in Figure 3, which displays, for the three models, box plots of the posterior mean estimates of the relative risk in the raised-risk areas over the 100 replicates for Simu 2 with true relative risks of 3 and 2. When the true relative risk is 3, the MIX model is clearly performing better than the other two models, whereas for a relative risk of 2 and the lowest SF, the MIX model is the model that produces the most smoothing.