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\begin{center}   
{\Large \textbf{STA 312f2022 Formulas}}
\footnote{This formula sheet 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/312f22} {\texttt{http://www.utstat.toronto.edu/brunner/oldclass/312f22}}}\\
\vspace{1 mm}
\end{center}


\noindent
\hspace{-10mm}\begin{tabular}{lllll} 
$Pr(A|B) = \frac{Pr(A \cap B)}{Pr(B)}$ & ~~~~~ &
$Pr(A) = \sum_{j=1}^k Pr(A|B_j)Pr(B_j)$  & ~~~~~ &
$Pr(B|A) = \frac{Pr(A|B)Pr(B)}
                  {Pr(A|B)Pr(B)+Pr(A|B^c)Pr(B^c)}$  \\ 
 & & \\
\end{tabular}

\noindent
\hspace{-10mm}\begin{tabular}{lll} 
\textbf{Bernoulli} & ~~~~~ & $P(y) = \pi^y (1-\pi)^{1-y}$ for $y = 0,1$  \\
 & & \\
\textbf{Binomial}  & ~~~~~ & $P(y) = \binom{n}{y} \pi^y (1-\pi)^{n-y}$ for $y = 0, \ldots, n$  \\
 & & \\
\textbf{Poisson}   & ~~~~~ & $P(y) = \frac{e^{-\lambda} \lambda^y}{y!}$ for $y = 0, \ldots $  \\
 & & \\
\textbf{Hypergeometric} & ~~~~~ & $P(y) = \frac{ \binom{M}{y}\binom{N-M}{n-y} } {\binom{N}{n}}$,
where $\binom{a}{b}$ must make sense.
\end{tabular}

% 

\begin{center}
\hspace{-10mm}\begin{tabular}{cccl}
$Z_1 = \frac{\sqrt{n}(p-\pi_0)}{\sqrt{\pi_0(1-\pi_0)}}$ & 
$Z_2 = \frac{\sqrt{n}(p-\pi_0)}{\sqrt{p(1-p)}}$ &
$p \pm z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}$ &
\begin{minipage}{3in}
\begin{verbatim}
> qnorm(0.975)
[1] 1.959964
> qnorm(0.995)
[1] 2.575829
\end{verbatim}
\end{minipage}
\end{tabular}
\end{center}

\vspace{5mm}

\noindent
\hspace{-10mm}\begin{tabular}{lll} 
\textbf{Multinomial} & & \\
$P(n_1, \ldots, n_c) = \binom{n}{n_1~\cdots~n_c} \pi_1^{n_1} \cdots \pi_c^{n_c}$ & 
~~~~~ &
$\ell(\boldsymbol{\pi}) =  \prod_{i=1}^n \pi_1^{y_{i,1}} \pi_2^{y_{i,2}} \cdots \pi_c^{y_{i,c}}
                        = \pi_1^{n_1} \pi_2^{n_2} \cdots \pi_c^{n_c}$ \\ 
 & & \\
$G^2 = -2 \log \left(   
           \frac{\max_{\beta \in \mathscr{B}_0} \ell(\beta)}
                {\max_{\beta \in \mathscr{B}} \ell(\beta)}
           \right)
        = -2\log\left(\frac{\ell(\widehat{\beta}_0)}{\ell(\widehat{\beta})} \right)$ &
~~~~~ &
$G^2 = 2 \sum_{j=1}^c n_j\log \left(\frac{n_j}{n\widehat{\pi}_j}\right) 
        = 2 \sum_{j=1}^c n_j\log \left(\frac{n_j}{\widehat{\mu}_j}\right)$ \\
 & & \\
$X^2 = \sum_{j=1}^c \frac{(n_j-\widehat{\mu}_j)^2}{\widehat{\mu}_j}$ & 
~~~~~ &
$n_{i+} = \sum_{j=1}^J n_{ij}$ ~~ $n_{+j} = \sum_{i=1}^I n_{ij}$ ~~
$\widehat{\mu}_{ij} = \frac{n_{i+}n_{+j}}{n}$ \\
 & & \\
 Odds = $\frac{\pi}{1-\pi}$ & 
~~~~~ & 
$\theta = \frac{\pi_{11}\pi_{22}}{\pi_{12}\pi_{21}}$ \\
 & & \\
 
