\documentclass[11pt]{article} 
%\usepackage{amsbsy} % for \boldsymbol and \pmb 
\usepackage{graphicx} % To include pdf files!
\usepackage{amsmath}
\usepackage{amsbsy}
\usepackage{amsfonts}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref} % For links
\usepackage{fullpage}
%\pagestyle{empty} % No page numbers


\begin{document}
%\enlargethispage*{1000 pt} 

\begin{center}   
{\Large \textbf{STA 312f23 Assignment Seven}}\footnote{This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/brunner/oldclass/312f23} {\texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f23}}}
\vspace{1 mm}
\end{center}

\noindent
The paper and pencil part of this assignment is not to be handed in. It is practice for Quiz~7 on November 3d. The R part may be handed in as part of the quiz. \textbf{Bring hard copy of your printout to the quiz}. Do not write anything on your printout in advance except possibly your name and student number. %\vspace{5mm}

\begin{enumerate} 

\item You have done these exercises before. It's just a warm-up. For the Weibull distribution (see formula sheet), show that
    \begin{enumerate}
        \item $E(T^k) = \frac{\Gamma(1+\frac{k}{\alpha})}{\lambda^k}$
        \item Median = $\frac{[\log(2)]^{1/\alpha}}{\lambda}$
        \item $h(t) = \alpha\lambda^\alpha t^{\alpha-1}$
    \end{enumerate}

\item \label{standGumbel} Let $T$ have a standard exponential distribution (that is, $\lambda=1$), and let $Z=\log(T)$. Derive the density of $Z$. We will call this the \emph{standard Gumbel}, even though the term is also applied to $-\log(T)$.

\item Let $Z$ have the standard Gumbel density of Question~\ref{standGumbel}. 
    \begin{enumerate}
        \item Find the moment-generating function of $Z$, $M_z(t)=E(e^{Zt})$. 
        \item Differentiating the moment-generating function and setting $t=0$, show that $E(Z) = \Gamma^\prime(1)$ (derivative of the gamma function).
        \item Find the median of $Z$.
        \item Find the mode of $Z$.
    \end{enumerate}

\item \label{genGumbel} Let $Z$ have the standard Gumbel density of Question~\ref{standGumbel}, and let $Y = \sigma Z + \mu$, where $\sigma>0$. 
    \begin{enumerate}
        \item Find the density of $Y$. 
        \item Find the expected value of $Y$.
        \item Given that the variance of the standard Gumbel is $\frac{\pi^2}{6}$ (you don't have to prove this), what is the variance of $Y$?
        \item Find the median of $Y$.
        \item Find the mode of $Y$.
    \end{enumerate}

\newpage

\item Let $T \sim$ Weibull($\alpha,\lambda$), and $Y=\log(T)$. 
    \begin{enumerate}
        \item First, find the density of $Y$. 
        \item Now re-parameterize, meaning express the parameters in a different, equivalent way. Let $\alpha = \frac{1}{\sigma}$ and $\lambda = e^{-\mu}$. Write the resulting density of $Y$.
        \item If necessary, re-write the density of $Y$ so that it has the form of the Gumbel density of Question~\ref{genGumbel}.
    \end{enumerate}
The point of all this is that if you believe the distribution of a set of failure time data could be Weibull (a popular choice), you can log-transform the data and apply a Gumbel model. The Gumbel distribution may be preferable because the parameters $\mu$ and $\sigma$ are easy to interpret.

\item High School History classes from across Ontario are randomly assigned to either a discovery-oriented or a memory-oriented curriculum in Canadian history. At the end of the year, the students are given a standardized test and the median score of each class is recorded. Please consider a regression model with these variables:
    \begin{itemize}
        \item[$X_1$] Equals 1 if the class uses the discovery-oriented curriculum, and equals 0 the class it uses the memory-oriented curriculum.
        \item[$X_2$] Average parents' education for the classroom
        \item[$X_3$] Average parents' income for the classroom
        \item[$X_4$] Number of university History courses taken by the teacher
        \item[$X_5$] Teacher's final cumulative university grade point average
        \item[$Y~$] Class median score on the standardized history test.
    \end{itemize}
The full regression model has $E[Y|\mathbf{x}] = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_4 + \beta_5x_5$. For each question below, give the null hypothesis in terms of $\beta$ values. Also give $E[Y|\mathbf{x}]$ for the \emph{restricted} (reduced) model you would use to answer each question. Don't re-number the variables. 
    \begin{enumerate}
        \item If you control for parents' education and income and for teacher's university background, does curriculum type affect test scores? 
        \item Allowing for parents' education and income and for curriculum type, is teacher's university background (two variables) related to students' test performance?
        \item Correcting for teacher's university background and for curriculum type, are parents' education and income (considered simultaneously) related to students' test performance?
        \item Adjusting for curriculum type, teacher's university background and parents' education, is parents' income related to students' test performance?
    \end{enumerate}

\pagebreak

\item \label{workout} In a study comparing the effectiveness of different exercise programmes, volunteers were randomly assigned to one of three exercise programmes ($A$, $B$, $C$) or put on a waiting list and told to work out on their own. Aerobic capacity is the body's ability to process oxygen. Aerobic capacity was measured before and after 6 months of participation in the program (or 6 months of being on the waiting list). The response variable was improvement in aerobic capacity. The explanatory variables were age (a covariate) and treatment group. Note that treatment group is a variable with four categories; wait list is one of them. 

