2009 STAT/BIOSTAT 571: Coursework 7 To be handed in by the start of the lecture on Friday 6th March, 2009. Table 1 contains pharmacokinetic data on 10 subjects who have been administered a dose of D = 30mg of the drug Cadralazine. Let Zij denote the drug concentration (in mg/ml) at time tij, j = 1,...,ni, i = 1,...,10. A one-compartment model would suggest that, in the absence of a stochastic component, the concentration at time t would be given by D V exp(−kt), where V is the volume of distribution, and k is the elimination rate. A plausible statistical model is Yij = logZij = logD −logVi −kitij + epsilon1ij, where epsilon1ij ∼iid N(0,σ2epsilon1), and Vi and ki are subject i specific volume and elimination parameters, assumed to arise from distributions with median(Vi) = V and median(ki) = k. 1. Consider marginal linear models of the form E[Yij] = β0 + β1tij, for j = 1,...,ni, i = 1,...,10. Give point and interval estimates for the population medians V and k, using GEE and LMEM approaches. 2. Do the assumptions required for valid inference in the previous question appear valid here? As part of this exercise you should obtain individual fits to each of the ten subjects, using the estimated random effects. 3. Suppose that we obtain data {zstar,tstar} from a new subject who has received a dose of 30mg of Cadralazine, with zstar = (z1,...,znstar)T and tstar = (t1,...,tnstar)T. Let βstar = (βstar0,βstar1) represent the parameters of this new subject. Using the results of the LMEM analysis construct a prior distribution for βstar, and hence derive the posterior distribution for βstar. (You may assume that σ2epsilon1 is known and equal to hatwideσ2epsilon1, the estimate from the LMEM analysis.) 1 4. Suppose that the new subject provides a single log concentration zstar1 at time tstar1 = 2 hours, and that we wish to administer a further dose of size Dstar at 24 hours (by which time you may assume the previous dose has been eliminated), with the aim of attaining a concentration of 1.5mg/ml at 26 hours. Based on the previous question, provide a plot of the optimal dose Dstar versus zstar1 (with log1.0 ≤ zstar1 ≤ log2.5), explaining the method you use. 5. Now consider the nonlinear model Yij = logD −θ0i −exp(θ1i)tij + epsilon1ij, with epsilon1ij ∼iid N(0,σ2epsilon1), and where θi = θ0i θ1i ∼iid N2 (θ,D), with θ = θ0 θ1 , and D = σ 2 00 σ 2 01 σ201 σ211 , is the population distribution of θi, i = 1,...,10. Fit this model using the nlme() software, and briefly summarize your analysis in language understandable to a non-statistician. 6. Carry out a Bayesian analysis of the previous model, using the bivariate normal prior N2 0 0 , 10 3 0 0 103 , forθ, and a Wishart prior forD−1 with r = 5 and such that the mean forD is 0.04 0 0 0.04 . Assume pi(σ2epsilon1) ∝ σ−2epsilon1 . Report the 2.5%, 50% and 97.5% points of the posterior distributions for the mode, median, and mean of the population distribution for V star, the volume of an individual randomly sampled from the population distribution. 7. Write down the population distributions of the half-life tstar1/2 = log2/kstar and the clearance Clstar = V star×kstar where tstar1/2, kstar and Clstar have analagous definitions to that of V star in the previous part. Using the results from either the nlme or Bayes analysis, estimate the probability that a subject selected from the population has a half-life greater than 3.7 hours. (You should include the nlme() and WinBUGS code in your write-up.) 2 Time point tij Subject 2 4 6 8 10 24 28 32 1 1.09 0.75 0.53 0.34 0.23 0.02 2 2.03 1.28 1.20 1.02 0.83 0.28 3 1.44 1.30 0.95 0.68 0.52 0.06 4 1.55 0.96 0.80 0.62 0.46 0.08 5 1.35 0.78 0.50 0.33 0.18 0.02 6 1.08 0.59 0.37 0.23 0.17 7 1.32 0.74 0.46 0.28 0.27 0.03 0.02 8 1.63 1.01 0.73 0.55 0.41 0.01 0.06 0.02 9 1.26 0.73 0.40 0.30 0.21 10 1.30 0.70 0.40 0.25 0.14 Table 1: Pharmacokinetic concentration data, following administration of the drug Cadralazine. 3