Short answer questions.


\begin{enumerate}
\item{}
Each of the following diagnostic plots
is used to diagnosis potential problems with ordinary
regression models (Normal errors).
Identify the problem that each plot
is used to detect. Also, for each plot describe the appearance of the
plot when no problem is present and when the problem is present.
\begin{enumerate}
\item{}
A normal probability plot of the residuals.
\item{}
A plot of residuals versus lagged residuals.
\end{enumerate}

\marks{5}

\vspace{15mm}


\item{}
Multicollinearity is one problem that can occur in a dataset.
\begin{enumerate}
\item
What is meant by multicollinearity?
\item
What diagnostic is used to detect multicollinearity? How does this diagnostic
indicate multicollinearity is present?
\end{enumerate}

\marks{5}

\newpage

\item{}
Beetles were exposed to gaseous carbon disulphide and 
their mortality rates recorded after five hours:
\begin{center}
\begin{tabular}{ccccccccccccccc}
Obs & & Dose & & \multicolumn{1}{c}{Deaths} & & Total & \phantom{xxxx} &
Obs & & Dose & & \multicolumn{1}{c}{Deaths} & & Total \\[2pt]
\cline{1-7} \cline{9-15}
1 & & 1.6907 &  &   6 &  & 59 & & 2 & & 1.7242 &  &  13 &  & 60 \\
3 & & 1.7552 &  &  18 &  & 62 & & 4 & & 1.7842 &  &  28 &  & 56 \\
5 & & 1.8113 &  &  52 &  & 63 & & 6 & & 1.8389 &  &  53 &  & 59 \\
7 & & 1.8610 &  &  61 &  & 62 & & 8 & & 1.8839 &  &  60 &  & 80 \\
\end{tabular}
\end{center}

A logistic regression model that relates the probability 
of death to the level of dose  was estimated using $R$:
\begin{small}
\begin{verbatim}
      > beetle.fit<-glm(Deaths/Total~Dose,family=binomial, weights=Total)
      > summary(beetle.fit)
                  Estimate Std. Error z value Pr(>|z|)    
      (Intercept)  -40.971      3.658  -11.20   <2e-16 ***
      Dose          23.040      2.044   11.28   <2e-16 ***
\end{verbatim}
\end{small}


\begin{enumerate}
\item{}
Write down the fitted model in the logistic form ($\hat{\pi}= \ldots$) 
and in the logit form ($\mbox{logit} \, \hat{\pi}= \ldots$).
\item{}
According to the fitted model what effect does increasing the level 
of {\sf Dose} by 0.1 units have on the odds of death?
\item{}
What level of Dose corresponds to 
an estimated probability of death of 0.80?
\end{enumerate}
\marks{5}

\vspace{5mm}

\item
The following diagnostics plots were obtained for the fitted model 
in part (c). Note that the observation numbers in 
the plots match those in the data table.  

\centerline{\resizebox{0.75\textwidth}{0.75\textheight}{\includegraphics{short.pdf}}}


Do these plots indicate any problems with the fitted model?
(For each plot state the 
problem that it is used detect and state whether or not that problem is 
present.) 
If you identify a problem 
briefly describe what action you would take to address it.
\marks{5}
\end{enumerate}
