########################### # PK Hierarchical Linear Model # This is the linearized version of the PK model, where we model Z = log(concentration) as a # simple linear function of dosage time X for each patient i = 1, ..., N=10. # See Carlin and Louis (2009, Ch 2, Example 2.13). ########################### model { for(i in 1:N){ for(j in 1:T){ Z[i,j] ~ dnorm (mu[i,j] , tauZ ) Y[i,j] <- exp(Z[i,j]) mu[i,j] <- beta[i, 1] + beta[i, 2]* X[j] } beta[i , 1:2] ~ dmnorm (betamu[1:2] , betaUpsilon[1:2, 1:2]) } betamu[1:2] ~ dmnorm( meen[1:2] , prec[1:2 ,1:2]) betaUpsilon[1:2 , 1:2] ~ dwish(R[1:2, 1:2], 2) beta.s2[1:2, 1:2] <- inverse(betaUpsilon[1:2, 1:2]) for (i in 1 : 2) {beta.s[i] <- sqrt(beta.s2[i, i]) } # tauZ ~ dgamma(0.001, 0.001) # usual WinBUGS vague gamma prior # sigmaZ <- 1/tauZ tauZ <- 1/(sigmaZ * sigmaZ) # vague Gelman prior sigmaZ ~ dunif(0.01, 100) } # end of linear PK program ## Inits: list( tauZ =20 , betaT=structure(.Data = c(0.01, 0,0, 0.01), .Dim = c(2, 2)), betamu=c( -3, 3 )) # gamma list( sigmaZ =1, betaUpsilon=structure(.Data = c(0.1, 0,0, 0.1), .Dim = c(2, 2)), betamu=c(0, 0)) # Gelman ## Data: # NOTE: Obs 6 for Patient 8 (apparent data entry error) set to NA! # Z = log(concentration) list(X=c(2,4,6,8,10,24,28,32), N=10, T=8, Z = structure( .Data = c(0.086, -0.288, -0.635, -1.079, -1.47, -3.912, NA, NA, 0.708, 0.247, 0.182, 0.02, -0.186, -1.273, NA, NA, 0.365, 0.262, -0.051, -0.386, -0.654, -2.813, NA, NA, 0.438, -0.041, -0.223, -0.478, -0.777, -2.526, NA, NA, 0.3, -0.248, -0.693, -1.109, -1.715, -3.912, NA, NA, 0.077, -0.528, -0.994, -1.47, -1.772, NA, NA, NA, 0.278, -0.301, -0.777, -1.273, -1.309, -3.507, -3.912, NA, 0.489, 0.01, -0.315, -0.598, -0.892, NA, -2.813, -3.912, 0.231, -0.315, -0.916, -1.204, -1.561, NA, NA, NA, 0.262, -0.357, -0.916, -1.386, -1.966, NA, NA, NA), .Dim = c(10,8)), meen = c(0,0) , prec = structure(.Data = c(1.0E-6, 0, 0, 1.0E-6), .Dim = c(2, 2)), R = structure(.Data = c(0.1, 0, 0, 0.1), .Dim = c(2, 2)) ) # Results: node mean sd MC error 2.5% median 97.5% start sample tauZ 48.04 10.63 0.08587 29.47 47.23 71.18 15001 30000 node mean sd MC error 2.5% median 97.5% start sample betamu[1] 0.5504 0.07132 5.741E-4 0.4101 0.5504 0.6923 15001 30000 betamu[2] -0.1734 0.04032 2.264E-4 -0.2538 -0.1732 -0.09241 15001 30000 node mean sd MC error 2.5% median 97.5% start sample beta.s[1] 0.1882 0.05423 3.962E-4 0.1099 0.179 0.3205 15001 30000 beta.s[2] 0.1226 0.0312 2.027E-4 0.07893 0.117 0.2001 15001 30000 node mean sd MC error 2.5% median 97.5% start sample beta[1,1] 0.4513 0.08549 5.126E-4 0.282 0.4517 0.6174 15001 30000 beta[1,2] -0.1839 0.007822 4.595E-5 -0.1992 -0.1839 -0.1684 15001 30000 beta[2,1] 0.6701 0.0866 5.276E-4 0.5003 0.6695 0.8406 15001 30000 beta[2,2] -0.08176 0.007807 4.652E-5 -0.09707 -0.0818 -0.06642 15001 30000 beta[3,1] 0.7337 0.08823 5.682E-4 0.5603 0.7335 0.9074 15001 30000 beta[3,2] -0.1449 0.007888 4.572E-5 -0.1603 -0.1449 -0.1294 15001 30000 beta[4,1] 0.5682 0.08541 5.018E-4 0.4014 0.568 0.7361 15001 30000 beta[4,2] -0.1302 0.00778 4.222E-5 -0.1454 -0.1302 -0.1149 15001 30000 beta[5,1] 0.4795 0.08555 5.251E-4 0.3111 0.4793 0.6469 15001 30000 beta[5,2] -0.1888 0.00784 4.478E-5 -0.2044 -0.1888 -0.1735 15001 30000 beta[6,1] 0.4877 0.1175 8.276E-4 0.2556 0.4881 0.7176 15001 30000 beta[6,2] -0.236 0.01869 1.263E-4 -0.2728 -0.236 -0.1992 15001 30000 beta[7,1] 0.3314 0.08402 5.465E-4 0.1664 0.3312 0.4959 15001 30000 beta[7,2] -0.1579 0.005713 3.701E-5 -0.1694 -0.1578 -0.1468 15001 30000 beta[8,1] 0.5541 0.07686 4.27E-4 0.4033 0.5543 0.7057 15001 30000 beta[8,2] -0.1324 0.004736 2.517E-5 -0.1417 -0.1323 -0.123 15001 30000 beta[9,1] 0.565 0.1176 7.793E-4 0.3344 0.5641 0.7992 15001 30000 beta[9,2] -0.2199 0.01883 1.223E-4 -0.2577 -0.2199 -0.1829 15001 30000 beta[10,1] 0.6606 0.121 8.314E-4 0.4268 0.6579 0.9041 15001 30000 beta[10,2] -0.2582 0.01917 1.234E-4 -0.2964 -0.2582 -0.2207 15001 30000 node mean sd MC error 2.5% median 97.5% start sample Z[1,7] -4.698 0.2211 0.001421 -5.13 -4.696 -4.261 20001 20000 Z[1,8] -5.437 0.2439 0.001726 -5.918 -5.438 -4.96 20001 20000 Z[2,7] -1.619 0.2205 0.001582 -2.052 -1.62 -1.188 20001 20000 Z[2,8] -1.946 0.2439 0.001722 -2.422 -1.947 -1.461 20001 20000 Z[3,7] -3.322 0.2221 0.001519 -3.759 -3.321 -2.881 20001 20000 Z[3,8] -3.902 0.2454 0.001757 -4.382 -3.902 -3.412 20001 20000 Z[4,7] -3.079 0.2198 0.001387 -3.512 -3.08 -2.649 20001 20000 Z[4,8] -3.597 0.2438 0.001533 -4.071 -3.597 -3.109 20001 20000 Z[5,7] -4.808 0.2238 0.001537 -5.244 -4.809 -4.373 20001 20000 Z[5,8] -5.566 0.2464 0.001634 -6.051 -5.565 -5.084 20001 20000 Z[6,6] -5.178 0.3817 0.002857 -5.931 -5.179 -4.433 20001 20000 Z[6,7] -6.123 0.4513 0.0034 -7.009 -6.121 -5.247 20001 20000 Z[6,8] -7.068 0.5213 0.004193 -8.103 -7.064 -6.047 20001 20000 Z[7,8] -4.721 0.1955 0.00144 -5.106 -4.721 -4.341 20001 20000 Z[8,6] -2.623 0.1688 0.001209 -2.952 -2.623 -2.29 20001 20000 Z[9,6] -4.715 0.3856 0.002935 -5.487 -4.712 -3.961 20001 20000 Z[9,7] -5.592 0.454 0.003544 -6.49 -5.593 -4.696 20001 20000 Z[9,8] -6.472 0.5228 0.003982 -7.515 -6.47 -5.44 20001 20000 Z[10,6] -5.535 0.3918 0.002774 -6.304 -5.534 -4.76 20001 20000 Z[10,7] -6.57 0.4621 0.003528 -7.498 -6.572 -5.675 20001 20000 Z[10,8] -7.602 0.5338 0.004128 -8.658 -7.598 -6.556 20001 20000