MODEL 0 is non-spatial only with gamma random effects and no covariate

model
{
  for (i in 1 : N) {
      Y[i]  ~ dpois(mu[i])
      mu[i] <- E[i]*exp(beta0)*theta[i] 
      RR[i] <- exp(beta0)*theta[i]
      theta[i] ~ dgamma(alpha,alpha)
  } 
# 
  alpha ~ dgamma(1,1)
  beta0 ~ dflat()  
# Functions of interest:
  sigma.theta <- sqrt(1/alpha)   # standard deviation of non-spatial
  base <- exp(beta0)
}

DATA 0

list(N = 56,  
Y   = 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))

INIT 0
list(alpha = 1, beta0 = 0, 
theta=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))

OUTPUT:
	 node	 mean	 sd	 MC error	2.5%	median	97.5%	start	sample
	RR[1]	4.07	1.297	0.01877	1.959	3.92	7.001	4000	6001
	RR[2]	4.105	0.6469	0.008642	2.938	4.068	5.48	4000	6001
	RR[3]	3.006	0.858	0.01159	1.607	2.915	4.937	4000	6001
	RR[4]	2.875	0.8995	0.01019	1.391	2.773	4.886	4000	6001
	RR[5]	3.016	0.7406	0.01114	1.754	2.955	4.668	4000	6001
	RR[6]	2.68	0.8865	0.01325	1.227	2.568	4.696	4000	6001
	RR[7]	2.975	0.5666	0.008305	1.994	2.929	4.236	4000	6001
	RR[8]	2.476	0.8492	0.01224	1.082	2.379	4.412	4000	6001
	RR[9]	2.398	0.8863	0.01208	0.9793	2.306	4.445	4000	6001
	RR[10]	2.763	0.5932	0.007403	1.715	2.716	4.053	4000	6001
	RR[11]	2.615	0.6887	0.008123	1.432	2.565	4.119	4000	6001
	RR[12]	2.242	0.8778	0.01361	0.859	2.133	4.297	4000	6001
	RR[13]	2.03	0.9582	0.01481	0.6056	1.872	4.299	4000	6001
	RR[14]	2.143	0.6804	0.007334	1.037	2.065	3.688	4000	6001
	RR[15]	2.083	0.4764	0.005953	1.261	2.041	3.077	4000	6001
	RR[16]	1.839	0.556	0.006439	0.9382	1.783	3.086	4000	6001
	RR[17]	1.626	0.8399	0.01122	0.4277	1.488	3.598	4000	6001
	RR[18]	1.612	0.5461	0.007131	0.7327	1.547	2.85	4000	6001
	RR[19]	1.599	0.4912	0.007916	0.7854	1.546	2.674	4000	6001
	RR[20]	1.555	0.5271	0.007354	0.7136	1.495	2.748	4000	6001
	RR[21]	1.505	0.3596	0.004303	0.8876	1.476	2.302	4000	6001
	RR[22]	1.372	0.2355	0.003404	0.9517	1.356	1.869	4000	6001
	RR[23]	1.28	0.3608	0.004843	0.6784	1.246	2.084	4000	6001
	RR[24]	1.286	0.4385	0.005744	0.5745	1.237	2.251	4000	6001
	RR[25]	1.242	0.2713	0.003548	0.7714	1.219	1.828	4000	6001
	RR[26]	1.228	0.2975	0.004119	0.7142	1.208	1.88	4000	6001
	RR[27]	1.201	0.4039	0.005305	0.556	1.15	2.128	4000	6001
	RR[28]	1.146	0.3335	0.004086	0.5753	1.118	1.878	4000	6001
	RR[29]	1.131	0.2697	0.003386	0.6715	1.11	1.716	4000	6001
	RR[30]	1.112	0.3124	0.004678	0.5936	1.082	1.806	4000	6001
	RR[31]	1.115	0.4328	0.005919	0.4502	1.055	2.102	4000	6001
	RR[32]	1.154	0.5397	0.008106	0.3577	1.067	2.43	4000	6001
	RR[33]	1.064	0.3632	0.004776	0.4821	1.019	1.902	4000	6001
	RR[34]	1.0	0.323	0.003728	0.4689	0.9648	1.725	4000	6001
