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\begin{center}
\Large\bf Nonlinear Regression: Some Ideas and a Little Guts \\
Breakout Session
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\begin{description}
\item[My Name:] ~~~\rule{0pt}{0.25in}\rule{3in}{.01in}
\item[My Group:] \begin{tabular}[t]{c}
\rule{0pt}{0.25in}\rule{3in}{.01in}\\
\rule{0pt}{0.25in}\rule{3in}{.01in}\\
\rule{0pt}{0.25in}\rule{3in}{.01in}
\end{tabular}
\end{description}

\begin{enumerate}

\item Explain {\em carefully} what {\tt child45} is, what it contains,
  and why it had to be made.

\vfill

\item Comment on the fitted nonlinear regression: How many iterations
  were required?  Are all the parameters estimated to be
  (significantly) different from zero?  Which parameter is most poorly
  estimated?  Are there any problems analogous to collinearity in this
  fit?

\vfill

\item If we had used the starting values {\tt start=list(alpha=1,
  beta=1, gamma=1)}, the {\tt nls()} function would have stopped with
  the message {\tt Problem in nls(X1945 \~{} alpha * exp( - gamma *
  exp( -..: singular gradient matrix}.  What is going wrong?

\vfill

\item The value for \verb|sum(child45$X1945-mean(child45$X1945)^2)|
is approximately 7529.  Fill in the ``?????'' and the ``??'' in the
  line that calculates $R^2$, and then calculate $R^2$.  Comment on
  this value.

\vfill

\clearpage

\item Fill in the ``??????'' in the {\tt lines()} command that makes
  the plot of the fitted Gompertz curve for the 145 birth cohort.


\vfill

\item Do the plots look good?  What kind of plot could you make, to
  explore whether there are other local minima besides the ones {\tt
  nls()} found?

\vfill


\item The logistic function for nonlinear regression is 
\[
f(x;\alpha.\beta,\gamma) = \displaystyle
\frac{\alpha}{1 + e^{-\beta (x-\gamma)}}
\]
\begin{enumerate}
\item Do you expect the logistic fit to be better or worse than the
  Gompertz fit?  Why or why not?

\vfill

\item If you have time, write out the {\tt nls()} command you would
  use to fit the logistic function to the 1945 data (including
  starting values, if you can).  {\em Hint for starting values: }
\[
        \log\left(\frac{f(x)/\alpha}{1-f(x)/\alpha}\right) 
        =  {\beta (x - \gamma)}
\]


\vfill


\end{enumerate}
\end{enumerate}

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