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\title{Ignoring Measurement Error: Convergence\footnote{See last slide for copyright information.}}
\subtitle{STA2053 Fall 2022}
\date{} % To suppress date


\begin{document}

\begin{frame}
  \titlepage
\end{frame}

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

\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Reliability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Additive measurement error}
\framesubtitle{A very simple model} 

\begin{center}
\includegraphics[width=1.5in]{Additive}
\end{center}  \pause

{\LARGE
\begin{displaymath}
     W = X + e 
\end{displaymath} 
} % End size

Where $E(X)=\mu_x$, $E(e)=0$, $Var(X)=\sigma^2_x$, $Var(e)=\sigma^2_e$, and $Cov(X,e)=0$. 
\end{frame}

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

\begin{frame}
\frametitle{Variance and Covariance}
\framesubtitle{$W = X + e$}  \pause

\begin{eqnarray*}
Var(W)   &=& Var(X) + Var(e) \\ 
         &=& \sigma^2_x + \sigma^2_e \\ 
         &&\\
Cov(X,W) &=& Cov\left(X,X+e\right) \\
         &=& Cov(X,X) + Cov(X,e) \\
         &=& \sigma^2_x + 0 \\
         &=& \sigma^2_x
\end{eqnarray*} 
\end{frame}

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

\begin{frame}
\frametitle{Definition of Reliability}
\framesubtitle{Psychometric} 
Reliability is the squared correlation between the observed variable and the latent variable (true score).
\end{frame}

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

\begin{frame}
\frametitle{Calculation of Reliability}
\framesubtitle{Squared correlation between observed and true score} 
{\small
\begin{eqnarray*}
  \rho^2 \pause &=& \left(\frac{Cov(X,W)}{SD(X) SD(W)}\right)^2 
               \pause \\
         &=& \left(\frac{\sigma^2_x}{\sqrt{\sigma^2_x} \sqrt{\sigma^2_x+\sigma^2_e}}\right)^2 \pause \\
         &=& \frac{\sigma^4_x}{\sigma^2_x (\sigma^2_x+\sigma^2_e)}
                \\
         &=& \frac{\sigma^2_x}{\sigma^2_x+\sigma^2_e} \\ \pause
         &=& \frac{Var(X)}{Var(W)}. \pause
\end{eqnarray*} % Have a phi-omega version in OpenSEM work.
Reliability is the proportion of the variance in the observed variable that comes from the latent variable of interest, and not from random error.
} % End size
\end{frame}

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

\begin{frame}
\frametitle{How to estimate reliability from data} \pause
%\framesubtitle{} 
  \begin{itemize}
    \item Correlate usual measurement with ``Gold Standard?" 
    \item Not very realistic, except maybe for some bio-markers. \pause
    \item One answer: Measure twice.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Measure twice}
\framesubtitle{Called ``equivalent measurements" because error variance is the same} 

\begin{center}
% Path diagram: Had to fiddle with this!
\begin{picture}(100,100)(150,0)           % Size of picture (does not matter), origin

    \put(197,000){$X$}
    \put(202,4){\circle{20}}

    \put(157,50){\framebox{$W_1$}}  
    \put(232,50){\framebox{$W_2$}}  

    \put(197,15){\vector(-1,1){25}} % X -> W1
    \put(209,15){\vector(1,1){25}}  % X -> W2

    \put(161,95){$e_1$} % x = V2+4
    \put(165,90){\vector(0,-1){25}} % e1 -> W1

    \put(236,95){$e_2$} % x = V3+4
    \put(240,90){\vector(0,-1){25}} % e2 -> W2

\end{picture}
\end{center}
 \pause
\begin{eqnarray}
    W_1 & = &  X + e_1  \nonumber \\ 
    W_2 & = &  X + e_2, \nonumber 
\end{eqnarray} 
where $E(X)=\mu_x$, $Var(X)=\sigma^2_x$,  $E(e_1)=E(e_2)=0$, $Var(e_1)=Var(e_2)=\sigma^2_e$,  
and  $X$, $e_1$ and $e_2$ are all independent.  
\end{frame}