	RR[35]	0.9489	0.2669	0.003294	0.4941	0.924	1.526	4000	6001
	RR[36]	0.9589	0.2931	0.003769	0.4695	0.9254	1.624	4000	6001
	RR[37]	0.9252	0.2575	0.003131	0.488	0.8995	1.502	4000	6001
	RR[38]	0.9181	0.2942	0.003855	0.446	0.889	1.58	4000	6001
	RR[39]	0.9248	0.335	0.004034	0.3926	0.8853	1.694	4000	6001
	RR[40]	0.8836	0.3698	0.004262	0.3132	0.8298	1.721	4000	6001
	RR[41]	0.5866	0.1686	0.002154	0.3017	0.5714	0.9754	4000	6001
	RR[42]	0.5729	0.1846	0.002617	0.2709	0.5513	0.9862	4000	6001
	RR[43]	0.6791	0.3494	0.004313	0.1686	0.6266	1.509	4000	6001
	RR[44]	0.4904	0.1786	0.002317	0.1994	0.4702	0.8849	4000	6001
	RR[45]	0.4027	0.08775	0.001215	0.2512	0.3959	0.5925	4000	6001
	RR[46]	0.5066	0.2299	0.002959	0.1598	0.4747	1.034	4000	6001
	RR[47]	0.5487	0.2774	0.003792	0.1449	0.505	1.195	4000	6001
	RR[48]	0.4498	0.2039	0.002677	0.1369	0.4188	0.924	4000	6001
	RR[49]	0.3321	0.06051	7.884E-4	0.2261	0.3286	0.4612	4000	6001
	RR[50]	0.3685	0.1334	0.001624	0.1603	0.3522	0.6725	4000	6001
	RR[51]	0.6	0.3539	0.004243	0.1112	0.5327	1.45	4000	6001
	RR[52]	0.5702	0.3425	0.005197	0.1034	0.5017	1.4	4000	6001
	RR[53]	0.4021	0.2446	0.00316	0.07137	0.3546	0.9934	4000	6001
	RR[54]	0.3327	0.2042	0.002271	0.05706	0.2924	0.8143	4000	6001
	RR[55]	0.3259	0.2533	0.003452	0.02491	0.2646	0.9605	4000	6001
	RR[56]	0.5814	0.4538	0.006368	0.04737	0.4723	1.745	4000	6001
	alpha	1.79	0.3985	0.007926	1.129	1.753	2.682	4001	6000
	beta0	0.3567	0.1188	0.005916	0.1315	0.353	0.5966	4000	6001
	
MODEL 7: Spatial exp correlation model + non-spatial + cubic covariate + 
dependent variance priors.
model
{
  for (i in 1 : N) {
      Y[i]  ~ dpois(mu[i])
      X1c[i] <- X[i]-mean(X[1:N])
      X2c[i] <- X1c[i]*X1c[i]
      X3c[i] <- X1c[i]*X1c[i]*X1c[i]
      log(mu[i]) <- log(E[i]) + beta0 + beta1*X1c[i] + beta2*X2c[i] + 
                    beta3*X3c[i] + V[i] + U[i]
      RR[i] <- exp(beta0 + beta1*X1c[i] + beta2*X2c[i]+ 
      	       beta3*X3c[i] + V[i] + U[i]) 
      V[i] ~ dnorm(0,tau.V)
      mean[i] <- 0
  }
 # Multivariate prior distribution for spatial random effects: 
  U[1:N] ~ spatial.exp(mean[], xm[], ym[], tau.U, phi, 1)
  tau.T ~ dgamma(1,0.0260)
 # tau.T ~ dgamma(1,0.1399)
  p ~ dbeta(1,1) # p is the proportion of the variance that is spatial
  sigma.U <- sqrt(p/tau.T)
  sigma.V <- sqrt((1-p)/tau.T)
  tau.V <- 1/(sigma.V*sigma.V)
  tau.U <- 1/(sigma.U*sigma.U)
# 
# This prior is derived by assuming that there is a 5% chance that the 
# correlations die to 0.5 in less that 5km, and a 95% chance that they die
# to 0.5 in less than 100km.
#
  dhalf ~ dlnorm(3.107,0.9106)
