Problems students are to do -
    - UG students - please do problems 1 and 2 below
    - G students -  please do problems 1 - 4 below

As always, list necessary assumptions, and include respective p-values in parentheses next to
your conclusion - e.g., one might conclude, "the data suggested a difference in the treatments
(p = 0.0013)."

1.  Huet, Bouvier, et al (Statistical Tools for Nonlinear Regression, p.2) use the Pasture Regrowth
    data from Ratkowsky (Nonlinear Regression, p.88) to fit a four-parameter sigmoidal growth
    model.  In the dataset, Y = pasture regrowth (since last grazing), and X = time, and for our present
    purposes, we can assume that the data are independent measurements.  The model function that
    these authors used to fit the data is rather complicated, and coming up with starting values for the
    model parameters is not easy, and can only come after we understand the roles they play.
    (a) Look at this SAS Program/Output and write down the assumed model function and discuss the roles
      of the model parameters which play a role in determining the upper and lower asymptotes of the
      model function; assume that q₄ is positive. Next, look only at the plot to obtain a guess of
      these asymptotes, and discuss how to get good starting values for q₁ and q₂. Next, and most
      challenging, get a starting value for q₃ and q₄ by showing how the fitted linear regression is
      related to the original model function.
    (b) Look at the NLIN output and comment on the estimated upper and lower asymptotes (using
       the parameter estimates). Also, based on the output, do a (two-tailed) Wald test that q₄ = 3 using
      a = 1%.   Redo this (Wald) test using a = 5%.  Clearly report your conclusions in each case.
    (c) Repeat your tests in part (b) using LR tests.
    (d) Examine the residuals, and comment on your findings. If the NLIN were to be rerun with
      the third point (x = 21) removed, would the estimate of the lower asymptote increase or decrease?
      Why?

2.  In Nonlinear Regression Analysis and its Applications (1988, p.269), Bates and Watts report data
    from Treloar (1974) regarding the "velocity" of an enzymatic reaction.  The number of counts per minute
    of radioactive product from the reaction was measured as a function of substrate concentration (ppm),
    and from these counts the initial rate, or "velocity," of the reaction was calculated (counts/min²).  The
    experiment was conducted once with the enzyme treated with Puromycin (treated = "yes") and once
    with the enzyme untreated (treated = "no").  The velocity is assumed to depend on the substrate
    concentration according to the Michaelis-Menton (MM2) equation.  It has been hypothesized that
    the "ultimate velocity parameter" (q₁) should be affected by introduction of the Puromycin,
    but not necessarily the "half-velocity parameter" (q₂). This SAS program/output may help us to
    answer these queries.
    (a) Give estimates for the MM2 model parameters for both the treated and untreated curves; display
       these in a 2x2 box.
    (b) Using Wald hypothesis tests, test the relevant hypotheses, reporting test statistics, p-values and
        conclusions.  Approximate p-values as best you can here and in the next part.
    (c) Using a full-and-reduced (likelihood-based) F-test, test whether the half-velocity parameters
        are the same, reporting the test statistic, p-value and conclusion.

3.  In "Calibration and assay development using the four-parameter logistic curve" (Chem. Intell.
    Lab. Systems, 1993, p.97), O'Connell et al fit the LL4 (four-parameter log-logistic) model
    function to radioimmunoassay (RIA) data. Click here for the data and analysis in SAS. This
    program first uses PROC NLIN and then three PROC NLMIXEDs.
    (a) Comment on the necessary assumptions and the roles of the parameter for the model
      fit in the NLIN. Look at the residual plot and comment on whether all necessary
      assumptions (of the NLIN) appear to be met.  The first PROC NLMIXED just fits the
      same homoskedastic (assumed constant variance) model.
    (b) Explain what is being done in the second PROC NLMIXED in terms of the new "model"
      and the roles of the parameters, and note the value of -2LL.  Perform a likelihood-based test of
      whether the extra parameter (r) in the variance is need, writing out your hypotheses, test
      statistic, p-value,  and conclusion.
    (c) It turns out that the third PROC NLMIXED involves another, more appropriate, way of
      modelling the variance.  Comparing this latter NLMIXED with the first NLMIXED, perform
      a likelihood-based hypothesis test testing for homoskedasticity, again writing out your hypotheses,
      test statistic, p-value,  and conclusion.  Finally, compare the parameter estimates and SEs for
      this latter (third) and the first NLMIXED - what has changed?

4. Examine this SAS program/output and determine what model is being fit. Write down the
    model function. What is the relevance of the parameter named PHI?  Write down it's estimate, do
    a (WALD) t-test that the true value of PHI is equal to -0.40 (using a = 5%), and write down the 95 %
    Wald Confidence Interval (WCI) for this parameter. Next, do the likelihood test of whether PHI is
    equal to -0.40 (using a = 5%).  It turns out that the true 95% confidence interval for PHI here is very
    different from the WCI; in your opinion is the true CI shifted to the right or to the left of the WCI? Why?