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{\Large \textbf{STA 302f15 Assignment Eleven}}\footnote{Copyright information is at the end of the last page.}
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\noindent
 Problem~\ref{computer} uses R. Please bring your printout for Problem~\ref{computer} to the quiz. \textbf{Do not write anything on the printout in advance of the quiz, except possibly your name and student number.} The other questions are preparation for the quiz, and are not to be handed in. 

\begin{enumerate} 


\item For the general linear model with normal errors, 
    \begin{enumerate}
        \item What is the distribution of $\mathbf{C}\widehat{\boldsymbol{\beta}}$? Note $\mathbf{C}$ is $q \times (k=0,1)$, and the rows of $\mathbf{C}$ are linearly independent.
        \item If $H_0: \mathbf{C}\boldsymbol{\beta} = \mathbf{t}$ is true, what is the distribution of $\frac{1}{\sigma^2}(\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})^\prime
            (\mathbf{C}(\mathbf{X}^\prime \mathbf{X})^{-1}\mathbf{C}^\prime)^{-1}
            (\mathbf{C}\widehat{\boldsymbol{\beta}}-\mathbf{t})$? Please locate support for your answer on the formula sheet.  For full marks, don't forget the degrees of freedom.
        \item What other facts on the formula sheet allow you to establish the $F$ distribution for the general linear test? The distribution is \emph{given} on the formula sheet, so of course you can't use that. In particular, how do you know numerator and denominator are independent?
    \end{enumerate}

\item \label{tsq} Suppose you wish to test the null hypothesis that a \emph{single} linear combination of regression coefficients is equal to zero. That is, you want to test $H_0: \mathbf{a}^\prime\boldsymbol{\beta} = 0$. Referring to the formula sheet, verify that $F=T^2$. Show your work. 

\item Starting from the formula sheet, show that the $F$ test for comparing full and reduced models may be written
\begin{displaymath}
    F = \left( \frac{p}{1-p} \right) \left( \frac{n-k-1}{q}  \right),
\end{displaymath}
where $p = \frac{R^2(full)-R^2(reduced)}{1-R^2(reduced)}$. 
Show your work. It may help to compare  $SST$ from the full model to $SST$  from the reduced model before you begin the calculation. 

\item \label{a2} That quantity denoted by $p$ in the last question has a useful interpretation. It's the proportion of \emph{remaining} variation in the dependent variable that is explained when the independent variables in the second set are added to the model. That is, the variables in the reduced model explain $R^2(reduced)$, so they fail to explain $1-R^2(reduced)$. Then the variables in the second set are added to the reduced model, yielding the full model --- and $R^2$ goes up. The quantity $p$ expresses this improvement as a proportion of what improvement was possible. Of course it is not the same as the $p$-value.

Derive another formula for $p$, writing $p$ in terms of $F$, $n$, $k$ and $q$. Show your work. This formula can give an idea of how strong a set of results is, when all you are given is an $F$ or $t$ statistic and the degrees of freedom.  The answer is on the formula sheet; prove it. 

\item \label{moresat} In an extended version of the SAT data, the dependent variable is first-year university Grade Point Average (GPA) again. The independent variables are
    \begin{itemize}
        \item[$x_1=$] Verbal SAT score
        \item[$x_2=$] Math SAT score
        \item[$x_3=$] High school Grade Point Average
        \item[$x_4=$] Mother's education, in years
        \item[$x_5=$] Father's education, in years
        \item[$x_6=$] Total family income,
    \end{itemize}
and also Location of the family home: City, Suburbs or Country.
    \begin{enumerate}
        \item First, write the regression equation. It is up to you which dummy variable variable scheme you use, as long as the regression planes are parallel.
        \item Make a table with one row for each location of the family home, showing how your dummy variables are defined. Make one more column showing $E(y|\mathbf{x})$ for each location.
        \item For each of the following questions, do three things: Give the null hypothesis in the form of a statement about the $\beta$ values, Give  the $\mathbf{C}$ and $\mathbf{t}$ matrices in $H_0: \mathbf{C}\boldsymbol{\beta} = \mathbf{t}$, and Give $E(y|\mathbf{x})$ for the reduced model (note that expected $y$ for the full model is always the same).
            \begin{enumerate}
                \item Correcting for all other variables, is location of the family home related to first-year GPA?
                \item Controlling for all other variables, is either Verbal SAT score or Math SAT score (or both) related to GPA?
                \item When you allow for all the other variables, is family income a useful predictor of GPA?
                \item Controlling for all other variables, does expected GPA change faster as a function of Verbal SAT, or does it change faster as a function of Math SAT?
                \item Once you correct for the two SAT scores and High School marks, do any of the family variables matter? 
                \item Correcting for all other variables, does expected GPA change faster as a function of Mother's education, or does it change faster as a function of father's education?
                \item Holding all the other variables constant at fixed values, is Math SAT related to first-year university GPA?
                \item Once you allow for location of the family home, do any of the other predictors matter?
            \end{enumerate}

