This project concerns ongoing research between myself as statistical
consultant and a scientist in at the Center for Dairy Forage Research.
The key questions concern carbohydrate digestion in the rumen of cows.
Specifically, what is the effect of different levels of casein
infusion (a rumen degradable protein) on microbial protein synthesis
using branched chain amino acids (BCAA) as an indicator of protein
flow.  Note that I am NOT asking you to do a full-blown analysis here
but only a part, a begining.

There were five treatment levels of casein infusion,

\begin{center}
\begin{tabular}{lccccc}
TRT & 1 & 2 & 3 & 4 & 5 \\
g/day & 0 & 500 & 1000 & 1500 & 2000
\end{tabular}
\end{center}

\noindent
which were assigned to cows in 5$\times$5 Latin squares.  That is,
five cows were assigned the five treatment levels over 5 periods in
such a way that exactly one of the five cows received any given
treatment level in any period.  Cows were (as one would hope) randomly
assigned within this arrangement.

In each of the 5 periods, there were 3 subperiods.  Each subperiod
involved infusing the casein into a different region, in the same
order for all cows and all treatments.  These were

\begin{center}
\begin{tabular}{ccl}
subperiod & code & region and amount \\
1 & A100 & abomasum at 100\% \\
2 & A10 & abomasum at 10\% \\
3 & R100 & rumen at 100\%
\end{tabular}
\end{center}

\noindent
(Abomasum is part of the intestine, if you are interested.)
Previous studies suggest that response to A10 should be about the same
as R100, and both should be much lower than A100.  To allow for
possible carryover, cows were left on a treatment for 5--7 days and
only measurements on the last two days were used.

This is a complicated design, with Latin squares, split plots
(subperiods within periods) and possible carryover from previous
treatments.  There is one more problem.  One of the cows became sick
during the first period and had to be removed from the study.  It was
replaced by cow 675, which consequently was not used for period 1.
Thus the design is unbalanced!

It is NOT necessary to perform a combined analysis across subperiods
for this midterm.  Instead, analyze each subperiod separately (this
gets rid of the split plot, which you have not studied yet!).

You may want to
consider balanced subsets or other approaches appropriate for
unbalanced designs.  Check assumptions and address problems
accordingly, either by transformations or by careful discussion of
interpretation.  Once you have analyzed each subperiod,
compare your results to address the secondary questions about
subperiod treatments implied above.  Use plots as appropriate.

Throughout your report, document your steps so that someone else
(another statistician) could reproduce your results.  Justify your
choices.  Include relevant results from the computer but do not be
excessive.  Graphs may be helpful.  Your report should be at most 6--8
pages in length.  Be sure to address the key question, and
summarize your findings in a way which could be understood by another
statistician.  (Imagine that that person must explain your findings to
the scientist.)

This is a return to the problem examined for the midterm, but from
a different perspective.  Recall that each cow is given 5 levels of
concentration of casein infusion, and that in each period this is
injected three times, twice in the abomasum (A100,A10) and then in the
rumen (R100).  It was thought that A10 and R100 should be similar.
This question investigates that relationship.

Your task is to determine the effect of casein treatment on the rumen
(R100) adjusted for the abomasum (A10).  Be sure to carefully write
down your model with all assumptions, and remark on the analysis as
presented in {\tt fin3.prt}.  It is not necessary to formally check
for linear trend with concentration ({\tt TRT}), but you may want to
comment on any evidence for treatment differences that arise.
