PMC:1247195 / 36620-42690 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/1247195","sourcedb":"PMC","sourceid":"1247195","source_url":"https://www.ncbi.nlm.nih.gov/pmc/1247195","text":"Discussion\nThis comprehensive simulation study highlights some important points to be considered in interpreting any disease-mapping exercise based on hierarchical Bayesian procedures. First, the necessary control of false positives is indeed achieved using any of the models described. However, this is accompanied by a strong smoothing effect that renders the detection of localized increases in risk nearly impossible if these are not based on large (3-fold or more) excess risks or, in the case of more moderate (2-fold) excess risks, substantial expected counts of approximately 50 or more. Thus, in any study it is important to report the range of expected counts across the map and to calibrate any conclusions regarding the relative risks with respect to these expected counts.\nIn general Bayes procedures offer a tradeoff between bias and variance reduction of the estimates. Particularly in cases where the sample size is small, they produce a set of point estimates that have good properties in terms of minimizing squared error loss (Carlin and Louis 2000). This variance reduction is attained through borrowing information resulting from the adopted hierarchical structure, leading to Bayes point estimates shrunk toward a value related to the distribution of all the units included in the hierarchical structure. The effect of shrinkage is thus dependent on the prior structure that has been assumed and conditional on the latter being close to the true model in some sense. Consequently, different prior structures will lead to different shrinkage. Note that the desirable properties of the estimates thus obtained will depend on the ultimate goal of the estimation exercise. If producing a set of point estimates of the relative risk is the aim, then posterior means of the relative risk are best in squared error loss terms. However, if the goal is to estimate the histogram or the ranks of the area relative risks, different loss function should be considered. The desirability and difficulty of simultaneously achieving these triple goals has been discussed by Shen and Louis (1998) and has been illustrated in spatial case studies by Conlon and Louis (1999) and Stern and Cressie (1999). In our study, we focus on the goal of estimating the overall spatial pattern of risk, which involves producing and interpreting a set of point estimates that will not only give a good indication of the presence of heterogeneity in the relative risks but also highlight where on the map this heterogeneity arises and whether this is linked to isolated high- and/or low-risk areas or to more general spatial aggregation of areas of similar high or low risk. Inference about the latter will depend on the sensitivity and specificity of the posterior risk estimates, as discussed in this article. If the goal is purely the testing of heterogeneity, other methods could be used, such as the Potthoff-Whittinghill test or scan statistics [see Wakefield et al. (2000) for review] that test for particular prespecified patterns of overdispersion. Conversely, if the aim is a local study around a point source, then again, the disease-mapping framework is not appropriate, and focused models that make use of the additional information about the location of the putative cluster of high risk are required (Morris and Wakefield 2000).\nWe have shown that besides reporting and mapping the mean posterior relative risk, the whole posterior distribution can be usefully exploited to try to detect true raised-risk areas. For the BYM model, decision rules based on computing the probability that the relative risk is above 1 with a cutoff between 70 and 80% gives a specific rule. With this type of rule an average expected count of 20 in each of the raised-risk areas leads to a 50% chance of detecting a true relative risk of 1.5, but at least a 75% chance if the true relative risk is 2. For the same scenarios, the posterior mean relative risks are 1.05 and 1.23, respectively, showing that the posterior probabilities rather than the mean posterior relative risks are crucial for interpreting results from the BYM model. On the other hand, 3-fold increases in the relative risk are detected almost certainly with average expected counts of only 5 per area, although the mean of the posterior distribution is typically smoothed to about half the true excess. Note that the performance of the BYM model does improve when the risk is raised in a small group of contiguous areas with similar expected counts rather than in a single area because of the way spatial correlation is taken into account in these models.\nWe found no notable difference in performance between the BYM model, which uses a Gaussian distribution, and the L1 BYM version, which uses a heavier-tailed, double-exponential distribution. This finding is in agreement with that of an earlier simulation study (Best et al. 1999) that compared these two models. However, there were some clear differences between the BYM models and the spatial allocation model MIX. The performance of the latter model is characterized by an all-or-none feature in the sense that it tends to allocate the true raised-risk areas to either an elevated risk group or to a background group, depending on how much uncertainty is present in the data. If the information from the data is sufficient (i.e., moderate-size expected counts and/or high true excess risks) the MIX model is able to separate the raised-risk and background areas quite well, producing considerably less smoothing of the raised-risk estimates than BYM. When the information in the data is sparse, uncertainty in the groupings leads to more smoothing than the BYM. This type of dichotomy makes any decision rule exploiting the posterior distribution of the relative risks hard to calibrate and less useful than for the BYM model. The MIX model is best used for providing estimates of the underlying magnitude of the relative risks if those are clearly raised rather than as a tool for detecting the presence of areas with excess risk in a decision rule context.","divisions":[{"label":"title","span":{"begin":0,"end":10}},{"label":"p","span":{"begin":11,"end":785}},{"label":"p","span":{"begin":786,"end":3332}},{"label":"p","span":{"begin":3333,"end":4609}}],"tracks":[]}