%         Tables to go with STA441 Regression with SAS: Cars

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\title{Tables of $E(Y|\mathbf{x})$ for SAS Example Two\footnote{See last slide for copyright information.}}
\subtitle{STA441 Spring 2020}
\date{} % To suppress date

\begin{document}

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\begin{frame}
\frametitle{Just Country}
\begin{itemize}
    \item $c_1 = 1$ if Country = U.S., zero otherwise
    \item $c_2 = 1$ if Country = Europe, zero otherwise
    \item $E[Y|\boldsymbol{X}=\boldsymbol{x}] = \beta_0 + \beta_1c_1 + \beta_2 c_2$
\end{itemize}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
\textbf{Country} & $c_1$ & $c_2$ &  $E(Y|\mathbf{x}) = \beta_0 + \beta_1c_1 + \beta_2 c_2$ \\ \hline
United States   &   1   &   0   &  $\mu_1$ =     $\beta_0 + \beta_1$    \\ \hline
Europe          &   0   &   1   &  $\mu_2$ =     $\beta_0 + \beta_2$    \\ \hline
Japan           &   0   &   0   &  $\mu_3$ =     $\beta_0$              \\ \hline
\end{tabular}
\end{center}} 
\end{frame}

\begin{frame}
\frametitle{With weight and length}
\begin{itemize}
    \item $c_1 = 1$ if Country = U.S., zero otherwise
    \item $c_2 = 1$ if Country = Europe, zero otherwise
    \item $x_1$ = weight
    \item $x_2$ = length
    \item $E[Y|\boldsymbol{X}=\boldsymbol{x}] = \beta_0 + \beta_1c_1 + \beta_2 c_2 + \beta_3 x_1 + \beta_4 x_2$. \pause 
\end{itemize}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
\textbf{Country} & $c_1$ & $c_2$ &  \hspace{15mm} $E(Y|\mathbf{x}) $\\ \hline
United States   &   1   &   0   &  $\mu_1$ = $(\beta_0+\beta_1)+\beta_3x_1+\beta_4x_2$    \\ \hline
Europe          &   0   &   1   &  $\mu_2$ = $(\beta_0+\beta_2)+\beta_3x_1+\beta_4x_2$    \\ \hline
Japan           &   0   &   0   &  $\mu_3$ = ~~~~~$\beta_0$~~~~~$+\beta_3x_1 +\beta_4x_2$              \\ \hline
\end{tabular}
\end{center}} \pause 
Parallel regression planes.
\end{frame}

\begin{frame}
\frametitle{With product terms for interactions}
\begin{itemize}
    \item $c_1 = 1$ if Country = U.S., zero otherwise
    \item $c_2 = 1$ if Country = Europe, zero otherwise
    \item $x_1$ = weight
    \item $x_2$ = length
    \item $E[Y|\boldsymbol{X}=\boldsymbol{x}] = \beta_0 + \beta_1c_1 + \beta_2 c_2 + \beta_3 x_1 + \beta_4 x_2$ \\
\hspace{21mm} $+~ \beta_5c_1x_1 + \beta_6c_2x_1 + \beta_7c_1x_2 + \beta_8c_2x_2$. 
\end{itemize}
{\begin{center}
{\small
\begin{tabular}{|c|c|c|l|} \hline
\textbf{Country} & $c_1$ & $c_2$ &  \hspace{15mm} $E(Y|\mathbf{x}) $\\ \hline
United States   &   1   &   0   &  $\mu_1$ = $(\beta_0+\beta_1) + (\beta_3+\beta_5)x_1 + (\beta_4+\beta_7)x_2$    \\ \hline
Europe          &   0   &   1   &  $\mu_2$ = $(\beta_0+\beta_2) + (\beta_3+\beta_6)x_1 + (\beta_4+\beta_8)x_2$    \\ \hline
Japan           &   0   &   0   &  $\mu_3$ = ~~~~~$\beta_0$~~~~~$+~~~~~\beta_3~~~~x_1 +~~~~~\beta_4~~~~~x_2$              \\ \hline
\end{tabular}
} % End size
\end{center}} 
Non-parallel regression planes.
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\frametitle{Copyright Information}

This slide show was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistics, 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/441s20} {\footnotesize \texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/441s20}}

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