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64 HIERARCHICAL MIXTURE MODELSwhether alternatively it straddles 1 (and so is not definitively high risk). The reframedformat describe above is adopted, with expected SIDS deaths defined as E i ˆ RB i , andtaking S i Poi(n i E i ). The relative risks n i ˆ exp (f i ) are modelled via a DPP with aceiling of M ˆ 10 clusters, and with an N(0, 1) baseline prior on f i . The prior on r j is asin (30) with a ˆ 1 preset.A three chain run (with initial values randomly generated from the priors) showsconvergence of the non-empty cluster total J at around iteration 1000 and the summaryis based on iterations 1000±5000. This shows the relative risk clearly highest in Ansoncounty, and with the Swain county risk straddling 1 (Table 2.11). The distribution ofnon-empty clusters shows six clusters as the most frequent (Figure 2.1). One may wishto assess sensitivity of inferences to different baseline densities (e.g. Student t with mean0, variance 1 and low degrees of freedom) or to alternative values of a.Example 2.8 Exponential mixtures for patient length of stay distributions The work ofHarrison and Millard (1991) and McClean and Millard (1993) relates to lengths of stayof patients in hospital, with a particular focus on patient lengths of stay in geriatricdepartments. Lengths of stay of other classes of patients have also attracted discretemixture analysis (especially psychiatric patients), as well as other patient characteristicssuch as age at admission (Welham et al., 2000). Lengths of stay in geriatric departmentsexhibit pronounced skewness, but exponential mixture models with relatively few componentshave been found effective in modelling them. The analysis here shows the utilityof Bayesian sampling estimation in deriving densities for structural or system parameters.Table 2.11DPP prior on SIDS deaths; selected parametersNon-empty clusters Mean St. devn. 1% Median 99%J 5.99 1.55 3 6 9Relative RisksAnson (county 4) 3.07 1.26 1.44 2.68 6.76Halifax (county 42) 1.94 0.48 1.03 1.89 3.48Robeson (county 78) 1.80 0.35 1.07 1.78 2.65Swain (county 87) 1.27 0.54 0.55 1.09 3.010.30.20.10.01 5 10Figure 2.1Number of DPP clusters in SIDS deaths analysis