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\begin{document}

\mbox{}

\vspace{-.8in}


\begin{center}
\bf \large

Ph.\ D.\ Qualifying Exam\\
Saturday, August 21, 2004

\end{center}

\vspace{.1in}

Please submit solutions to at most {\bf seven} problems.
You have four hours.  No one is expected to answer all the
problems correctly.  Partial credit will be given.  All problems
are worth an equal amount of credit.

\vspace{.2in}

{\bf Put your solution to each problem on a separate sheet of paper.}

\vspace{.2in}

\topic{Applied Statistics}


\QQ Consider the following fractional factorial experimental
design:
 
 \vspace{0.1in}
 
 \begin{tabular}{c|ccccccc|}\hline
 & \multicolumn{7}{c|}{$d_1$}  \\
 Run & 1 & 2 & 3 & 4=12 & 5=13 & 6=23 & 7=123 \\ \hline
 1   &$-$&$-$&$-$ & +    &  +   &   +  &  $-$  \\
 2   &$-$&$-$& +  & +    & $-$  &  $-$ &   +   \\
 3   &$-$& + &$-$ &$-$   &  +   &  $-$ &   +   \\
 4   &$-$& + & +  &$-$   & $-$  &   +  &  $-$  \\
 5   & + &$-$&$-$ &$-$   & $-$  &   +  &   +   \\
 6   & + &$-$& +  &$-$   &  +   &  $-$ &  $-$  \\
 7   & + & + &$-$ & +    & $-$  &  $-$ &  $-$  \\
 8   & + & + & +  & +    &  +   &   +  &   +   \\ \hline
 & \multicolumn{7}{c|}{$d_2$}  \\
 Run &$-$1 & $-$2 & $-$3 & $-4$ & $-5$ & $-6$ & $-7$ \\ \hline
 9   & + & + & +  &$-$   & $-$  &  $-$ &   +   \\
 10  & + & + &$-$ &$-$   &  +   &   +  &  $-$  \\
 11  & + &$-$& +  & +    & $-$  &   +  &  $-$  \\
 12  & + &$-$&$-$ & +    &  +   &  $-$ &   +   \\
 13  &$-$& + & +  & +    &  +   &  $-$ &  $-$  \\
 14  &$-$& + &$-$ & +    & $-$  &   +  &   +   \\
 15  &$-$&$-$& +  &$-$   &  +   &   +  &   +   \\
 16  &$-$&$-$&$-$ &$-$   & $-$  &  $-$ &  $-$  \\  \hline
 \end{tabular}
 
\begin{parts}
\PP 
 What is the defining relation of design $d_1$? What is the resolution of design $d_1$?
 List the aliases of main effects 1 and 2 in this design.

\PP
Design $d_2$ is obtained  by ``folding over" design $d_1$. 
It can be shown that $d_2$ has generators $4=-12$, $5=-13$, $6=-23$, and $7=123$.
What is the defining relation of design $d_2$? What is the resolution of design $d_2$?
 List the aliases of main effects 1 and 2 in  design $d_2$.

\PP
Denote by $d$ the augmented design consisting of $d_1$ and $d_2$. The augmented design $d$ has
generators $5=234$, $6=134$, and $7=123$.
What is the defining relation of design $d$? What is the resolution of design $d$?
 List the aliases of main effects 1 and 2 in  design $d$.

\end{parts}

\pagebreak

\mbox{}

\vspace{-.8in}

\QQ In a complete factorial experiment, consider the following
random-effects model:
\begin{eqnarray*}
Y_{ijk} & = & \mu +\alpha_i + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}, \\
& & \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1,\ldots, r, 
\end{eqnarray*}
where  $\mu$ is the overall mean, the random effects $\alpha_i$, $\beta_j$,
and $(\alpha\beta)_{ij}$ are assumed to be independent and normally
distributed with means of zero and variances $\sigma^2_{\alpha}$, $\sigma^2_{\beta}$,
and $\sigma^2_{\alpha\beta}$, respectively. The random errors $\{\epsilon_{ijk}\}$ are assumed to be independent and normally distributed with mean zero and variance $\sigma^2$.
Furthermore, $\{\alpha_i\}$, $\{\beta_j\}$, $\{(\alpha\beta)_{ij}\}$,  and $\{\epsilon_{ijk}\}$
are assumed to be independent of one another.


\begin{parts}
\PP
What are the distributions of $\bar{Y}_{ij\cdot}$, $\bar{Y}_{i\cdot\cdot}$, 
and  $\bar{Y}_{\cdot j \cdot}$?
Find the covariance  Cov($\bar{Y}_{ij\cdot}, \bar{Y}_{uv\cdot}$).


\PP
Find unbiased estimates for the variances  $\sigma^2_{\alpha\beta}$
and $\sigma^2$ based on the observations $\{Y_{ijk}, \hspace{0.2in} i=1,\ldots, a;
\hspace{0.1in} j=1, \ldots, b; \hspace{0.1in} k=1, \ldots,, r\}$.