# 
# This prior is derived by assuming that there is a 5% chance that the 
# correlations die to 0.5 in less that 5km, and a 95% chance that they die
# to 0.5 in less than 20km.
#
#  dhalf ~ dlnorm(2.303,0.4214)
  phi <- 0.6931/dhalf
  beta0 ~ dflat()  
  beta1 ~ dflat()
  beta2 ~ dflat()
  beta3 ~ dflat()
}

DATA 7
list(N = 56,  
Y   = 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),
X = c(0.16,0.16,0.10,0.24,0.10,0.24,0.10, 0.07, 0.07,0.16, 
              0.07,0.16,0.10,0.24, 0.07,0.16,0.10, 0.07, 0.07,0.10,
              0.07,0.16,0.10, 0.07, 0.01, 0.01, 0.07, 0.07,0.10,0.10,
              0.07,0.24,0.10, 0.07, 0.07, 0,0.10, 0.01,0.16, 0,
              0.01,0.16,0.16, 0, 0.01, 0.07, 0.01, 0.01, 0, 0.01,
              0.01, 0, 0.01, 0.01,0.16,0.10),
xm = c(
162.1894, 385.7761, 293.9555, 377.9338, 220.6786,
340.1739, 324.9915, 442.2445, 194.5176, 367.6924,
112.8916, 247.7566, 289.5922, 227.9563, 342.3574,
351.3505, 280.4916, 341.6081, 249.6855, 359.5902,
348.7138, 388.7655, 180.4228, 295.4908, 333.1159,
312.0605, 290.1701, 359.4153, 291.3727, 303.4219,
257.4402, 264.9711, 336.4464, 258.0319, 227.1801,
234.5294, 218.3428, 279.1010, 235.0805, 254.1736,
250.8301, 287.1202, 292.3773, 288.0333, 320.5682,
257.8758, 276.9737, 281.9644, 267.8444, 342.226,
274.8713, 257.8069, 265.5934, 267.8921, 321.4991,
322.1780),
ym =c(834.7496, 852.3782, 946.0722, 650501,
870.9356, 1015.154, 842.0317, 1168904, 781.3746,
828.219, 903.1592, 924.9536, 842.3052, 561.1628,
713.0808, 792.1617, 801.0356, 628.6406, 825.8545,
610.6554, 760.2982, 812.7655, 699.6693, 635.7658,
701.8189, 691.102, 586.6673, 669.4746, 746.2605,
670.1395, 605.9585, 568.3428, 658.671, 716.452,
598.2521, 668.0481, 641.4785, 670.285, 697.044,
677.589, 657.4675, 680.7535, 699.3761, 665.2905,
671.6064, 631.046, 640.8285, 654.6629, 666.7073,
736.4561, 678.8585, 683.7104, 646.5754, 682.2943,
640.1429, 589.9408))

INIT 7
list(tau.T = 1, p=0.5,beta0 = 0, beta1 = 0, beta2 = 0, beta3 =0, dhalf =1,
V=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),
U=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))
	RR[1]	4.033	1.358	0.01468	1.963	2.186	2.48	3.054	3.84	4.792	5.836	6.547	7.2	10000	40001
	RR[2]	4.003	0.6413	0.004393	2.866	3.022	3.209	3.555	3.964	4.41	4.842	5.127	5.373	10000	40001
	RR[3]	3.357	0.88	0.007384	1.896	2.075	2.304	2.729	3.271	3.887	4.514	4.938	5.333	10000	40001
	RR[4]	3.427	1.046	0.008706	1.732	1.929	2.194	2.672	3.305	4.048	4.821	5.326	5.801	10000	40001
	RR[5]	3.277	0.7481	0.004418	2.0	2.166	2.365	2.741	3.217	3.742	4.269	4.604	4.914	10000	40001
	RR[6]	3.512	1.063	0.006877	1.765	1.98	2.251	2.75	3.399	4.151	4.925	5.437	5.904	10000	40001
	RR[7]	3.092	0.5565	0.003316	2.111	2.251	2.41	2.697	3.056	3.441	3.825	4.065	4.287	10000	40001
	RR[8]	2.707	0.9097	0.01297	1.279	1.442	1.65	2.052	2.59	3.237	3.917	4.372	4.788	10000	40001
	RR[9]	2.558	0.8203	0.006955	1.28	1.428	1.619	1.968	2.448	3.021	3.646	4.085	4.48	10000	40001
	RR[10]	2.77	0.5452	0.003246	1.821	1.949	2.103	2.385	2.727	3.112	3.495	3.74	3.951	10000	40001
