Short answer questions.
\begin{enumerate}
\item{}
Leave one out diagnostics can be used to assess the influence that each observation has
on a fitted regression model. Explain what the following occurrences would
indicate:
\begin{enumerate}
\item{}
One observation has a much larger value of Cook's distance than any other observation.
\item{}
An observation has a value of  Deviance Changes  of 4.6. 
\item{}
An observation has a value of {\sf DFBETA} of 1.08 for a particular explanatory variable.
\marks{5}
\end{enumerate}

\vspace{30mm}

\item{}
The ``dissimilarity index'' is sometimes used when selecting a model for 
contingency table data.
\begin{enumerate}
\item
Describe a situation where the dissimilarity index would be useful.
\item{}
What does the dissimilarity index measure and how is it used? 
\marks{5}
\end{enumerate}


\newpage
\item{}
The following data comes from a case-control study investigating the 
occurrence of esophageal cancer.
A sample of 200 cases were matched with 775 controls. The following table
classifies the cases and controls by their annual alcohol consumption.

\begin{center}
\begin{tabular}{cccc}
 & \multicolumn{2}{c}{Annual Alcohol consumption} \\ 
 &\phantom{xx}  $\ge 80\;\mbox{g}$ \phantom{xx} &\phantom{xx}  $<80\;\mbox{g}$ \phantom{xx}  \\ \cline{2-3}
Cases & 96 & 104  \\
Controls & 109 & 666 \\ 
\end{tabular}
\end{center}

Create a 95\% confidence interval for the odds ratio that compares the odds of 
esophageal cancer for alcohol consumption $\ge 80\;\mbox{g}$ to that for alcohol 
consumption $<80\;\mbox{g}$. Explain what this interval indicates.
Note that if $Z\sim\mbox{N}(0,1)$, then $\mbox{Pr}(-1.96 \le Z \le 1.96) = 0.95$.
\marks{5}

\vspace{25mm}

\item
An experiment was conducted to test the compressive strength of an alloy fastener used 
in the construction of aircraft. Samples of fasteners were tested at ten different pressure loads increasing 
from 2500 psi to 4300 psi in steps of 200 psi.  The number of fasteners failing at each load was recorded.
\begin{center}
\begin{tabular}{ccc}
Load & Number tested & Number failing \\
2500 & 50 & 10 \\
2700 & 70 & 17 \\
2900 &100 & 30 \\
3100 & 60 & 21 \\
3300 & 40 & 18 \\
3500 & 85 & 43 \\
3700 & 90 & 54 \\
3900 & 50 & 33 \\
4100 & 80 & 60 \\
4300 & 65 & 51 \\
\end{tabular}
\end{center}

A logistic regression model was fitted and the following output obtained:
\begin{verbatim}
Coefficients:
                   Value   Std. Error   t value 
(Intercept) -5.339711512 0.5456929729 -9.785194
       load  0.001548434 0.0001575379  9.828962
\end{verbatim}

\begin{enumerate}
\item Write down the fitted model in both its logistic form and in its logit 
      form.
\item Use the fitted model to estimate the load at which 50\% of fasteners fail.
\end{enumerate}
\marks{5}
\end{enumerate}








