ST790R -- Fall 2009

  Homework #5 -- due Tuesday, 27 October 2009
  (last edited Tuesday, 20 October)
   
  Exercises in Textbook: (*just turn in these)
   8.7, 8.8, 8.13, 9.13*, 9.14, 9.15*

  R Exercises:

  *1) (Variation on 8.8) Consider maximizing the function 
    f(t) = y*t - n*log(1+exp(t)) for n=3 and y=2.
   a) Find the root of the derivative of f(t) using "uniroot."
   b) Find the maximum of f(t) using "optimize."
   c) Write a function h(t) that computes the Newton step, with
  t0 as its argument and returning the result of a Newton step t1.
   d) Find the maximum of f(t) using Newton's method.
   e) Find the three (oops, not 2) points where h(h(t))=t.
   f) Graph h(h(t)) vs t near the two outer points.
   d) Argue that if we use a starting value outside the interval 
  formed by those two points in (e), then Newton will diverge.

  *2) Fit a nonlinear regression model to the data
    in one of the files "sub1.dat" through "sub10.dat" using the last
    digit of your NCSU id.  The files are in the directory "yafune" 
    and are times (ti)(hours) and drug concentrations (yi)(ng/ml) for 
    subjects in a Phase I trial.  Fit a model of the form
     log(yi) dist Normal(log(f(ti)), sigma^2) where 
              /  a(1-exp(-bt))+c*(1-exp(-dt))  for t <= T=3
      f(t) = {
              \  a(1-exp(-bT))exp(-b(t-T))+c(1-exp(-dT))exp(-d(t-T))
    for t > T=3.  Note that if we define m(t) = min(t,T), then we can
    rewrite f(t) in one line as
        f(t) = a(1-exp(-bm(t)))exp(-b(t-m(t))) + 
                   c(1-exp(-dm(t)))exp(-d(t-m(t)))
 
    Note that getting starting values can be a challenge.  Let b > d
    so that the first term decays more quickly.  So for large values
    of t, fit just the second term -- just a simple linear regression.
    Then adjust the response and fit the first few observations for 
    the first piece.
   
    Some options:
      a) instead of logs, fit yi dist Normal( f(ti), sigma^2 )
      b) another dataset -- there are more subjects