	RR[11]	2.931	0.7241	0.00673	1.71	1.87	2.054	2.411	2.87	3.372	3.892	4.222	4.534	10000	40001
	RR[12]	2.68	0.8891	0.00908	1.277	1.44	1.648	2.039	2.561	3.197	3.862	4.298	4.736	10000	40001
	RR[13]	2.626	0.8835	0.005277	1.263	1.421	1.626	2.0	2.507	3.115	3.763	4.23	4.698	10000	40001
	RR[14]	2.046	0.6436	0.005312	1.018	1.139	1.291	1.589	1.964	2.423	2.896	3.224	3.528	10000	40001
	RR[15]	1.848	0.4064	0.003497	1.162	1.248	1.358	1.559	1.812	2.096	2.391	2.576	2.746	10000	40001
	RR[16]	1.816	0.4777	0.003913	1.032	1.126	1.249	1.479	1.766	2.1	2.449	2.679	2.897	10000	40001
	RR[17]	1.947	0.6926	0.00395	0.8953	1.011	1.164	1.456	1.846	2.318	2.863	3.224	3.596	10000	40001
	RR[18]	1.29	0.3754	0.003287	0.694	0.7666	0.8573	1.025	1.241	1.503	1.782	1.976	2.153	10000	40001
	RR[19]	1.952	0.4882	0.003565	1.126	1.229	1.366	1.605	1.907	2.253	2.59	2.813	3.019	10000	40001
	RR[20]	1.405	0.4151	0.003103	0.7407	0.8225	0.9191	1.109	1.355	1.644	1.953	2.162	2.356	10000	40001
	RR[21]	1.529	0.3239	0.002559	0.9695	1.045	1.134	1.301	1.503	1.728	1.956	2.103	2.241	10000	40001
	RR[22]	1.509	0.2405	0.001697	1.074	1.135	1.211	1.342	1.496	1.663	1.824	1.925	2.013	10000	40001
	RR[23]	1.343	0.3294	0.001886	0.7866	0.8602	0.9479	1.109	1.313	1.543	1.778	1.933	2.077	10000	40001
	RR[24]	0.9852	0.2857	0.003036	0.5333	0.5867	0.6544	0.7819	0.9487	1.149	1.362	1.505	1.646	10000	40001
	RR[25]	1.03	0.218	0.001733	0.6632	0.7099	0.7666	0.8747	1.01	1.165	1.319	1.419	1.512	10000	40001
	RR[26]	0.889	0.2152	0.002086	0.5417	0.5832	0.6358	0.7351	0.8638	1.016	1.176	1.279	1.378	10000	40001
	RR[27]	1.071	0.3053	0.00228	0.5757	0.633	0.7094	0.8541	1.036	1.251	1.477	1.623	1.757	10000	40001
	RR[28]	1.204	0.2914	0.001843	0.7117	0.7742	0.8532	0.9975	1.179	1.382	1.591	1.722	1.845	10000	40001
	RR[29]	1.184	0.2478	0.001388	0.7568	0.8128	0.8828	1.01	1.164	1.338	1.512	1.621	1.723	10000	40001
	RR[30]	0.9959	0.2368	0.002526	0.6025	0.6505	0.7124	0.8276	0.9736	1.137	1.309	1.422	1.527	10000	40001
	RR[31]	0.9553	0.2851	0.002207	0.5036	0.5602	0.6261	0.7524	0.9168	1.12	1.333	1.476	1.615	10000	40001
	RR[32]	1.443	0.5101	0.004964	0.6503	0.7365	0.851	1.076	1.378	1.733	2.119	2.374	2.62	10000	40001
	RR[33]	1.031	0.2699	0.002038	0.5822	0.6397	0.7105	0.8397	1.004	1.193	1.391	1.518	1.636	10000	40001
	RR[34]	1.039	0.269	0.002545	0.5893	0.6468	0.716	0.848	1.014	1.203	1.395	1.522	1.638	10000	40001
	RR[35]	0.9711	0.2301	0.001627	0.5803	0.6307	0.6923	0.8083	0.9508	1.113	1.275	1.384	1.481	10000	40001
	RR[36]	0.6079	0.1778	0.001731	0.329	0.3624	0.4029	0.4809	0.5837	0.7097	0.8447	0.9362	1.021	10000	40001
	RR[37]	0.9505	0.2232	0.001418	0.57	0.6197	0.6813	0.7923	0.9306	1.087	1.245	1.345	1.437	10000	40001
	RR[38]	0.5708	0.1563	0.002052	0.332	0.36	0.3945	0.4594	0.5479	0.6561	0.778	0.8592	0.9402	10000	40001
	RR[39]	0.914	0.2644	0.003053	0.4828	0.5357	0.6011	0.7267	0.8857	1.068	1.263	1.395	1.519	10000	40001
	RR[40]	0.4795	0.1539	0.00138	0.2495	0.276	0.3084	0.3719	0.4559	0.5598	0.679	0.7646	0.8479	10000	40001
	RR[41]	0.5137	0.12	6.833E-4	0.3125	0.3379	0.3705	0.4281	0.5021	0.586	0.6727	0.7292	0.7802	10000	40001
