SED navigation bar go to SED home page go to SED projects page go to NIST home page SED Home Page SED Contacts SED Projects SED Products and Publications Search SED Pages

Bayesian Analysis of the SRM 1946 chlorinated data
(Blaza Toman)
Introduction The SRM 1946 chlorinated data, shown in Table 1, consists of means and standard deviations from multiple methods for 62 different compounds. It is of interest to calculate a consensus mean and a measure of uncertainty for each compound. The classical method of analysis is using the DATAPLOT consensus mean procedure. A Bayesian hierarchical model provides an alternative solution to this problem.
The Model For each compound i, there are several average values and sample standard deviations . We assume that
  • Ybar(ij) ~ N(delta(ij),sigma(ij)/n(ij))
  • s**2(ij) ~ Gamma((n(ij) - 1)/2,2*sigma(ij)/(n(ij) - 1)
We further assume that
  • delta(ij) ~ N(mu(i),tau
  • mu(i) ~ N(0,10000)
  • tau ~ Gamma(10000,10000)
  • sigma(ij) ~ Gamma(10000,10000)
This model assumes that the averages for each compound and method are averages of a random sample from a Normal distribution with a particular mean and variance . The means are themselves random variables, distributed according to common distributions. The mean parameters of these distributions are the consensus means. The model further assumes that the sample standard deviations represent a random sample from a Gamma distribution. The parameter of this distribution, , is itself a random variable.
The BUGS Program The above model can be applied to the data given in Table 1 and the consensus means can be estimated using statistical software called BUGS. This software performs Markov Chain Monte Carlo simulation which results in a random sample of draws from the posterior distributions of the . This simulated sample is then used to give an estimate of the consensus mean and a measure of the posterior precision. The code for the BUGS program is as follows: 
model pcb;
const
k=6;
var
theta[k],sigma[k],mu,tau,vb,Y[k],a,b, method[k],ss[k], dv[k], div[k],
 sg2[k], pg2[k], gg2[k], df[k];
data method, Y, dv, ss in "summr.dat";
inits in "summrgen.in";
{ mu ~ dnorm(0, 1.0E-4);
  tau ~ dgamma(1.0E-4, 1.0E-4);
  b ~ dgamma(1.0E-4, 1.0E-4);
  a ~ dexp(1.0);
 

for (i in 1:k) { theta[i] ~ dnorm(mu,tau);
                                }

for (i in 1:k) { sg2[i] ~ dgamma(a,b);
                    }

for (i in 1:k) {
          div[i]<- (dv[i]*dv[i]);
          df[i] <- (ss[i]-1)/2.;
          pg2[i] <- df[i] * sg2[i];
          gg2[i] <- ss[i] * sg2[i];
          Y[i] ~ dnorm(theta[i], gg2[i]);

