Applied Bayesian Modelling - Free

ENSEMBLE ESTIMATES: POOLING OVER SIMILAR UNITS 45X{(O i ^LE i ) 2 =^LE i }iThe overall mean relative risk in this case is expected to be approximately 368/361.2,and a posterior mean ^L ˆ 1:019 is accordingly obtained. The chi square statisticaverages 195, with median 193.8, and shows clear excess dispersion. The above momentestimator (2.18) for regional variability in hepatitis rates, ^t 2 , has mean 0.594.A fixed effects model might be adopted to allow for such variations. Here theparameters l i are drawn independently of each other (typically from flat gamma priors)without reference to an overall density. In practice, this leads to posterior estimates veryclose to the corresponding maximum likelihood estimate of the relative incidence ratefor the ith region. These are obtained simply asR i ˆ O i =E iAlternatively, a hierarchical model may be adopted involving a Gamma prior G(a,b) forheterogeneous relative risks l i , with the parameters a and b themselves assigned flatprior densities confined to positive values (e.g. Gamma, exponential). So withl i G(a, b) andwhere J 1 , J 2 , K 1 and K 2 are known, thena G(J 1 , J 2 ), b G(K 1 , K 2 )O i Poi(l i E i )Here take J i ˆ K i ˆ 0:001 for i ˆ 1, 2. Running three chains for 20 000 iterations,convergence is apparent early (at under 1000 iterations) in terms of Gelman±Rubinstatistics (Brooks and Gelman, 1998). While there is a some sampling autocorrelation inthe parameters a and b (around 0.20 at lag 10 for both), the posterior summaries onthese parameters are altered little by sub-sampling every tenth iterate, or by extendingthe sampling a further 10 000 iterations.In terms of fit and estimates with this model, the posterior mean of the chi squarestatistic comparing O i and m i ˆ l i E i is now 23, so extra-variation in relation to availabledegrees of freedom is accounted for. Given that the l i 's are smoothed incidenceratios centred around 1, it would be anticipated that E(l) 1. Accordingly, posteriorestimates of a and b are found that are approximately equal, with a ˆ 2:06 and b ˆ 2:1;hence the variance of the l i 's is estimated at 0.574 (posterior mean of var(l)) and 0.494(posterior median). Comparison (Table 2.1) of the unsmoothed incidence ratios, R i , andthe l i , shows smoothing up towards the mean greatest for regions 16, 17 and 19, eachhaving the smallest total ( just two) of observed cases. Smoothing is slightly less for area23, also with two cases, but higher expected cases (based on a larger population at riskthan in areas 16, 17 and 19), and so more evidence for a low `true' incidence rate.Suppose we wish to assess whether the hierarchical model improves over thehomogenous Poisson model. On fitting the latter an average deviance of 178.2 isobtained or a DIC of 179.2; following Spiegelhalter et al. (2002) the AIC is obtainedas either (a) the deviance at the posterior mean D(u) plus 2p, or (b) the mean devianceplus p. Comparing D and D(u) under the hierarchical model suggests an effectivenumber of parameters of 18.6, since the average deviance is 119.7, but the deviance atthe posterior mean (defined in this case by the posterior averages of the l i 's) is 101.1.The DIC under the gamma mixture model is 138.3, a clear gain in fit over thehomogenous Poisson model.