	RR[42]	0.6488	0.161	0.002016	0.3728	0.4077	0.4521	0.5346	0.6363	0.7477	0.8603	0.9322	0.9988	10000	40001
	RR[43]	0.7985	0.2583	0.003448	0.3869	0.4358	0.4976	0.615	0.7675	0.9471	1.141	1.266	1.392	10000	40001
	RR[44]	0.3961	0.1054	8.623E-4	0.2253	0.246	0.2724	0.3207	0.3844	0.4574	0.5341	0.586	0.6362	10000	40001
	RR[45]	0.4434	0.08251	7.423E-4	0.2954	0.3163	0.3413	0.3859	0.4385	0.4954	0.5511	0.5868	0.6204	10000	40001
	RR[46]	0.6734	0.1929	0.001987	0.3497	0.3895	0.4414	0.5359	0.6554	0.7908	0.9287	1.018	1.1	10000	40001
	RR[47]	0.4317	0.1361	0.001136	0.2168	0.2424	0.2757	0.3359	0.415	0.5078	0.6083	0.6792	0.7461	10000	40001
	RR[48]	0.4273	0.1198	8.324E-4	0.2283	0.2548	0.2866	0.3432	0.4151	0.4969	0.5839	0.6409	0.6946	10000	40001
	RR[49]	0.3334	0.05428	3.205E-4	0.2363	0.2499	0.2658	0.2953	0.3305	0.3683	0.4044	0.4274	0.4486	10000	40001
	RR[50]	0.5354	0.1375	0.0016	0.2976	0.3288	0.3668	0.4384	0.5244	0.621	0.7167	0.7797	0.8354	10000	40001
	RR[51]	0.4625	0.1433	0.001229	0.2357	0.2662	0.3007	0.3638	0.4436	0.5396	0.646	0.7217	0.797	10000	40001
	RR[52]	0.4113	0.1365	0.00105	0.2014	0.2257	0.2588	0.317	0.3923	0.4843	0.5861	0.658	0.7305	10000	40001
	RR[53]	0.4018	0.1245	9.594E-4	0.2006	0.2259	0.2579	0.3155	0.3872	0.4709	0.562	0.6277	0.6906	10000	40001
	RR[54]	0.4251	0.1259	0.001103	0.216	0.2442	0.2779	0.3373	0.4119	0.499	0.5887	0.6485	0.7088	10000	40001
	RR[55]	0.6505	0.2298	0.003425	0.2833	0.3262	0.3818	0.4874	0.6227	0.782	0.9559	1.069	1.176	10000	40001
	RR[56]	0.9446	0.3532	0.002919	0.4005	0.4622	0.545	0.6942	0.8929	1.14	1.404	1.597	1.774	10000	40001
	beta0	0.7421	0.2673	0.01361	0.2517	0.3317	0.4226	0.567	0.73	0.9051	1.076	1.183	1.293	10000	40001
	beta1	-0.6314	2.673	0.08623	-5.904	-4.964	-4.011	-2.443	-0.6578	1.192	2.792	3.77	4.569	10000	40001
	beta2	-68.82	25.63	0.8337	-118.8	-111.0	-101.5	-86.31	-68.75	-51.31	-36.12	-26.87	-19.16	10000	40001
	beta3	567.1	260.1	9.313	54.28	139.6	235.1	389.4	566.7	744.4	898.6	995.2	1080.0	10000	40001
	dhalf	93.43	56.46	1.295	32.32	37.37	43.8	57.4	78.82	112.1	158.6	199.0	243.1	10000	40001
	p	0.7925	0.1478	0.005081	0.44	0.5092	0.586	0.7051	0.82	0.9092	0.9605	0.9789	0.9872	10000	40001
	phi	0.009636	0.004882	1.029E-4	0.002852	0.003484	0.004371	0.006184	0.008794	0.01207	0.01583	0.01855	0.02144	10000	40001
	sigma.U	0.4888	0.1145	0.003098	0.3004	0.3269	0.3565	0.4108	0.4763	0.5522	0.636	0.6921	0.7453	10000	40001
	sigma.V	0.2297	0.0941	0.00338	0.05879	0.07797	0.1057	0.163	0.228	0.2922	0.3515	0.3882	0.422	10000	40001
	tau.T	3.661	1.346	0.03115	1.585	1.826	2.121	2.707	3.471	4.401	5.442	6.131	6.79	10000	40001
MODEL ICAR_3 is non-spatial and spatial ICAR with cubic covariate and new nghbors
model
{
  for (i in 1 : N) {
      Y[i]  ~ dpois(mu[i])
      X1c[i] <- X[i]-mean(X[1:N])
      X2c[i] <- X1c[i]*X1c[i]
      X3c[i] <- X1c[i]*X1c[i]*X1c[i]
      log(mu[i]) <- log(E[i]) + beta0 + beta1*X1c[i]  + 
                    beta2*X2c[i] + beta3*X3c[i] + V[i] + U[i]
      RR[i] <- exp(beta0 + beta1*X1c[i] + 
                   beta2*X2c[i] + beta3*X3c[i] + V[i] + U[i])  
      V[i] ~ dnorm(0,tau.V)
  }