\end{parts}


\QQ In a non-linear regression analysis, the true model has the
form:
 \begin{displaymath} 
y_i = f(\beta_0 +\beta_1 x_{1i} + \cdots + \beta_p x_{pi}) +
\epsilon_i
\end{displaymath}
 where $f(\cdot)$ is an unknown function and $\{\epsilon_i\}$ are
 iid random variables.  Suppose that instead of the true model, a
 linear model with the form
\begin{displaymath} 
y_i = \beta_0 +\beta_1 x_{1i} + \cdots + \beta_p x_{pi} +
\epsilon_i
\end{displaymath}
is fitted to the data and the residuals $\{\hat{\epsilon}_i\}$
are calculated.

\begin{parts}

\PP
 What would a plot of $\hat{y}_i$ versus $\hat{\epsilon}_i$ look like in this problem?
 
\PP
  Give two ways to improve your fitting and state your justifications.

\end{parts}



\QQ An experiment analyzes imperfection rates for two processes A
and B used in fabricating wafers for computer chips. Let $Y_1$
and $Y_2$ be the numbers of imperfections for Processes A and B,
respectively. $Y_1$ and $Y_2$ are independent Poisson random
variables with means $\mu_A$ and $\mu_B$, respectively.
 
\begin{parts}
  
  \PP Suppose a log-linear model $\log(\mu) = \alpha +\beta x$ is
  fitted with $x=1$ for treatment B and $x=0$ for treatment A.
  Show that $\exp(\beta) = \mu_B/\mu_A$.  Suggest a test for
  testing the null hypothesis $H_0: \hspace{0.1in} \mu_A =
  \mu_B$.
 
\PP
Show that $Y= Y_1 + Y_2$ is Poisson with mean $\mu_A + \mu_B$. Conditionally, 
show that $Y_1|(Y=n)$ is binomial with $\pi = \mu_A/(\mu_A + \mu_B)$.

\PP
Based on the result in (b), suggest a test for testing the
 null hypothesis $H_0: \hspace{0.1in} \mu_A = \mu_B$.

\end{parts}



\topic{Probability}



\QQ Let $N$ be a positive integer-valued random variable and let
$X_1,X_2,\ldots $ be random variables with finite expectations.

\begin{parts}

\PP Define $X_N(\omega)=X_{N(\omega)}(\omega)$.
Show that $X_N$ is a random variable.

\PP Suppose that $N$ is bounded and let $\{N_n$, $n=1,2,\ldots\}$
be a sequence of positive integer-valued random variables such
that $N_n\nearrow N$ almost surely.  Show that
$$\lim _{n\to \infty} E(X_{N_n}) =E(X_N).$$

\end{parts}




\QQ Let $X_1,X_2,\ldots$ be identically distributed random
variables.  Show that
\begin{parts}
  
  \PP $\displaystyle \frac{X_n}{n} \to 0$ in probability.
  
  \PP If the random variables have finite mean, then
  $\displaystyle \frac{X_n}{n} \to 0$ almost surely.
  
  \PP Part (b) is false if the condition that the random
  variables have finite mean is omitted.

\end{parts}


\QQ Let $P_0,\, P_1$ be two probability measures on $({\cal X},
{\cal B})$.  They are said to be singular with respect to one
another if there is a set $A \in {\cal B}$ such that
$P_0(A)=1, P_1(A)=0$, which will also mean that
$P_0(A^c)=0,\, P_1(A^c)=1$.

\begin{parts}

\PP
  Suppose that there is a sequence of random variables $\{T_n\}$
  on $({\cal X},{\cal B})$ such that
  \begin{eqnarray*}
   &&  T_n \rightarrow 0\,\  a.s. {\rm\ under \ } P_0\,, {\rm \ and}\\
   &&  T_n \rightarrow 1\,\  a.s. {\rm\ under \ } P_1.
  \end{eqnarray*}
Show that $P_0$ and $P_1$ are singular with respect to one
another.

\PP Show that the same conclusion is true if
\begin{eqnarray*}
&& T_n \rightarrow 0\, {\rm\ in \ probability 
\  under \ } P_0\,, {\rm \ and}\\
&& T_n \rightarrow 1\, {\rm\ in \ probability \ under \ } P_1.
\end{eqnarray*}

\end{parts}

\pagebreak

\topic{Theoretical Statistics}

\QQ Consider the linear model $Y=X \beta+e$, where $e\sim N(0,
\sigma^2I)$. Let $M$ be the perpendicular projection operator
onto $C(X)$, the range or column space of $X$. Define $\hat e
=(I-M)Y$.
 
\begin{parts}

\PP Find $\mbox{Cov}(\hat e,MY)$
 
\PP Find $E(\hat e^T\hat e)$

\PP Consider the following particular regression model:
\begin{displaymath}
  \begin{array}{cccc}
Y_1=\beta_0+\beta_1+e_1\,, & Y_2=\beta_0+\beta_1+e_2\,, &
Y_3=\beta_0+\beta_2+e_3\,, & Y_4=\beta_0+\beta_2+e_4\,. 
  \end{array}
\end{displaymath}
Show that the parameter $\beta_1-\beta_2$ is estimable.
  