          div[i] ~ dgamma(df[i], pg2[i]); 
   } 

     }
Results The results of the computations of the above program are given in Table 2. For each compound, the mean is given along with the standard deviation, the 95% HPD interval and the median of the posterior distribution. Each simulated sample is of size 1000.
Results Table
Table 2: SRM 1946 chlorinated data
Compound Mean Standard Deviation 2.5% : 97.5% Confidence Interval Median Sample
PCB28 0.1966E+01 0.1682E+00 0.1672E+01 0.2330E+01 0.1958E+01 1000
PCB31 0.1565E+01 0.2154E+00 0.1121E+01 0.1969E+01 0.1564E+01 1000
PCB52 0.8037E+01 0.6438E+00 0.6673E+01 0.9392E+01 0.8032E+01 1000
PCB49 0.3763E+01 0.2713E+00 0.3188E+01 0.4264E+01 0.3769E+01 1000
PCB 44 0.4557E+01 0.5149E+00 0.3455E+01 0.5628E+01 0.4536E+01 1000
PCB 66 0.1067E+02 0.1563E+01 0.7818E+01 0.1402E+02 0.1063E+02 1000
PCB 95 0.1132E+02 0.1149E+01 0.9111E+01 0.1371E+02 0.1130E+02 1000
PCB 101 0.3422E+02 0.1711E+01 0.3069E+02 0.3756E+02 0.3424E+02 1000
PCB 99(check 0.1941E+02 0.6257E+01 0.6133E+01 0.3101E+02 0.1963E+02 1000
PCB 87 0.9243E+01 0.1009E+01 0.7236E+01 0.1120E+02 0.9259E+01 1000
PCB 110 0.2267E+02 0.1543E+01 0.1928E+02 0.2561E+02 0.2265E+02 1000
PCB 151 M 0.1554E+01 0.5135E+00 0.6764E+00 0.2303E+01 0.1556E+01 1000
PCB 149 0.2893E+02 0.3960E+01 0.2103E+02 0.3592E+02 0.2906E+02 1000
PCB 118 0.5186E+02 0.4094E+00 0.5100E+02 0.5274E+02 0.5183E+02 1000
PCB 153 0.1667E+03 0.6979E+01 0.1522E+03 0.1799E+03 0.1669E+03 1000
PCB 105 0.1975E+02 0.5485E+00 0.1878E+02 0.2088E+02 0.1972E+02 1000
PCB 138 0.1114E+03 0.1331E+02 0.8094E+02 0.1340E+03 0.1126E+03 1000
PCB 138/1 0.1493E+03 0.5689E+01 0.1362E+03 0.1597E+03 0.1497E+03 1000
PCB 187 0.5494E+02 0.1106E+01 0.5263E+02 0.5743E+02 0.5486E+02 1000
PCB 183 0.2158E+02 0.1568E+01 0.1855E+02 0.2486E+02 0.2155E+02 1000
PCB 128 0.2256E+02 0.1267E+01 0.2023E+02 0.2545E+02 0.2242E+02 1000
PCB 156 0.9457E+01 0.3443E+00 0.8770E+01 0.1017E+02 0.9454E+01 1000
PCB 180 0.7379E+02 0.2457E+01 0.6875E+02 0.7851E+02 0.7393E+02 1000
PCB 170 0.2483E+02 0.1321E+01 0.2206E+02 0.2734E+02 0.2486E+02 1000
PCB 194 0.1279E+02 0.8745E+00 0.1101E+02 0.1447E+02 0.1279E+02 1000
PCB 195 0.5232E+01 0.2953E+00 0.4660E+01 0.5764E+01 0.5232E+01 1000
PCB 206 0.5340E+01 0.2623E+00 0.4779E+01 0.5855E+01 0.5334E+01 1000
PCB 209 0.1267E+01 0.1332E+00 0.9710E+00 0.1533E+01 0.1263E+01 1000
alpha-HCH 0.5664E+01 0.1411E+00 0.5414E+01 0.5985E+01 0.5648E+01 1000
HCB 0.7116E+01 0.6217E+00 0.5903E+01 0.8176E+01 0.7145E+01 1000
gamma-HCH 0.1108E+01 0.1578E+00 0.8202E+00 0.1430E+01 0.1098E+01 1000
hept epox 0.5739E+01 0.3851E+00 0.5006E+01 0.6523E+01 0.5744E+01 1000
oxychlord 0.1878E+02 0.1047E+01 0.1673E+02 0.2104E+02 0.1876E+02 1000
trans-chl 0.8319E+01 0.6155E+00 0.7105E+01 0.9684E+01 0.8291E+01 1000
2,4-DDE 0.1002E+01 0.2084E+00 0.6345E+00 0.1415E+01 0.9963E+00 1000
cis-chlor 0.3223E+02 0.1325E+01 0.2935E+02 0.3470E+02 0.3237E+02 1000
trans-nonach 0.9902E+02 0.4955E+01 0.8770E+02 0.1088E+03 0.9934E+02 1000
dieldrin 0.3237E+02 0.2768E+01 0.2673E+02 0.3834E+02 0.3238E+02 1000
44-DDE skip- 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0
24-DDD 0.2199E+01 0.2071E+00 0.1860E+01 0.2538E+01 0.2194E+01 1000
44-DDD 0.1740E+02 0.2251E+01 0.1310E+02 0.2181E+02 0.1743E+02 1000
24-DDT 0.2169E+02 0.3804E+01 0.1514E+02 0.2912E+02 0.2163E+02 1000
cis-nonac 0.5869E+02 0.2691E+01 0.5317E+02 0.6400E+02 0.5875E+02 1000
44-DDT 0.3798E+02 0.4170E+01 0.3174E+02 0.4764E+02 0.3716E+02 1000
mirex 0.6316E+01 0.5493E+00 0.5230E+01 0.7311E+01 0.6347E+01 1000
PCB 63 0.1257E+01 0.2540E+00 0.8177E+00 0.1704E+01 0.1263E+01 1000
PCB 74 0.4736E+01 0.4088E+00 0.4000E+01 0.5467E+01 0.4733E+01 1000
PCB 70 0.1473E+02 0.8178E+00 0.1337E+02 0.1626E+02 0.1473E+02 1000
PCB 56 0.5663E+01 0.1184E+01 0.3829E+01 0.7576E+01 0.5647E+01 1000
PCB 107 0.8931E+01 0.3291E+00 0.8373E+01 0.9658E+01 0.8893E+01 1000
PCB 146 0.2967E+02 0.2891E+01 0.2311E+02 0.3531E+02 0.2971E+02 1000
PCB 132 0.5725E+01 0.8580E+00 0.4393E+01 0.6842E+01 0.5698E+01 1000
PCB 163 0.3164E+02 0.4728E+00 0.3067E+02 0.3248E+02 0.3168E+02 1000
PCB 158 0.7498E+01 0.6754E+00 0.6069E+01 0.8705E+01 0.7514E+01 1000
PCB 157 0.3768E+01 0.3397E+01 -0.3223E+01 0.1082E+02 0.3761E+01 1000
PCB 174 0.9075E+01 0.8881E+00 0.7329E+01 0.1086E+02 0.9065E+01 1000
PCB 193 0.5694E+01 0.5214E+00 0.4643E+01 0.6660E+01 0.5689E+01 1000
PCB 77 0.3265E+00 0.2467E-01 0.2783E+00 0.3713E+00 0.3268E+00 1000
PCB 126 0.3792E+00 0.1864E-01 0.3456E+00 0.4142E+00 0.3789E+00 1000
PCB 169 0.1061E+00 0.2143E-01 0.7418E-01 0.1464E+00 0.1057E+00 1000

Date created: 9/20/2001
Last updated: 9/20/2001
Please email comments on this WWW page to sedwww@nist.gov.

[ SED Home |  Bayesian Home |  Previous |  Next ]