# ICAR prior distribution for spatial random effects: 
  U[1:N] ~ car.normal(adj[], weights[], num[], tauomega.U)
  for(k in 1:sumNumNeigh) {
      weights[k] <- 1
  }
#
# We simulate for the total variance and then transform to marginal
# spatial and non-spatial, and then transform to omega
#
  tau.T ~ dgamma(1,0.0260)
#  tau.T ~ dgamma(1,0.1399)
  p ~ dbeta(1,1)
  sigma.Z <- sqrt(p/tau.T)
  omega.U <- sigma.Z/sqrt(1.164)
  sigma.V <- sqrt((1-p)/tau.T)
  tau.V <- 1/(sigma.V*sigma.V)
  tauomega.U <- 1/(omega.U*omega.U)
  beta0 ~ dflat()  
  beta1 ~ dflat()
  beta2 ~ dflat()  
  beta3 ~ dflat()
# Parameters of interest
  sd.U <- sd(U[1:N])
  vratio <- sd.U*sd.U/(sd.U*sd.U+sigma.V*sigma.V) 
}
DATA ICAR_1
list(N = 56,  
Y   = 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),
X = c(0.16,0.16,0.10,0.24,0.10,0.24,0.10, 0.07, 0.07,0.16, 
              0.07,0.16,0.10,0.24, 0.07,0.16,0.10, 0.07, 0.07,0.10,
              0.07,0.16,0.10, 0.07, 0.01, 0.01, 0.07, 0.07,0.10,0.10,
              0.07,0.24,0.10, 0.07, 0.07, 0,0.10, 0.01,0.16, 0,
              0.01,0.16,0.16, 0, 0.01, 0.07, 0.01, 0.01, 0, 0.01,
              0.01, 0, 0.01, 0.01,0.16,0.10),
num = c(3, 2, 2, 3, 4, 2, 5, 1, 5, 4, 
1, 2, 3, 3, 2, 6, 6, 6, 5, 3, 
3, 2, 4, 8, 3, 3, 4, 4, 11, 6, 
7, 3, 4, 9, 4, 2, 4, 6, 3, 4, 
5, 5, 4, 5, 4, 6, 6, 4, 9, 2, 
4, 4, 4, 5, 6, 5
),
adj = c(
19, 9, 5, 
10, 7, 
12, 6,
28, 20, 18, 
19, 12, 11, 1, 
3,8,
17, 16, 13, 10, 2, 
6,
29, 23, 19, 17, 1, 
22, 16, 7, 2, 
5,
5, 3, 
19, 17, 7, 
35, 32, 31, 
29, 25, 
29, 22, 21, 17, 10, 7, 
29, 19, 16, 13, 9, 7, 
56, 55, 33, 28, 20, 4, 
17, 13, 9, 5, 1, 
56, 18, 4, 
50, 29, 16, 
16, 10, 
39, 34, 29, 9, 
56, 55, 48, 47, 44, 31, 30, 27, 
29, 26, 15, 
43, 29, 25, 
56, 32, 31, 24, 
45, 33, 18, 4, 
50, 43, 34, 26, 25, 23, 21, 17, 16, 15, 9, 
55, 45, 44, 42, 38, 24, 
47, 46, 35, 32, 27, 24, 14, 
31, 27, 14, 
55, 45, 28, 18, 
54, 52, 51, 43, 42, 40, 39, 29, 23, 
46, 37, 31, 14, 
41, 37, 
46, 41, 36, 35, 
54, 51, 49, 44, 42, 30, 
40, 34, 23, 
52, 49, 39, 34, 
53, 49, 46, 37, 36, 
51, 43, 38, 34, 30, 
42, 34, 29, 26, 
49, 48, 38, 30, 24, 
55, 33, 30, 28, 
53, 47, 41, 37, 35, 31, 
53, 49, 48, 46, 31, 24, 
49, 47, 44, 24, 
54, 53, 52, 48, 47, 44, 41, 40, 38, 
29, 21, 
54, 42, 38, 34, 
54, 49, 40, 34, 
49, 47, 46, 41, 
52, 51, 49, 38, 34, 
56, 45, 33, 30, 24, 18, 
55, 27, 24, 20, 18
),
sumNumNeigh = 240))

INIT ICAR_3
list(tau.T = 1, p=0.5, beta0 = 0, beta1 = 0, beta2 = 0, beta3 = 0,
V=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),
U=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))
RR[1]	3.506	1.152	0.01419	1.774	1.961	2.209	2.675	3.33	4.137	5.032	5.646	6.247	10000	40001
	RR[2]	4.088	0.6661	0.004441	2.905	3.062	3.261	3.619	4.048	4.51	4.96	5.247	5.504	10000	40001
	RR[3]	3.262	0.8793	0.005272	1.813	1.996	2.219	2.632	3.167	3.797	4.434	4.849	5.231	10000	40001
	RR[4]	2.78	0.8753	0.00746	1.386	1.549	1.761	2.149	2.675	3.291	3.941	4.376	4.776	10000	40001
	RR[5]	3.224	0.7053	0.004398	2.034	2.188	2.378	2.721	3.158	3.647	4.159	4.49	4.794	10000	40001
	RR[6]	4.098	1.172	0.007563	2.166	2.408	2.699	3.251	3.981	4.8	5.654	6.221	6.741	10000	40001
