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\begin{center}   
{\Large \textbf{STA 441s20 Formulas}}\\   
\vspace{10 mm}
\end{center}

\noindent
\renewcommand{\arraystretch}{2.0}
\begin{tabular}{lcc}
$y = \beta_0 + \beta_1 x_1 + \cdots + \beta_{p-1} x_{p-1} + \epsilon$ 
& $SST=SSR+SSE$ &
$R^2 = \frac{SSR}{SST}$
\\
$a = \frac{R^2_F - R^2_R}{1-R^2_R}$ 
& $a = \frac{sF}{n-p+sF}$ & 
$F = \left( \frac{n-p}{s} \right) \left( \frac{a}{1-a} \right)$
\\
\multicolumn{3}{l}{\parbox{6.5in}{If an overall test has $s$ numerator degrees of freedom and critical value $c$, the   critical value of a Scheff\'e follow-up test with $r$ degrees of freedom is $\left(\frac{s}{r}\right) \cdot c$.}} \\ 
&& \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\ln\left(\frac{\pi}{1-\pi}\right) = \beta_0 + \beta_1 x_1 + \cdots + \beta_{p-1} x_{p-1} = L$
&&
$\pi = \frac{e^L} {1+e^L}$
\\
&& \\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\ln\left(\frac{\pi_1}{\pi_3} \right ) = 
     \beta_{0,1} + \beta_{1,1} x_1 + \ldots + \beta_{p-1,1} x_{p-1} = L_1$
&&
$\pi_1  = \frac{e^{L_1}}{1+e^{L_1}+e^{L_2}}$
\\
$\ln\left(\frac{\pi_2}{\pi_3} \right ) = 
     \beta_{0,2} + \beta_{1,2} x_1 + \ldots + \beta_{p-1,2} x_{p-1} = L_2$
&& 
$\pi_2 = \frac{e^{L_2}}{1+e^{L_1}+e^{L_2}} $
\\
&&
$\pi_3 = \frac{1}{1+e^{L_1}+e^{L_2}}$
\\
\end{tabular}
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\vspace{5mm}


\begin{displaymath}
 \boldsymbol{\mu} = 
     \left( \begin{array}{c}
        \mu_1  \\
        \mu_2 \\
        \vdots     \\
        \mu_k  \\  
    \end{array} \right) =
     \left( \begin{array}{c}
        E[y_1|\textbf{X=x}]  \\
        E[y_2|\textbf{X=x}] \\
        \vdots     \\
        E[y_k|\textbf{X=x}]  \\  
    \end{array} \right) =
     \left( \begin{array}{c c c}
       \beta_{0,1} + \beta_{1,1}x_1 + & \cdots & + \beta_{p-1,1}x_{p-1}   \\
       \beta_{0,2} + \beta_{1,2}x_1 + & \cdots & + \beta_{p-1,2}x_{p-1}   \\
                \vdots        & \vdots &         \vdots         \\
       \beta_{0,k} + \beta_{1,k}x_1 + & \cdots & + \beta_{p-1,k}x_{p-1}   \\
    \end{array} \right) 
\end{displaymath}
\vspace{3mm}

\begin{center}
\begin{tabular}{cccc} 
Unknown   &  Compound Symmetry & Autoregressive \\
(\texttt{type=un}) &       (\texttt{type=cs})    &  (\texttt{type=ar(1)}) \\
&& \\
    $\left( \begin{array}{c c c c}
                 \sigma^2_1 & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} \\
                 \sigma_{1,2} & \sigma^2_2 & \sigma_{2,3} & \sigma_{2,4} \\
                 \sigma_{1,3} & \sigma_{2,3} & \sigma^2_3 & \sigma_{3,4} \\
                 \sigma_{1,4} & \sigma_{2,4} & \sigma_{3,4} & \sigma^2_4 
    \end{array} \right)$
&   $\left( \begin{array}{c c c c}
                 \sigma^2+\sigma^2_1 & \sigma^2_1 & \sigma^2_1 & \sigma^2_1 \\
                 \sigma^2_1 & \sigma^2+\sigma^2_1 & \sigma^2_1 & \sigma^2_1 \\
                 \sigma^2_1 & \sigma^2_1 & \sigma^2+\sigma^2_1 & \sigma^2_1 \\
                 \sigma^2_1 & \sigma^2_1 & \sigma^2_1 & \sigma^2+\sigma^2_1 
    \end{array} \right)$
&   $\sigma^2
     \left( \begin{array}{c c c c}
                 1 & \rho & \rho^2 & \rho^3 \\
                 \rho & 1 & \rho & \rho^2 \\
                 \rho^2 & \rho & 1 & \rho \\
                 \rho^3 & \rho^2 & \rho & 1 
    \end{array} \right)$   \\
\end{tabular}
\end{center}

\vspace{15mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\vspace{3mm} \hrule \vspace{3mm}

\noindent
This formula sheet was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto Mississauga. 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:
\begin{center}
\href{http://www.utstat.toronto.edu/~brunner/oldclass/441s20} {\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/441s20}}
\end{center}

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