In a small clinical trial there are ten subjects, whose survival times
(in days) are: 15 3 46 623 126 64 1350 23 279 1024 (with mean 355.3).
One possible model for these data would specify that they are all
INDEPENDENT exponentially-distributed random variables X1,
..., X10, with unknown parameter lambda, or EACH with
pdf
f(x) = (lambda) e-lambda x, x>0.
We would like to learn more about lambda.
- Give the posterior density function xi(lambda|x) for the
rate parameter lambda, starting with uniform prior
xi(lambda)=1, normalized to be a probability density
function (as a function of lambda on the interval
0<lambda<oo), so that the integral
Int{0<lambda<oo} xi(lambda|x) = 1.
Your xi(lambda|x) will be the product of a constant, a
power of lambda, and the exponential of a negative
constant times lambda; you must give the values
of the constants and the power. Use Mathematica or Maple to
evaluate the required integral; in your solution, give the
Mathematica or Maple expression you used to evaluate it.
- Plot xi(lambda|x) as a function of lambda, using
your choice of S-Plus, Matlab, Mathematica, or Maple. Turn in
the plot and the computer code used to generate it. Identify
(both numerically and graphically) the point where xi(lambda|x)
attains its maximum value.
- Give the name of this distribution and the values of any
parameters. You may find this list of probability
distributions (ps, pdf) useful.
- Give the mean and standard deviation of this
distribution. Again, the distribution list may be helpful: you
are not asked to derive the mean or variance (no integration
is necessary).
- Find an expression for the probability p of one-year
survival, as a function of lambda, and find its
expectation E[p|x] under the posterior distribution of
lambda, correct to two decimal places, using S-Plus or
Mathematica. Explain in your own words what are the meanings
of these probabilities, in the context of the clinical trial.
- Using S-Plus, find a symmetric 90% posterior interval for
lambda and, from this, a symmetric 90% posterior
interval for the probability p of one-year survival.
- In fact these are the survival times of the first ten subjects
in the Stanford Heart Transplant trial; your data last week
were from these same subjects, in this same trial. How does
today's estimate of p compare to last week's? Which
requires more assumptions? Which uses more information from
the data? Which do you prefer, and why?