	RR[7]	3.024	0.5345	0.004128	2.082	2.213	2.371	2.65	2.988	3.357	3.726	3.962	4.176	10000	40001
	RR[8]	2.809	0.9397	0.005401	1.321	1.497	1.713	2.133	2.694	3.352	4.065	4.529	4.973	10000	40001
	RR[9]	2.329	0.6642	0.007161	1.29	1.411	1.57	1.86	2.239	2.7	3.204	3.551	3.886	10000	40001
	RR[10]	2.719	0.545	0.003182	1.788	1.909	2.059	2.335	2.67	3.05	3.446	3.689	3.915	10000	40001
	RR[11]	2.984	0.7489	0.004161	1.726	1.882	2.08	2.452	2.914	3.439	3.979	4.327	4.635	10000	40001
	RR[12]	2.735	0.8931	0.008939	1.324	1.49	1.7	2.099	2.617	3.246	3.92	4.379	4.794	10000	40001
	RR[13]	2.543	0.8346	0.005384	1.255	1.414	1.597	1.957	2.421	2.997	3.642	4.081	4.515	10000	40001
	RR[14]	2.001	0.6157	0.00429	1.019	1.134	1.277	1.559	1.926	2.357	2.821	3.134	3.424	10000	40001
	RR[15]	2.038	0.4477	0.003017	1.272	1.371	1.495	1.721	2.0	2.313	2.627	2.832	3.018	10000	40001
	RR[16]	1.839	0.4395	0.004078	1.102	1.195	1.314	1.53	1.794	2.103	2.418	2.62	2.822	10000	40001
	RR[17]	2.151	0.6171	0.004239	1.154	1.286	1.442	1.722	2.077	2.497	2.95	3.265	3.567	10000	40001
	RR[18]	1.243	0.3403	0.003145	0.7023	0.7679	0.8505	1.001	1.199	1.434	1.691	1.863	2.037	10000	40001
	RR[19]	2.117	0.4879	0.00588	1.271	1.386	1.522	1.777	2.08	2.419	2.754	2.981	3.189	10000	40001
	RR[20]	1.317	0.3869	0.00287	0.6998	0.7723	0.8667	1.042	1.27	1.542	1.824	2.019	2.212	10000	40001
	RR[21]	1.512	0.3227	0.002539	0.9553	1.031	1.122	1.286	1.484	1.713	1.938	2.084	2.218	10000	40001
	RR[22]	1.475	0.2435	0.001374	1.035	1.1	1.174	1.305	1.46	1.629	1.795	1.898	1.99	10000	40001
	RR[23]	1.314	0.2981	0.001968	0.8049	0.8723	0.9563	1.103	1.287	1.499	1.706	1.844	1.971	10000	40001
	RR[24]	1.01	0.2611	0.002802	0.5895	0.6425	0.7083	0.8276	0.9786	1.157	1.352	1.489	1.609	10000	40001
	RR[25]	1.102	0.2331	0.001485	0.7055	0.7576	0.8197	0.9364	1.082	1.246	1.409	1.512	1.617	10000	40001
	RR[26]	1.011	0.2372	0.001693	0.617	0.6663	0.7287	0.8429	0.9876	1.153	1.325	1.44	1.543	10000	40001
	RR[27]	1.02	0.2791	0.002166	0.5655	0.6244	0.6925	0.8207	0.9889	1.182	1.387	1.523	1.652	10000	40001
	RR[28]	1.093	0.2642	0.001966	0.6503	0.7094	0.7799	0.906	1.068	1.251	1.442	1.567	1.687	10000	40001
	RR[29]	1.326	0.2421	0.002386	0.8886	0.9512	1.025	1.158	1.312	1.481	1.642	1.745	1.834	10000	40001
	RR[30]	0.9866	0.2286	0.002022	0.6029	0.6519	0.7127	0.8239	0.9649	1.124	1.287	1.395	1.494	10000	40001
	RR[31]	0.9489	0.255	0.002245	0.5373	0.5878	0.6527	0.7693	0.9182	1.095	1.282	1.412	1.53	10000	40001
	RR[32]	1.549	0.5211	0.00339	0.7191	0.8214	0.9395	1.178	1.485	1.848	2.244	2.497	2.743	10000	40001
	RR[33]	0.9907	0.2619	0.001986	0.5614	0.6129	0.6809	0.8064	0.9627	1.145	1.337	1.46	1.582	10000	40001
	RR[34]	1.03	0.2417	0.002952	0.6188	0.6751	0.7421	0.8595	1.01	1.176	1.349	1.463	1.568	10000	40001
	RR[35]	0.9081	0.2125	0.00146	0.5473	0.595	0.6516	0.7567	0.8885	1.037	1.189	1.288	1.377	10000	40001
	RR[36]	0.6083	0.1944	0.00164	0.3103	0.3421	0.3859	0.4689	0.5815	0.718	0.8645	0.9655	1.064	10000	40001
	RR[37]	0.9182	0.2162	0.001637	0.5515	0.5998	0.6574	0.7645	0.8988	1.05	1.206	1.305	1.397	10000	40001
