Problems students are to do -

    - UG students, please do problems 1,2 and 3 below
    - G students, please do problems 1 - 4 below

Students needing help on this assignment are strongly encouraged to contact a classmate
or Dr. O'Brien!  Please don't wait until the last minute.

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  Finney (1978, p.18) provides data regarding the tolerances of cats for tinctures of two variants of
strophanthus and ouabain.  The table below shows the fatal doses, or tolerances, with the data recorded
as quantities per kg body weight; the last row of the table provides the means.  We wish to use these
data to estimate (a) the relative potency of Strophanthus1 to Strophanthus2, and (b) the relative potency
of Strophanthus2 to Ouabain.
 

Strophanthus1	Strophanthus2	Ouabain
15.5	24.2	52.3
15.8	18.5	99.1
17.1	20.0	47.6
14.4	22.7	65.1
12.4	17.0	66.8
18.9	14.7	57.6
23.4	22.0	49.3
---	---	45.8
---	---	66.9
mean = 16.8	mean = 19.9	mean = 61.2

(a) Using this SAS program/output, make the first comparison (Stroph1 versus Stroph2), by
providing 90%, 95%, and 99% Wald and PLCIs for the relative potency.  Comment on the
necessary assumption here regarding the respective treatment variances and the reasonable-
ness of this assumption for these data.  Give the p-value for just accepting equivalent potency
for the two drugs for both the Wald and PL situations.

(b) Using this SAS program/output, answer each of the questions asked in part (a) but here
for the second comparison, viz, comparing Strophanthus2 with Ouabain.

2. Return to the data discussed in exercise 4 of Homework # 4 - our goal now is to assess
the relative potency of dichlorodiphenyltrichloroethane (DDT) at 2.0% w/v to g-benzene hexachloride
(gBHC) used at 1.5% w/v. This SAS program/output will help; the program runs Proc NLMixed
four times, followed by some matrix code in SAS/IML and a graph

(a) Use the first or the second NLMIXED to give parameter estimates of the slope (b, called "bet"
in the program) and the LD50 (g, called "gam" in the program) for each of the curves (one for
DDT and one for gBHC).

(b) In contrasting the output of the second and third NLMIXEDs, perform a likelihood-based test
of parallelism.  Clearly write out your hypotheses, conclusion and p-value.

(c) Assuming parallelism of the two curves, provide 90%, 95% and 99% Wald and PLCIs for
the relative potency.  Do you believe the treatments are equally potent?  Support your claim(s).

3. Data provided in Giltinan, Capizzi and Malani (1988), "Diagnostic tests for simliar action of two
compounds," Applied Statistics, resulted from an experiment designed to investigate how two
insecticides (A and B) may act in combination.  Of interest here is whether the insecticides interact
to produce enhanced performance, a reduction in performance or neither.  The data, corresponding
to the number of dead insects corresponding to a fixed number of insects tested (usually 30) for
20 combinations of the insecticides are analyzed in this SAS program/output.

(a) Comment on the experimental design used for this experiment (identify whether it is a ray,
factorial or other design, how many interior rays if any and support points per ray, etc.)  How many
experimental units (insects) were used for the study?  Does the chosen design seem like an efficient
way to use the experimental units, or would you have chosen another strategy?  If so, what and why?
If not, why do you like this one?

(b) Does this analysis suggest enhanced effect, a reduction in performance or neither of these?
Use only the results of the first two NLMixed's for this part, identifying the model that is being
used here. Support your claim(s) using two separate test statistics and p-values (approximations
are okay) and suggest which of these tests preferred and why.

(c) In examining how well or otherwise the predicted and actual proportions coincide, comment on
the adequacy of the above-used model for these data.

4. Returning to the previous exercise and SAS program/output,

(a) describe the model(s) that are being fit in the third and fourth NLMixed's, and decide which
one is best for these data; give a test statistic and p-value if possible.

(b) Connect the first and fourth NLMixed's (is either a special case of the other?) and decide
which is better; give a test statistic and p-value if possible.

(c) Calculate the predicted values of the LD50's along the interior rays that the model used in the
first NLMixed gives and compare these with the LD50's for the fourth NLMixed. Are they close?

(d) Extra Credit: in the general setting, give necessary and sufficient conditions for the Finney and
Separate-ray models to be equivalent when the chosen design is a 2+3 ray design and when the
logistic curve is fit as for this exercise.