ST758 -- Fall 2011

  Homework #5 -- due Tuesday, 25 October 2011 (revised 15,17,20 October)
   
  Exercises in Textbook: (*just turn in these)
   9.13*, but use matrices a) matrix(c(4,3,3,4),2,2) and
    b) matrix(c(2,-1,-1,1),2,2).  Compare by taking a few starting
    points away from the origin and look at the distance of the 
    iterates from the origin (solution).
   9.15

  R Exercises:

    In the files 'gardner1.dat' and 'gardner2.dat' are data from a study 
    of gallery construction by pine beetles. Rows (observations) are 
    galleries (dug by beetles); columns are 
       X1i = temperature (1 or 2)
       X2i = treatment (1 or 2) 
       ai = length of galllery
       Yi = number of egg niches
       stuff = something to be ignored for now
    Theory says that the number of egg niches should be Poisson with a rate 
    proportional to the length; researchers are interested on whether the
    temperature or treatment has an effect.

  *1) Fit a Poisson regression model with rate 
       lambdai = ai * exp(beta0 + beta1*X1i + beta2*X2i)
    so si = ai = intensity, X1i and X2i are the covariates.
    Perform a likelihood ratio test of H: beta1=beta2=0 against the
    alternative that either or both are not zero at level 0.05.
    (Write a function to compute the (un-)log-likelihood, and
     minimize it using nlm.  You can use 'glm' only to check.)
    
  *2) Fit a nonlinear regression model with mean lambdai as above, but
    with variances also lambdai.  So the goal is to minimize a nonlinear
    generalized least squares, or minimize the sum of squares of
      
         Yi/sqrt(lambdai) - sqrt(lambdai)
    Again test the same hypothesis that the covariates have no effect,
    using a LRT (F test with SSE(Reduce) and SSE(Full)) at level 0.05.
    (Use 'nls' here.)

  *3) Compute sandwich standard errors using the C4 expression
    on p.246.  (You can drop the T(betahat) terms to simplify.)

  Which file to use?  If you were born on an odd day of the month, use
  'garder1.dat' and use 'gardner2.dat' for even.  
  Files are in rfiles directory.