Consider a regression model with an intercept, and no interaction between age and treatment group.
            \begin{enumerate}
                \item Make a table showing how you would set up indicator dummy variables for treatment group. Make Waiting List the reference category
                \item Write the regression model. Please use $x$ for age, and make its regression coefficient $\beta_1$.
                \item In terms of $\beta$ values, what null hypothesis would you test to find out whether, allowing for age, the three exercise programmes differ in their effectiveness?
                \item Write the null hypothesis for the preceding question as $H_0:  \mathbf{L}\boldsymbol{\beta}=\mathbf{0}$. Just give the $\mathbf{L}$ matrix. 
                \item In terms of $\beta$ values, what null hypothesis would you test to find out whether Programme $B$ was better than the waiting list? 
                \item In terms of $\beta$ values, what null hypothesis would you test to find out whether Programmes $A$ and $B$ differ in their effectiveness? 
                \item Suppose you wanted to estimate the difference in average benefit between programmes $A$ and $C$ for a 27 year old participant. Give your answer in terms of $\widehat{\beta}$ values.
                \item Is it safe to assume that age is independent of the other explanatory variables? Answer Yes or No and briefly explain.
            \end{enumerate}


% 305f15 Final exam has a nice version of this question with proper spacing for the answers. 
\item \label{computer} Pigs are routinely given large doses of antibiotics even when they show no signs of illness, to protect their health under unsanitary conditions. Pigs were randomly assigned to one of three antibiotic drugs. Dressed weight (weight of the pig after slaughter and removal of head, intestines and skin) was the response variable.  The explanatory variables are Drug type, Mother's live adult weight and Father's live adult weight.


Data are in the file
\href{http://www.utstat.toronto.edu/brunner/data/legal/pigweight.data.txt} {\texttt{pigweight.data.txt}}. You can get a copy with
{\footnotesize
\begin{verbatim}
oink = read.table("http://www.utstat.toronto.edu/~brunner/data/legal/pigweight.data.txt").
\end{verbatim}
} % End size

    \begin{enumerate}
        \item Write the regression equation for the full model, including $\epsilon_i$. 
        \item Make a table with one row for every drug, and with columns showing how the dummy variables were defined. Make another column giving $E(y|\mathbf{x})$ for each drug.
        \item Fit the model (meaning estimate the parameters), and display the results of \texttt{summary}. What proportion of variation in the response variables is explained by the explanatory variables?
       \item Predict the dressed weight of a pig getting Drug 2, whose mother weighed 140 pounds, and whose father weighed 185 pounds. Your answer is a single number. 
        \item This parallel planes regression model specifies that the differences in expected weight for the different drug treatments are the same for every possible combination of mother's weight and father's weight. Give a 95\% confidence interval for the difference in expected weight between drug treatments 2 and 3. The final answer is a pair of numbers, a lower confidence limit and an upper confidence limit. There is an easy way and a less easy way. The lecture slides illustrate both ways.
        \item In symbols, give the null hypotheses you would test to answer the following questions. Your answers are statements involving the $\beta$ values from your regression equation. 
        \begin{enumerate} % There were more questions in an earlier draft.
            \item Allowing for mother's weight and father's weight, does type of drug have an effect on the expected weight of a pig? 
            \item Controlling for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 1 or Drug 2?
            \item  Adjusting for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 1 or Drug 3? 
            \item  Correcting for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 2 or Drug 3? 
        \end{enumerate}
        \item For each of the questions below, give the value of the $t$ or $F$ statistic (a number from your printout), and indicate whether or not you reject the null hypothesis. The numbers may or may not be part of the default output from \texttt{summary}.
        \begin{enumerate}
            \item Controlling for mother's weight and father's weight, does type of drug have an effect on the expected weight of a pig?
            \item Adjusting for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 1 or Drug 2?
            \item  Allowing for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 1 or Drug 3?
            \item  Controlling for mother's weight and father's weight, which drug helps the average pig gain more weight, Drug 2 or Drug 3?
            \item Allowing for which drug they were given, does expected weight of a pig increase faster as a function of the mother's weight, or does it increase faster as a function of the father's weight?
        \end{enumerate}

        \item We can assume that farmers want their pigs to weigh a lot. In plain, non-statistical language, can you offer some advice to a farmer based on these data? Remember, the farmer must be able to understand your answer or it is worthless.

    \end{enumerate} % End computer question

    \noindent
    Please bring your printout to the quiz. \textbf{Your printout should show \emph{all} R input and output, 
    and \emph{only} R input and output}. Do not write anything on your printouts except your name and student 
    number. 




% \pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{enumerate} % End of all the questions



\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%