Applied Bayesian Modelling - Free

MODEL ASSESSMENT AND SENSITIVITY 23sented by d(a i ) ˆ 0:5ja i 1j ± see Example 1.4. Specific models, such as those introducinglatent data, lead to particular types of Bayesian residual (Jackman, 2000). Thus, ina binary probit or logit model, underlying the observed binary y are latent continuousvariables z, confined to negative or positive values according as y is 0 or 1. Theestimated residual is then z ^bx i analogously to a Normal errors model.Example 1.3 Lung cancer in London small areas As an example of the possibleinfluence of prior specification on regression coefficients and random effects, considera small area health outcome: female lung cancer deaths y i in the three year period1990±92 in 758 London small areas 19 (electoral wards). If we focus first on regressioneffects, there is overwhelming accumulated evidence that ill health and mortality (especiallylung cancer deaths) are higher in more deprived, lower income areas. Havingallowed for the impact of age differences via indirect standardisation (to provideexpected deaths E i ) variations in this type of mortality are expected to be positivelyrelated to a deprivation score x i , which is in standard form (zero mean, variance 1). Thefollowing model is assumedy i Poi(m i )m i ˆ E i r ilog (r i ) ˆ b 1b 2 x iThe only parameters, b 1 and b 2 , are assigned diffuse but proper N(0,1000) priors. Sincethe sum of observed and expected deaths is the same and x is standardised, one mightexpect b 1 to be near zero. Two sets initial values of adopted b ˆ (0, 0) and b ˆ (0, 0:2)with the latter the mean of a trial (single chain) run. A two chain run then shows earlyconvergence via Gelman-Rubin criteria (at under 250 iterations) and from iterations250±2500 pooled over the chains a 95% credible interval for b 2 of (0.18,0.24) isobtained.However, there may well be information which would provide more informativepriors. Relative risks r i between areas for major causes of death (from chronic disease)reflect, albeit imperfectly, gradients in risk for individuals over attributes such asincome, occupation, health behaviours, household tenure, ethnicity, etc. These gradientstypically show at most five fold variation between social categories except perhapsfor risk behaviours directly implicated in causing a disease. Though area contrasts mayalso be related to environmental influences (usually less strongly) accumulated evidence,including evidence for London wards, suggests that extreme relative contrasts in standardmortality ratios (100 r i ) between areas are unlikely to exceed 10 or 20 (i.e. SMRsranging from 30 to 300, or 20 to 400 at the outside). Simulating with the knowncovariate x i and expectancies E i it is possible to obtain or `elicit' priors consistentwith these prior beliefs. For instance one might consider taking a N(0,1) prior on b 1and a N(0.5,1) prior on b 2 . The latter favours positive values, but still has a large part ofits density over negative values.Values of y i are simulated (see Model 2 in Program 1.3) with these priors; note thatinitial values are by definition generated from the priors, and since this is pure simulationthere is no notion of convergence. Because relative risks tend to be skewed, themedian relative risks (i.e. y i =E i ) from a run of 1000 iterations are considered as19 The first is the City of London (1 ward), then wards are alphabetic within boroughs arranged alphabetically(Barking, Barnet, . . . . ,Westminster). All wards have five near neighbours as defined by the nearest wards interms of crow-fly distance.