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

\begin{frame}
\frametitle{Reliability equals the correlation between two equivalent measurements}  
\framesubtitle{This is a population correlation} 
{\small
\begin{eqnarray*}
    Corr(W_1,W_2) & = &  \frac{Cov(W_1,W_2)}{SD(W_1)SD(W_2)} \\  \pause
    & &  \\
    & = & \frac{Cov(X+e_1,X+e_2) }
               {\sqrt{\sigma^2_x+\sigma^2_e}\sqrt{\sigma^2_x+\sigma^2_e}} \\  \pause
    & & \\
    & = & \frac{Cov(X,X)+Cov(X,e_2)+Cov(e_1,X)+Cov(e_1,e_2)}
               {\sigma^2_x+\sigma^2_e}  \pause \\  
    & & \\
    & = & \frac{\sigma^2_x}{\sigma^2_x+\sigma^2_e},    
\end{eqnarray*}
which is the reliability. 
} % End size
\end{frame}

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

\begin{frame}
\frametitle{Estimate the reliability: Measure twice for a sample of size $n$} 
\framesubtitle{With a well-chosen time gap} 
Calculate $r = \frac{\sum_{i=1}^n (W_{i1}-\overline{W}_1)(W_{i2}-\overline{W}_2)}
               {\sqrt{\sum_{i=1}^n (W_{i1}-\overline{W}_1)^2} 
                \sqrt{\sum_{i=1}^n (W_{i2}-\overline{W}_2)^2}}$. \pause

\vspace{5mm}

  \begin{itemize}
    \item Test-retest reliability \pause
    \item Alternate forms reliability \pause
    \item Split-half reliability 
  \end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Measurement Error and Consistency}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Measurement error in the response variable only}
%\framesubtitle{} 
\begin{center}
\begin{picture}(100,100)(150,0)           % Size of picture (does not matter), origin

    \put(197,000){$Y$}
    \put(202,4){\circle{20}}

    \put(157,50){\framebox{$X$}}  
%    \put(168,25){{\footnotesize $\beta_1$}} % Label the arrow X -> Y
    \put(182,30){{\footnotesize $\beta_1$}} % Label the arrow X -> Y

    \put(235,50){\framebox{$V$}}  

    \put(167,42){\vector(1,-1){25}} % X -> Y
    \put(212,17){\vector(1,1){25}}  % Y -> V

    \put(240,95){$e$} 
    \put(243,90){\vector(0,-1){25}} % e -> V

    \put(244,01){$\epsilon$} 
    \put(242,03){\vector(-1,0){25}} % e -> V

\end{picture} \pause
\end{center}

True model:
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    V_i &=&  \nu + Y_i + e_i
\end{eqnarray*}  \pause
Naive model: $V_i = \beta_0 + \beta_1 X_i + \epsilon_i$
\end{frame}

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

\begin{frame}
\frametitle{Is $\widehat{\beta}_1$ consistent?}
\framesubtitle{Ignoring measurement error in $Y$} 
First calculate $Cov(X_i,V_i)$. Under the true model. \pause
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    V_i &=&  \nu + Y_i + e_i,
\end{eqnarray*}  \pause
\begin{eqnarray*}
    Cov(X_i,V_i) &=& Cov(X_i,\beta_1 X_i + \epsilon_i) \pause \\
                 &=&   \beta_1 \sigma^2_x
\end{eqnarray*}  

\end{frame}

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

\begin{frame}
\frametitle{Target of $\widehat{\beta}_1$ as $n \rightarrow \infty$}
\framesubtitle{Have $Cov(X_i,V_i) = \beta_1 \sigma^2_x$ and $Var(X_i) = \sigma^2_x$}  \pause

\begin{eqnarray*}
\widehat{\beta}_1 &=& \frac{\sum_{i=1}^n(X_i-\overline{X})(V_i-\overline{V})}
                           {\sum_{i=1}^n(X_i-\overline{X})^2}  \\   \pause
                  &=& \frac{\widehat{\sigma}_{x,v}}{\widehat{\sigma}^2_x} \\  \pause
                  &\stackrel{a.s.}{\rightarrow}&
                       \frac{Cov(X_i,V_i)}{Var(X_i)}  \\  \pause
                  &=& \frac{\beta_1 \sigma^2_x}{\sigma^2_x}  \\  \pause
                  &=& \beta_1       
\end{eqnarray*}
Consistent.
\end{frame}

