The survival times (in years) of female breast cancer patients after mastectomy are given in Survival. txt.
For 1 < i < n, let Xi, the survival time of the ith patient, follow a Log-Normal distribution, i.e.
Xi ~ IN(N, 02). We are interested in estimating &
= eH+0?/2, the expected lifetime of female breast
cancer patients after mastectomy. Suppose also that we have the following prior distributions for u and
T(4) ~ N(a0,1/ w0),
-2|10) ~ Exp(No)
where u and o-2 (and thus g?) are assumed to be apriori independent and ao and wo are assumed to be
known hyperparameters. A is unknown so a Gamma(Bo, v0 hyperprior, with both ßo and yo known, is
also assumed.
1. Denote Yi = log (Xi). Write down the likelihood function L(u, 02 y) and use it to find the conditional
posterior distributions of mo?, 10, y; g2 u, 10, y and do 4, 02, y, where y are the log-transformed
survival time data.
2. Write a Gibbs sampler program to obtain samples from the joint distribution of (u, o2).
3. Apply the Gibbs sampler to the 100 observations from x in Survival. txt. Obtain the Monte Carlo
estimate of » (with standard deviation) through your Gibbs sample.
Now, suppose that some of the survival times are missing due to right censoring (this can happen if pa-
tients survive beyond the length of a study). Then we can rearrange the data as 21,
., I'm, 25419.
where af = c (e is the censoring time). Similarly, y = (Y1, ..., Um, Um+1%..-, UM), where y = log(c). The
amended data, including censored survival times, can be found in Survivalcensored.txt.
4. Let 2 = (am+1,..., En) represent the true but unobserved values of ym+1%..-, un. Write down the
likelihood function L(4, 2 w*) based on the augmented data w*
= (y,2) and use it to find the
conditional posterior distributions of ,02, An, W*: 024, 10, w* and 10 1, 02, w*
5. Give an expression (up to proportionality) for T(Zil4,0?,y,2;-) and hence deduce the distribution
of zilM, 02, y, zi-
, where zi_
., Zi-1.2it1.
6. Write a data augmentation Gibbs sampler program to obtain samples from the joint distribution
of (u. 02).