Bayesian Simultaneous Intervals for Small Areas: An Application to Mapping Mortality Rates in U.S. Health Service Areas Public
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It is customary when presenting a choropleth map of rates or counts to present only the estimates (mean or mode) of the parameters of interest. While this technique illustrates spatial variation, it ignores the variation inherent in the estimates. We describe an approach to present variability in choropleth maps by constructing 100(1-alpha)% simultaneous intervals. The result provides three maps (estimate with two bands). We propose two methods to construct simultaneous intervals from the optimal individual highest posterior density (HPD) intervals to ensure joint simultaneous coverage of 100(1-alpha)%. Both methods exhibit the main feature of multiplying the lower bound and dividing the upper bound of the individual HPD intervals by parameters 0<gamma_1,gamma_2<1 to ``stretch' the interval until the simultaneous probability content is 100(1-alpha)%. We employ the Nelder-Mead minimization algorithm to solve a system of nonlinear equations involving the probability content and an optimality condition. Our Single-gamma Method, where gamma_1=gamma_2, optimizes over the probability content only, while the Double-gamma Method includes an optimality condition. For our example, we found that these methods are comparable, appearing that the optimality condition adds very little information. For illustrative purposes we apply our methods to chronic obstructive pulmonary disease (COPD) mortality rates from 1988--92, subset White Males age group 65 and older, for the continental United States consisting of 798 Health Service Areas (HSA).
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