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

\begin{frame}
\frametitle{Why did it work?} \pause
%\framesubtitle{} 

\begin{eqnarray*}
Y_i &=& \beta_0 + \beta_1 X_i  + \epsilon_i \\
V_i &=& \nu +  Y_i + e \\ \pause
   &=& \nu + (\beta_0 + \beta_1 X_i  + \epsilon_i) + e_i \\ \pause
   &=& (\nu + \beta_0) + \beta_1 X_i  
                                + (\epsilon_i + e_i) \\ \pause
   &=& \beta_0^\prime +  \beta_1 X_i  + \epsilon_i^\prime   \pause
\end{eqnarray*} 
\begin{itemize}
    \item This is a re-parameterization. 
    \item Most definitely \emph{not} one-to-one. \pause
    \item $(\nu,\beta_0)$ is absorbed into $\beta_0^\prime$. \pause
    \item $(\epsilon_i, e_i)$ is absorbed into $\epsilon_i^\prime$. \pause
    \item Can't know everything, but all we care about is $\beta_1$ anyway.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Don't Worry}
{\Large
\begin{itemize}
    \item If a response variable appears to have no measurement error, assume it does have measurement error but the problem has been re-parameterized.  \pause
    \item Measurement error in $Y$ is part of $\epsilon$.
\end{itemize}

} % End size
\end{frame}

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

\begin{frame}
\frametitle{Measurement error in a single explanatory variable}
%\framesubtitle{} 
\begin{center}
% Path diagram: Had to fiddle with this!
\begin{picture}(100,100)(150,0)           % Size of picture (does not matter), origin

    \put(197,000){$X$}
    \put(202,4){\circle{20}}
    \put(210,30){{\footnotesize $\beta_1$}} % Label the arrow X -> Y

    \put(157,50){\framebox{$W$}}  
    \put(232,50){\framebox{$Y$}}  

    \put(197,15){\vector(-1,1){25}} % X -> W
    \put(209,15){\vector(1,1){25}}  % X -> Y

    \put(161,95){$e$} 
    \put(165,90){\vector(0,-1){25}} % e -> W

    \put(236,95){$\epsilon$} 
    \put(240,90){\vector(0,-1){25}} % epsilon -> Y

\end{picture}
\end{center} \pause

True model:
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    W_i &=&  X_i + e_i,  \nonumber
\end{eqnarray*}  
Naive model: $Y_i = \beta_0 + \beta_1 W_i + \epsilon_i$
\end{frame}

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

\begin{frame}
\frametitle{Target of $\widehat{\beta}_1$ as $n \rightarrow \infty$}
\framesubtitle{$Y_i = \beta_0 + \beta_1 X_i + \epsilon_i$ and $W_i = X_i + e_i$}  \pause
Have $Cov(W_i,Y_i) = \beta_1 \sigma^2_x$ and $Var(W_i) = \sigma^2_x+\sigma^2_e$  \pause
\begin{eqnarray*}
\widehat{\beta}_1 &=& \frac{\sum_{i=1}^n(W_i-\overline{W})(Y_i-\overline{Y})}
                           {\sum_{i=1}^n(W_i-\overline{W})^2} \\   \pause
                  &=& \frac{\widehat{\sigma}_{w,y}}{\widehat{\sigma}^2_w}\nonumber \\  \pause
                  &\stackrel{a.s.}{\rightarrow}&
                       \frac{Cov(W,Y)}{Var(W)} 
                       = \frac{\beta_1 \sigma^2_x}{\sigma^2_x+\sigma^2_e}
                       \\  \pause
                  &=& \beta_1 \left(\frac{\sigma^2_x}{\sigma^2_x+\sigma^2_e} \right)  
\end{eqnarray*}
\end{frame}

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

\begin{frame}
\frametitle{$\widehat{\beta}_1 \stackrel{a.s.}{\rightarrow} \beta_1 \left(\frac{\sigma^2_x}{\sigma^2_x+\sigma^2_e} \right)$}
\framesubtitle{$W_i =  X_i + e_i$}  \pause
\begin{itemize}
    \item $\widehat{\beta}_1$ converges to $\beta$ times the reliability of $W_i$. \pause
    \item It's inconsistent. \pause
    \item Because the reliability is less than one, it's asymptotically biased toward zero. \pause
    \item The worse the measurement of $X_i$, the more the asymptotic bias. \pause
    \item Sometimes called ``attenuation" (weakening). \pause
    \item If a good estimate of reliability is available from another source, one can ``correct for attenuation." \pause
    \item When $H_0:\beta_1=0$ is true, no problem. \pause
    \item False sense of security?
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Measurement error in two explanatory variables}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{MeReg2Path}
\end{center} 

Want to assess the relationship of $X_2$ to $Y$ controlling for $X_1$ by testing $H_0:\beta_2=0$.
\end{frame}

