Lips: spatial smoothing of cancer rates The rates of lip cancer in 56 counties in Scotland have been analysed by Clayton and Kaldor (1987) and Breslow and Clayton (1993). The form of the data includes the observed and expected cases (expected numbers based on the population and its age and sex distribution in the county), a covariate measuring the percentage of the population engaged in agriculture, fishing, or forestry, and the "position'' of each county expressed as a list of adjacent counties. County Observed Expected x SMR Adjacent cases cases (% in agric.) counties _________________________________________________________________ 1 9 1.4 16 652.2 5,9,11,19 2 39 8.7 16 450.3 7,10 ... ... ... ... ... ... 56 0 1.8 10 0.0 18,24,30,33,45,55 We note that the extreme SMRs (Standardised Mortality Ratios) are based on very few cases. Breslow and Clayton initially consider a random-effects Poisson model allowing for over-dispersion, where Oi,Ei are the observed and expected cancer incidence in the ith county. model { for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + beta0 + beta1*aff[i]/10 + phi[i] SMRhat[i] <- 100 * mu[i] / E[i] SMRraw[i] <- 100* O[i] / E[i] } phi[1:regions] ~ car.normal(adj[], weights[], num[], tau.c) # beta0 ~ dnorm(0.0, 1.0E-5) # vague prior on grand intercept beta0 ~ dflat() beta1 ~ dnorm(0.0, 1.0E-5) # vague prior on covariate effect tau.c ~ dgamma(1.0E-1,1.0E-1) # (1999, Bayesian Statistics 6) sd.c <- sd(phi[]) # marginal SD of clustering (spatial) effects } Data list(regions = 56, O = c( 9, 39, 11, 9, 15, 8, 26, 7, 6, 20, 13, 5, 3, 8, 17, 9, 2, 7, 9, 7, 16, 31, 11, 7, 19, 15, 7, 10, 16, 11, 5, 3, 7, 8, 11, 9, 11, 8, 6, 4, 10, 8, 2, 6, 19, 3, 2, 3, 28, 6, 1, 1, 1, 1, 0, 0), E = c( 1.4, 8.7, 3.0, 2.5, 4.3, 2.4, 8.1, 2.3, 2.0, 6.6, 4.4, 1.8, 1.1, 3.3, 7.8, 4.6, 1.1, 4.2, 5.5, 4.4, 10.5,22.7, 8.8, 5.6,15.5,12.5, 6.0, 9.0,14.4,10.2, 4.8, 2.9, 7.0, 8.5,12.3,10.1,12.7, 9.4, 7.2, 5.3, 18.8,15.8, 4.3,14.6,50.7, 8.2, 5.6, 9.3,88.7,19.6, 3.4, 3.6, 5.7, 7.0, 4.2, 1.8), aff = c(16,16,10,24,10,24,10, 7, 7,16, 7,16,10,24, 7,16,10, 7, 7,10, 7,16,10, 7, 1, 1, 7, 7,10,10, 7,24,10, 7, 7, 0,10, 1,16, 0, 1,16,16, 0, 1, 7, 1, 1, 0, 1, 1, 0, 1, 1,16,10), adj = c( 5, 9,11,19, 7,10, 6,12, 18,20,28, 1,11,12,13,19, 3, 8, 2,10,13,16,17, 6, 1,11,17,19,23,29, 2, 7,16,22, 1, 5, 9,12, 3, 5,11, 5, 7,17,19, 31,32,35, 25,29,50, 7,10,17,21,22,29, 7, 9,13,16,19,29, 4,20,28,33,55,56, 1, 5, 9,13,17, 4,18,55, 16,29,50, 10,16, 9,29,34,36,37,39, 27,30,31,44,47,48,55,56, 15,26,29, 25,29,42,43, 24,31,32,55, 4,18,33,45, 9,15,16,17,21,23,25,26,34,43,50, 24,38,42,44,45,56, 14,24,27,32,35,46,47, 14,27,31,35, 18,28,45,56, 23,29,39,40,42,43,51,52,54, 14,31,32,37,46, 23,37,39,41, 23,35,36,41,46, 30,42,44,49,51,54, 23,34,36,40,41, 34,39,41,49,52, 36,37,39,40,46,49,53, 26,30,34,38,43,51, 26,29,34,42, 24,30,38,48,49, 28,30,33,56, 31,35,37,41,47,53, 24,31,46,48,49,53, 24,44,47,49, 38,40,41,44,47,48,52,53,54, 15,21,29, 34,38,42,54, 34,40,49,54, 41,46,47,49, 34,38,49,51,52, 18,20,24,27,56, 18,24,30,33,45,55), weights = c( 1, 1,1,1, 1,1, 1,1, 1,1,1, 1,1,1,1,1, 1, 1, 1,1,1,1,1, 1, 1,1,1,1,1,1, 1, 1,1,1, 1, 1, 1,1, 1, 1,1, 1, 1,1,1, 1,1,1, 1,1,1, 1,1,1,1,1,1, 1, 1,1,1,1,1, 1,1,1,1,1,1, 1, 1, 1,1,1, 1,1,1, 1,1,1, 1,1, 1,1,1,1,1,1, 1,1,1,1,1,1,1,1, 1,1,1, 1,1,1,1, 1,1,1,1, 1,1,1,1, 1,1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1, 1,1,1,1,1,1,1, 1,1,1,1, 1,1,1,1, 1,1,1,1,1,1,1,1,1, 1,1,1,1,1, 1,1,1,1, 1,1,1,1,1, 1,1,1,1,1,1, 1,1,1,1,1, 1,1,1,1,1, 1,1,1,1,1,1,1, 1,1,1,1,1,1, 1,1,1,1, 1,1,1,1,1, 1,1,1,1, 1,1,1,1,1,1, 1,1,1,1,1,1, 1,1,1,1, 1,1,1,1,1,1,1,1,1, 1,1,1, 1,1,1,1, 1,1,1,1, 1,1,1,1, 1,1,1,1,1, 1,1,1,1,1, 1,1,1,1,1,1), num = c(4, 2, 2, 3, 5, 2, 5, 1, 6, 4, 4, 3, 4, 3, 3, 6, 6, 6 ,5, 3, 3, 2, 6, 8, 3, 4, 4, 4,11, 6, 7, 4, 4, 9, 5, 4, 5, 6, 5, 5, 7, 6, 4, 5, 4, 6, 6, 4, 9, 3, 4, 4, 4, 5, 5, 6) ) Inits list(tau.c = 1, beta0 = 0, beta1 = 0, phi=c(0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0))