%%%%%%%%%%%%%%%%%%%%%%%%% Logistic regression %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Logistic regression} & & \\
$\log\left( \frac{\pi_i}{1-\pi_i} \right) = 
\beta_0 + \beta_1 x_{i,1} + \ldots + \beta_k x_{i,k} = 
\mathbf{x}^\prime_i\boldsymbol{\beta}$  &
~~~~~ &
$\pi_i = \frac{e^{\mathbf{x}^\prime_i\boldsymbol{\beta}}}
{1 + e^{\mathbf{x}^\prime_i\boldsymbol{\beta}}}$  \\
 & & \\
If $\mathbf{z} \sim N_p(\boldsymbol{\mu}, \Sigma)$, then $\mathbf{Az}  \sim N_q(\mathbf{A}\boldsymbol{\mu},\mathbf{A}\boldsymbol{\Sigma} \mathbf{A}^\prime)$,
 & ~~~~~ &
and $w = (\mathbf{z}-\boldsymbol{\mu})^\prime
           \boldsymbol{\Sigma}^{-1}(\mathbf{z}-\boldsymbol{\mu}) \sim \chi^2(p)$ \\
 & & \\
$\widehat{\boldsymbol{\beta}}_n \stackrel{\cdot}{\sim} N_{k+1}(\boldsymbol{\beta},\mathbf{V}_n)$
 & ~~~~~ & 
$W_n = (\mathbf{L}\widehat{\boldsymbol{\beta}}_n-\mathbf{h})^\prime 
\left(\mathbf{L} \widehat{\mathbf{V}}_n \mathbf{L}^\prime \right)^{-1} 
(\mathbf{L}\widehat{\boldsymbol{\beta}}_n-\mathbf{h})
\stackrel{\cdot}{\sim} \chi^2(r)$ \\
&& if $H_0: \mathbf{L}\boldsymbol{\beta} = \mathbf{h}$ is true. \\
\end{tabular}
\newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hspace{-10mm}\begin{tabular}{lll} 
%%%%%%%%%%%%%%%%%%%%%%%%% Poisson regression %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Poisson regression} & & \\
$ \log(\lambda_i) = 
    \beta_0 + \beta_1 x_{i,1} + \ldots + \beta_{p-1} x_{i,p-1}$
& ~~~~~ & \\   
 & & \\
%%%%%%%%%%%%%%%%%%%%%%%%% Multinomial logit models %%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Multinomial logit regression} & & \\
\parbox{7 cm}{ 
\begin{eqnarray*}
\log\left(\frac{\pi_1}{\pi_3} \right ) & = & 
     \beta_{0,1} + \beta_{1,1} x_1 + \ldots + \beta_{k,1} x_k  = L_1 \\ \\
\log\left(\frac{\pi_2}{\pi_3} \right ) & = & 
     \beta_{0,2} + \beta_{1,2} x_1 + \ldots + \beta_{k,2} x_k = L_2
\end{eqnarray*}
} % End parbox
& ~~~~~ &
\parbox{7 cm}{ 
\begin{eqnarray*}
\pi_1     & = & \frac{e^{L_1}}{1+e^{L_1}+e^{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{eqnarray*}
} % End parbox
\\  
% Still okay?
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\end{tabular}
\begin{verbatim}

> df = 1:8
> CriticalValue = qchisq(0.95,df)
> round(rbind(df,CriticalValue),3)
               [,1]  [,2]  [,3]  [,4]  [,5]   [,6]   [,7]   [,8]
df            1.000 2.000 3.000 4.000  5.00  6.000  7.000  8.000
CriticalValue 3.841 5.991 7.815 9.488 11.07 12.592 14.067 15.507
\end{verbatim}





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