	RR[38]	0.5812	0.1598	0.001715	0.3326	0.3631	0.3998	0.4682	0.5584	0.6702	0.792	0.8734	0.9528	10000	40001
	RR[39]	0.9244	0.2668	0.002808	0.4879	0.5411	0.6085	0.7354	0.8942	1.083	1.277	1.407	1.528	10000	40001
	RR[40]	0.4784	0.1604	0.001444	0.241	0.2673	0.3013	0.3661	0.453	0.5613	0.688	0.7754	0.864	10000	40001
	RR[41]	0.5054	0.1201	7.311E-4	0.3046	0.3312	0.3623	0.4199	0.4935	0.5781	0.6635	0.7194	0.7733	10000	40001
	RR[42]	0.684	0.1676	0.002055	0.398	0.4345	0.4798	0.5656	0.6694	0.7867	0.9043	0.9822	1.052	10000	40001
	RR[43]	0.9068	0.2811	0.003401	0.4509	0.5056	0.5752	0.7058	0.8761	1.072	1.274	1.415	1.55	10000	40001
	RR[44]	0.3883	0.1055	8.296E-4	0.2175	0.2383	0.2637	0.3131	0.3768	0.4501	0.5268	0.5787	0.6266	10000	40001
	RR[45]	0.4139	0.07946	5.415E-4	0.2739	0.2931	0.317	0.3576	0.4087	0.4642	0.5181	0.5522	0.5841	10000	40001
	RR[46]	0.7246	0.1919	0.002073	0.3943	0.4382	0.493	0.5891	0.7092	0.8418	0.9764	1.063	1.146	10000	40001
	RR[47]	0.4386	0.1282	8.383E-4	0.2315	0.2586	0.2914	0.3496	0.4229	0.5098	0.6042	0.6675	0.7323	10000	40001
	RR[48]	0.4318	0.1257	7.434E-4	0.2275	0.2521	0.2843	0.3424	0.4179	0.5058	0.5976	0.6599	0.7114	10000	40001
	RR[49]	0.3341	0.0544	3.826E-4	0.237	0.25	0.2667	0.2959	0.3312	0.369	0.4055	0.4281	0.4485	10000	40001
	RR[50]	0.4755	0.13	8.952E-4	0.256	0.2854	0.3187	0.3819	0.4638	0.5542	0.6478	0.7068	0.7619	10000	40001
	RR[51]	0.4778	0.1601	0.001066	0.2303	0.2589	0.2968	0.3664	0.4557	0.5646	0.6851	0.768	0.8567	10000	40001
	RR[52]	0.3889	0.1335	0.00103	0.1854	0.2099	0.2397	0.2947	0.3695	0.4602	0.5607	0.6355	0.7077	10000	40001
	RR[53]	0.405	0.1281	9.061E-4	0.1997	0.2243	0.2569	0.3155	0.3894	0.4769	0.5718	0.638	0.7031	10000	40001
	RR[54]	0.426	0.1263	9.858E-4	0.2175	0.244	0.2771	0.3373	0.4124	0.4997	0.591	0.6542	0.7129	10000	40001
	RR[55]	0.6738	0.2188	0.003715	0.3162	0.3614	0.4157	0.5202	0.6488	0.801	0.963	1.071	1.168	10000	40001
	RR[56]	0.9052	0.2879	0.002646	0.4403	0.4989	0.5709	0.7018	0.872	1.069	1.278	1.425	1.57	10000	40001
	beta0	0.3139	0.1089	0.00224	0.09592	0.1338	0.1754	0.2422	0.3149	0.3857	0.4515	0.4911	0.5243	10000	40001
	beta1	-0.7491	2.625	0.08686	-5.923	-5.064	-4.103	-2.522	-0.7299	1.012	2.596	3.515	4.299	10000	40001
	beta2	-73.4	23.18	0.7162	-119.4	-111.6	-103.2	-88.77	-73.28	-58.07	-43.97	-35.73	-27.96	10000	40001
	beta3	677.8	245.9	8.604	201.5	271.7	361.5	513.3	677.8	843.2	993.0	1084.0	1166.0	10000	40001
	omega.U	0.4799	0.1046	0.002462	0.2964	0.3216	0.3521	0.4069	0.4722	0.5444	0.6168	0.664	0.7066	10000	40001
	p	0.8563	0.128	0.00456	0.5223	0.5918	0.6728	0.7966	0.8934	0.9541	0.9816	0.9903	0.9948	10000	40001
	sd.U	0.5363	0.07353	0.001788	0.387	0.4147	0.4434	0.4886	0.537	0.5851	0.629	0.655	0.6799	10000	40001
	sigma.V	0.187	0.08939	0.003448	0.04022	0.05529	0.07438	0.1202	0.1799	0.2443	0.3073	0.3449	0.38	10000	40001
	sigma.Z	0.5178	0.1129	0.002657	0.3198	0.347	0.3799	0.439	0.5095	0.5873	0.6655	0.7163	0.7623	10000	40001
	tau.T	3.534	1.339	0.02608	1.597	1.797	2.06	2.587	3.315	4.218	5.285	6.035	6.791	10000	40001
	vratio	0.8716	0.11	0.003894	0.5862	0.6518	0.7203	0.8193	0.9004	0.9552	0.9822	0.9903	0.995	10000	40001