\end{parts}

\QQ Let $X_1, X_2, \ldots, X_n$ be a random sample from the
density (pdf)
\begin{displaymath}
  f(x \mid \theta) = 
\left\{
    \begin{array}{ll}
\theta/x^2 & \mbox{for $x \geq \theta$}\\
0 & \mbox{for $x < \theta$}
    \end{array}
\right.
\end{displaymath}
where $\theta>0$.

\begin{parts}

  \PP Find the MLE of $\theta$.
  
  \PP Suppose $\theta$ has a prior distribution with density $
  \pi(\theta) = 1/\theta^2$ for $\theta \geq 1$ (and
  $\pi(\theta)=0$ for $\theta<1$).  Find the posterior mean of
  $\theta$.  Assume $n \geq 2$.

\end{parts}

\QQ Let $X_1, X_2, \ldots, X_n$ be iid $N(\theta,\theta^2)$ where
$\theta>0$.

\begin{parts}
  
  \PP What is the Cram\'er-Rao lower bound for the variance of an
  unbiased estimator of $\theta$\,?  For an unbiased estimator of
  $\theta^2$\,?
  
  \PP Use your answers to part (a) to show that $\bar{x}$ and
  $s^2$ (the sample mean and variance) are {\bf not}
  asymptotically efficient estimators of $\theta$ and $\theta^2$,
  respectively.
  
\item[Hint:] Depending on how you do the problem, you may find
  the following facts to be useful.  If $Y \sim N(\mu,\sigma^2)$,
  then $E(Y-\mu)^4 = 3 \sigma^4$.  If $U \sim \chi^2_k$, then
  $\mbox{Var}(U) = 2k$.

\end{parts}

\pagebreak

\mbox{}

\vspace{-.6in}

\topic{Biostatistics}

\QQ
The results of a {\bf cohort} study examining the relationship
between gender and coronary heart disease (CHD) is summarized in
the following table. In this study, 4,541 individuals were
followed 10 years and it was determined which of these
individuals actually experienced CHD.  For example, 236 out of
2,009 males developed the disease, 1,773 males did not develop
the disease, etc.


\begin{center}
\begin{tabular}{llccc}
\hline && \multicolumn{2}{c}{CHD} &  \\
& & Yes & No & Total \\
\hline & Yes & 236 & 1,773 & 2,009\\
Male &&&&\\
& No &142 & 2,390 & 2,532\\
\hline & Total& 378 & 4,163 & 4,541\\
\end{tabular}
\end{center}


Provide definitions and results for each of the following
statistics. {(\bf You do not have to actually do the calculations,
just give the correct definitions and put the right numbers into
a formula.)}

\begin{parts}

\PP The relative risk of CHD for males.

\PP The odds ratio of CHD for males.

\PP Recall that there are several types (definitions) of
attributable risk.  Define any one and provide the correct
estimator
.
\end{parts}

\QQ The following data are the results of seven
placebo-controlled randomized studies of the effect of aspirin in
preventing death after myocardial infarction.  Each row of the
table provides data from a single clinical trial.  For example,
the MRC-1 study randomized 615 patients to receive aspirin
treatment and 624 patients to receive placebo treatment.  In the
MRC-1 study, 49 deaths occurred in the patients receiving aspirin
treatment and 67 deaths occurred among the patients receiving
placebo.


\begin{center}
\begin{tabular}{|r||r||r||r||r|}
\hline
           &    \multicolumn{2}{|c|}{Aspirin}           &    \multicolumn{2}{|c|}{Placebo}            \\
\hline & \multicolumn{2}{|c|}{Status at} &
\multicolumn{2}{|c|}{Status at} \\
& \multicolumn{2}{|c|}{Study End} & \multicolumn{2}{|c|}{Study End}  \\
\hline
     Study &      Alive &     Dead &      Alive &     Dead \\
     MRC-1 &        566 &         49 &        557 &         67 \\
       CDP &        714 &         44 &        707 &         64 \\
     MRC-2 &        730 &        102 &        724 &        126 \\
      GASP &        285 &         32 &        271 &         38 \\
     PARIS &        725 &         85 &        354 &         52 \\
      AMIS &       2021 &        246 &       2038 &        219 \\
    ISIS-2 &       7017 &       1570 &       6880 &       1720 \\
    \hline
\end{tabular}

\end{center}

\noindent 
Explain the steps you would go through to produce a meta-analysis
of these results.  {(\bf Again, you do not have to actually do
  any calculations.)}






\end{document}

