Conditional Autoregressive (CAR) models for disease mapping: Lip cancer in Scotland

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.

[scotland1]We note that the extreme SMRs (Standardised Mortality Ratios) are based on very few cases.

We may smooth the raw SMRs by fitting a random-effects Poisson model allowing for spatial correlation, using the intrinsic conditional autoregressive (CAR) prior proposed by Besag, York and Mollie (1991). For the lip cancer example, the model may be written as:

Oi ~ Poisson(μi)
log μi =   og Ei + α0 + α1xi / 10 + bi

where α0 is an intercept term representing the baseline (log) relative risk of disease across the study region, xi is the covariate "percentage of the population engaged in agriculture, fishing, or forestry" in district i, with associated regression coefficient α1 and bi is an area-specific random effect capturing the residual or unexplained (log) relative risk of disease in area i. We often think of bi as representing the effect of latent (unobserved) risk factors.

To allow for spatial dependence between the random effects bi in nearby areas, we may assume a CAR prior for these terms. Technical details, including parameterisation and a discussion of suitable hyperpriors for the parameters of this model, are given in appendix 1. The car.normal distribution may be used to fit this model. The code for the lip cancer data is given below.

Model
         model {

# Likelihood
for (i in 1 : N) {
O[i] ~ dpois(mu[i])
log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * X[i]/10 + b[i]
# Area-specific relative risk (for maps)
RR[i] <- exp(alpha0 + alpha1 * X[i]/10 + b[i])
}

# CAR prior distribution for random effects:
b[1:N] ~ car.normal(adj[], weights[], num[], tau)
for(k in 1:sumNumNeigh) {
   weights[k] <- 1
}

# Other priors:
alpha0 ~ dflat()
alpha1 ~ dnorm(0.0, 1.0E-5)
tau ~ dgamma(0.5, 0.0005)             # prior on precision
sigma <- sqrt(1 / tau)            # standard deviation
         b.mean <- sum(b[])
}
Data
list(N = 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),
X = 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),
num = c(3, 2, 1, 3, 3, 0, 5, 0, 5, 4,
0, 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,
28, 20, 18,
19, 12, 1,

17, 16, 13, 10, 2,

29, 23, 19, 17, 1,
22, 16, 7, 2,

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 = 234)

Note that the data for the adjacency matrix (variables adj, num and SumNumNeigh) have been generated using the adj matrix option of the Adjacency Tool menu in GeoBUGS. By default, this treats islands as having no neighbours, and so the three areas representing the Orkneys, Shetland and the Outer Hebrides islands in Scotland have zero neighbours. You can edit the adjacency map of Scotland to include these areas as neighbours if you wish. The car.normal distribution sets the value of bi equal to zero for areas i that are islands. Hence the posterior relative risks for the Orkneys, Shetland and the Outer Hebrides in the present example will just depend on the overall baseline rate α0 and the covariate xi. Alternatively, you could specify a convolution prior for the area-specific random effects (Besag, York and Mollie 1991) which partitions the overall random effect for each area into the sum of a spatial component plus a non-spatial component. In this model, the islands will just have a non-spatial term for the random effect. See example on lung cancer in a London Health Authority for details of this model.

Inits for chain 1

list(tau = 1, alpha0 = 0, alpha1 = 0)
   
Inits for chain 2

list(tau = 10.0, alpha0 = 1.0, alpha1 = 1.0)


Note that the initial values for elements 6, 8 and 11 of the vector b are set to NA since these correspond to the three islands (Orkneys, Shetland and the Outer Hebrides). The values of b are set to zero by the car.normal prior for these 3 areas, and so they are not stochastic nodes.

               
               

[scotland2]




[scotland3]