    \end{enumerate}

\pagebreak

\item \label{computer} This is an expanded version of the \texttt{statclass} data used in Assignments~6 and~9. 
 At the R prompt, type 

{\scriptsize
\begin{verbatim}
statclass = read.table("http://www.utstat.utoronto.ca/~brunner/data/legal/LittleStatClassData2.txt")
\end{verbatim}
} % End size
Do \texttt{head(statclass)} to see what's there. Fit a regression model in which the dependent variable is mark on the Final Exam, and the independent variables are Sex, Race, Quiz Average, Computer Average, and mark on the Midterm test. 
    \begin{enumerate}
        \item What is the predicted Final Exam score for a Male student from Race $A$ with a Quiz average of 8.5, a Computer average of 5, and a Midterm mark of 60\%? The answer is a number. Be able to do this kind of thing on the quiz with a calculator from the output of \texttt{summary}.
        \item Obtain a 95\% prediction interval for the student described in the previous question. Do it the easy way.
        \item Controlling for the other independent variables, are there any differences in the average performance of students from the different racial groups? 
            \begin{enumerate}
                \item Please test the main hypothesis 2 different ways: using the general linear test, and also the full-reduced approach. Are your $F$-statistics the same? 
                \item What proportion of the variation in Final Exam mark is explained by the independent variables in the reduced model? The answer is a number from your printout.
                \item What proportion of the variation in Final Exam mark is explained by the independent variables in the full model? The answer is a number from your printout.
                \item After allowing for the other independent variables, what proportion of the \emph{remaining} variation in Final Exam score is explained by race? For reference, we are talking about the quantity $p$ from Question~\ref{a2}. There are two formulas for $p$ on the formula sheet. Get both of them from the numbers on your printout.
                \item Three comparisons are of interest, race $A$ versus $B$, $A$ versus $C$ and $B$ versus $C$. You already have two of them. Do the third one. Now you have three test statistics and associated $p$-values.
                \item In plain, non-statistical language, what do you conclude? 
            \end{enumerate}
        \item Allowing for other predictors, is the student's sex related to mark on the Final Exam? Give the test statistic, the $p$-value, and answer the question in plain, non-statistical language.

\pagebreak

        \item When other variables are held constant, an increase of one point (out of 10) in the Quiz Average results in an increase of \underline{\hspace{10mm}} in predicted mark on the Final Exam.
        \item Controlling for other independent variables in the model, is Quiz Average a useful predictor of mark on the Final Exam? Give the test statistic, the $p$-value, and answer the question in plain, non-statistical language. For full marks, give a directional conclusion if possible. You will not be reminded of this on our Final Exam.
        \item  After allowing for the other independent variables, what proportion of the \emph{remaining} variation in Final Exam score is explained by Quiz Average? This is a number you could obtain with a calculator from the \texttt{summary} output of the full model. There is no need to literally fit the reduced model, though you may check your answer that way if you wish. If you are stuck, look at Question~\ref{tsq}.
        \end{enumerate}

\end{enumerate}

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical 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/302f15} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f15}}

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