1. Steensma et al. (2005) presented the data in Table 4.4, from a randomized controlled trial comparing two dosing schedules for the drug erythropoiten. Serum hemoglobin (HGB, in g/dL) was recorded for N = 365 cancer patients with anemia over T = 22 weeks. The full data file is available at www.biostat.umn.edu/~brad/data/HGB_data.txt. We wish to fit a hierarchical simple linear regression model of the form Yij = β0i + β1i(Xj − μX) + ij , i = 1,..., 365, j = 1,..., 22, (4.43)

where Xj = j, the week index, ij iid

∼ N(0, τ), β0i iid

∼ N(μ0, τ0), and

β1i iid

∼ N(μ1, τ1). That is, as in Example 2.13 (and Example 7.2 in Chapter 7), we allow each subject’s HGB trajectory to have its own slope and intercept, but borrow strength from the ensemble by treating these as normal random effects. Note that we also center the week index around its own mean, μX = 11.5.

(a) Use WinBUGS to fit the above model, assuming vague priors for the hyperparameters τ, μ0, τ0, μ1, and τ1. Run multiple chains to assess convergence, and interpret the resulting posterior distributions for the grand slope μ1 and the individual-specific slopes β1i. How do the interpretations of these parameters differ? Use the compare function in WinBUGS to examine these for all participants. Are all participants’

HGB measurements improving over time?

(b) Note that many of the Yij are missing; under the assumption that they are missing at random, WinBUGS can impute them according to the fitted model. Monitor the hemoglobin values estimated for participant?