      mean   median   sd   MC_error   val2.5pc   val97.5pc   start   sample   ESS
   alpha0   -0.3006   -0.3014   0.1155   0.002089   -0.5257   -0.07243   1001   20000   3054
   alpha1   0.4474   0.4504   0.1204   0.002487   0.2039   0.6783   1001   20000   2345
   b[1]   1.131   1.134   0.2926   0.002901   0.5571   1.708   1001   20000   10168
   b[2]   1.06   1.063   0.1825   0.002477   0.6985   1.416   1001   20000   5426
   b[3]   1.072   1.084   0.2906   0.002372   0.4627   1.61   1001   20000   15010
   b[4]   0.3115   0.312   0.3019   0.003791   -0.274   0.9157   1001   20000   6340
   b[5]   1.022   1.027   0.2293   0.00201   0.5539   1.459   1001   20000   13025
   b[7]   0.9164   0.9183   0.1817   0.001819   0.5543   1.27   1001   20000   9979
   b[9]   0.6853   0.6848   0.27   0.002458   0.1558   1.221   1001   20000   12067
   b[10]   0.6612   0.6611   0.2103   0.002446   0.247   1.073   1001   20000   7393
   b[12]   0.7886   0.7992   0.3216   0.002875   0.1243   1.386   1001   20000   12516
   b[13]   0.7519   0.7559   0.3371   0.002852   0.07601   1.403   1001   20000   13973
   b[14]   -0.06239   -0.06424   0.3034   0.003381   -0.6611   0.5446   1001   20000   8052
   b[15]   0.6514   0.6556   0.2304   0.001754   0.1792   1.091   1001   20000   17243
   b[16]   0.3588   0.3604   0.2183   0.002008   -0.07919   0.7795   1001   20000   11823
   b[17]   0.566   0.5658   0.2671   0.002434   0.03871   1.093   1001   20000   12041
   b[18]   0.0958   0.09405   0.2562   0.002371   -0.4055   0.6138   1001   20000   11680
   b[19]   0.6615   0.669   0.2212   0.001801   0.2117   1.076   1001   20000   15074
   b[20]   0.1702   0.1734   0.293   0.002399   -0.4173   0.7324   1001   20000   14914
   b[21]   0.2766   0.28   0.2243   0.001725   -0.1779   0.7088   1001   20000   16899
   b[22]   -0.01555   -0.01304   0.1882   0.002336   -0.3927   0.3486   1001   20000   6488
   b[23]   0.0569   0.06053   0.2262   0.001884   -0.4001   0.4825   1001   20000   14422
   b[24]   -0.2216   -0.2222   0.2248   0.002068   -0.6605   0.2275   1001   20000   11819
   b[25]   0.4303   0.4342   0.2162   0.002177   -0.001143   0.8438   1001   20000   9869
   b[26]   0.304   0.3077   0.2348   0.002158   -0.1658   0.7513   1001   20000   11839
   b[27]   -0.09491   -0.09354   0.272   0.002363   -0.6394   0.4345   1001   20000   13252
   b[28]   -2.017E-4   0.002518   0.2414   0.001958   -0.485   0.4637   1001   20000   15192
   b[29]   0.1041   0.1085   0.1587   0.001256   -0.2179   0.4119   1001   20000   15961
   b[30]   -0.2978   -0.2968   0.2176   0.001773   -0.728   0.1279   1001   20000   15057
   b[31]   -0.2113   -0.21   0.24   0.002265   -0.6852   0.2614   1001   20000   11229
   b[32]   -0.3526   -0.3462   0.3271   0.003192   -1.016   0.278   1001   20000   10503
   b[33]   -0.2078   -0.203   0.2604   0.001981   -0.7282   0.2931   1001   20000   17267
   b[34]   -0.2943   -0.2919   0.2048   0.001792   -0.7075   0.1013   1001   20000   13056
   b[35]   -0.1955   -0.191   0.2339   0.001727   -0.6661   0.247   1001   20000   18330
   b[36]   -0.02304   -0.01787   0.3048   0.002711   -0.6334   0.5585   1001   20000   12641
   b[37]   -0.2905   -0.2861   0.2294   0.001778   -0.7577   0.1438   1001   20000   16639
   b[38]   -0.3446   -0.3464   0.248   0.002302   -0.8341   0.1458   1001   20000   11607
   b[39]   -0.3989   -0.3924   0.2746   0.002331   -0.961   0.1169   1001   20000   13880
   b[40]   -0.3523   -0.3487   0.3061   0.002561   -0.96   0.2426   1001   20000   14288
   b[41]   -0.4102   -0.4093   0.2378   0.002624   -0.8862   0.04511   1001   20000   8209
   b[42]   -0.7033   -0.6931   0.224   0.002017   -1.164   -0.2875   1001   20000   12336
   b[43]   -0.4121   -0.4001   0.2891   0.002514   -1.016   0.1262   1001   20000   13228
   b[44]   -0.4885   -0.4811   0.2518   0.002572   -1.003   -0.009489   1001   20000   9582
   b[45]   -0.5749   -0.5683   0.2001   0.002307   -0.99   -0.199   1001   20000   7528
   b[46]   -0.5265   -0.5182   0.2504   0.002362   -1.045   -0.0613   1001   20000   11234
   b[47]   -0.5276   -0.5229   0.2696   0.002726   -1.079   -0.009511   1001   20000   9774
   b[48]   -0.5965   -0.5868   0.2944   0.002984   -1.21   -0.05288   1001   20000   9737
   b[49]   -0.7056   -0.7003   0.1711   0.002617   -1.058   -0.3828   1001   20000   4272
   b[50]   -0.5116   -0.4981   0.295   0.003108   -1.127   0.03139   1001   20000   9010
   b[51]   -0.5575   -0.55   0.3319   0.003221   -1.229   0.07729   1001   20000   10617
   b[52]   -0.5596   -0.5501   0.3313   0.003075   -1.239   0.07035   1001   20000   11605
   b[53]   -0.6854   -0.6727   0.3225   0.00333   -1.358   -0.08382   1001   20000   9378
   b[54]   -0.6607   -0.6504   0.296   0.003587   -1.276   -0.1109   1001   20000   6809
   b[55]   -0.5257   -0.513   0.2628   0.002575   -1.073   -0.04021   1001   20000   10410
   b[56]   -0.2668   -0.2619   0.2997   0.002501   -0.8772   0.3076   1001   20000   14356
   b.mean   4.258E-17   0.0   1.0E-10   7.071E-13   -7.994E-15   8.216E-15   1001   20000   19999
   sigma   0.6664   0.6568   0.1198   0.002071   0.4604   0.9327   1001   20000   3348


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