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\begin{center}\bf
Finger Exercises --- Due Thursday, April 9, 1998
\end{center}

\begin{quote}\em
Please prepare answers for these exercises, to be turned in.  Your
grade will be based on how many questions you made a reasonably strong
effort to answer, {\bf not} how many questions you get right or
partially right.  There is no partial credit, but the grader may give
you additional written feedback on each question you attempt.

Feel free to discuss these exercises with each other and with me.  You
will get the most benefit from the grader's remarks, however, if the
work you turn in is your own.

\rm

This hw, like most material for this course, is posted on the World
Wide Web at URL \www{}; you can cut and paste data directly out of the
\TeX{} file if you wish.  Look for something under week11.  The files
you need are: {\tt ratcancer.dat} and {\tt motorettes.dat}.

\em

There are two problems.  Both problems have four parts.

\end{quote}

\begin{enumerate}

\item Please refer to the rats cancer data at the beginning of the
notes from Tue Mar 31, 1998.  The data set is also in {\tt
ratcancer.dat}.

\begin{enumerate}

\item Fit and plot Kaplan-Meier PL estimates of the survival curves
for the rats in the two groups.  Indicate in your plot which survival
curve goes with which group.

\item Is there evidence that these two groups have different survival
curves?  Use the Mantel-Haenszel test (a.k.a. the log-rank test) as
well as a modified Wilcoxon test, to test this hypothesis.  (see
documentation for {\tt survdiff} if you forget how to do this).

\item Now refit the data using a Cox proportional hazards model, and
plot the resulting survival curves, indicating which curve goes with
whih group.  Qualitatively compare the Cox PH curves with the
Kaplan-Meier PL curves from part (a).

\item Use an appropriate test based on the Cox model to decide whether
the two survival curves are the same.  What test(s) did you use
(i.e. give the statistical name(s) of the test(s))?  Compare your
answer with part (b) above.

\end{enumerate}

\item Please refer to the motorettes testing data from the beginning
of the notes from Tue Mar 31, 1998.  The data set is also on line in
{\tt motorettes.dat}.  Let us suppose that a normal operating
temperature for these motorettes is 25$^o$C.

\begin{enumerate}

\item Fit the Cox proportional hazards model to this data, regressing
survival time on temperature.  Should temp be kept in the model or can
it be dropped?  Justify.  

Prepare and examine appropriate residual plots; does the model look
like a reasonable fit?

\item Plot (on the same plot) the survival curves for temp$=$25, 150,
170, 190 and 220$^o$C.  What conclusions can you draw about survival
at the normal operating temperature, from this or other ways of
looking at the Cox model?

\item Use {\tt survreg} to fit a Weibull model to these data (note:
you can indicate censored observations just as you would with {\tt
survfit} or {\tt coxph}). 

Prepare and examine appropriate residual plots; does the model look
more or less acceptable than the Cox model?

\item Find a way to summarize (probably graphically but maybe not) the
survival function under the Weibull model at $25^{o}$C.  

In particular, find a value of $t$ such that $P[T>T] \approx 95\%$,
where $T$ is the time until failure for a motorette under the Weibull
model. 

\end{enumerate}

\end{enumerate}

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