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

\begin{frame}
\frametitle{Statement of the model}
\framesubtitle{Independently for $i=1, \ldots,n$}  
\begin{eqnarray}
    Y_i &=& \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \epsilon_i 
\nonumber \\ 
    W_{i,1} & = &  X_{i,1} + e_{i,1}  
\nonumber \\  
    W_{i,2} & = &  X_{i,2} + e_{i,2},  
\nonumber
\end{eqnarray}
{\footnotesize
where  
\begin{itemize}
    \item[] $E(X_{i,1})=\mu_1$, $E(X_{i,2})=\mu_2$, $E(\epsilon_i) = E(e_{i,1}) = E(e_{i,2}) = 0$,
    \item[] $Var(\epsilon_i)=\psi$, $Var(e_{i,1})=\omega_1$, $Var(e_{i,2})=\omega_2$, 
    \item[] The errors $\epsilon_i, e_{i,1}$ and $e_{i,2}$ are all independent,
    \item[] $X_{i,1}$ and $X_{i,2}$ are independent of $\epsilon_i, e_{i,1}$ and $e_{i,2}$, and
\end{itemize}
\begin{displaymath}
cov\left(
\begin{array}{c} X_{i,1} \\ X_{i,1} \end{array}
 \right) = \left(
\begin{array}{c c} 
\phi_{11} & \phi_{12} \\
\phi_{12} & \phi_{22}
\end{array} \right).
\end{displaymath} 
} % End size
Note
\begin{itemize}
    \item Reliability of $W_1$ is $\frac{\phi_{11}}{\phi_{11}+\omega_1}$.
    \item Reliability of $W_2$ is $\frac{\phi_{22}}{\phi_{22}+\omega_2} $.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{True Model versus Naive Model}
%\framesubtitle{Independently for $i=1, \ldots,n$}  \pause
%\vspace{1mm}
True model:
\begin{eqnarray}
    Y_i &=& \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \epsilon_i 
\nonumber \\ 
    W_{i,1} & = &  X_{i,1} + e_{i,1}  
\nonumber \\  
    W_{i,2} & = &  X_{i,2} + e_{i,2},  \pause
\nonumber
\end{eqnarray}

  
Naive model: $Y_i = \beta_0 + \beta_1 W_{i,1} + \beta_2 W_{i,2} + \epsilon_i$ \pause
\vspace{3mm}

\begin{itemize}
     \item Fit the naive model.
     \item See what happens to $\widehat{\beta}_2$ as $n \rightarrow \infty$ when the true model holds. \pause
     \item Start by calculating $cov(\mathbf{d}_i)$.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Covariance matrix of the observable data} 

% \vspace{5mm}
{\footnotesize
\begin{eqnarray*}
    \boldsymbol{\Sigma}
     &=& cov\left(\begin{array}{c}
        W_{i,1} \\   W_{i,2} \\ Y_i
        \end{array}\right) \\ \pause
     && \\   
     &=& \left(\begin{array}{rrr}
        \omega_{1} + \phi_{11} & \phi_{12} & \beta_{1} \phi_{11} +
        \beta_{2} \phi_{12} \\
        \phi_{12} & \omega_{2} + \phi_{22} & \beta_{1} \phi_{12} +
        \beta_{2} \phi_{22} \\
        \beta_{1} \phi_{11} + \beta_{2} \phi_{12} & \beta_{1} \phi_{12} +
        \beta_{2} \phi_{22} & \beta_{1}^{2} \phi_{11} + 2 \, \beta_{1}
        \beta_{2} \phi_{12} + \beta_{2}^{2} \phi_{22} + \psi
          \end{array}\right) % Last matrix pasted in from Sage!
\end{eqnarray*}
} % End size
\end{frame}

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

\begin{frame}
\frametitle{What happens to $\widehat{\beta}_2$ as $n \rightarrow \infty$?}
\framesubtitle{Interested in $H_0:\beta_2=0$}  \pause
{\footnotesize
\begin{eqnarray*}\label{abias}
\widehat{\beta}_2 
    &=& \frac{\widehat{\sigma}_{11}\widehat{\sigma}_{23} - 
          \widehat{\sigma}_{12}\widehat{\sigma}_{13}}
         {\widehat{\sigma}_{11}\widehat{\sigma}_{22} - 
          \widehat{\sigma}_{12}^2} \\ \pause
    &\stackrel{a.s.}{\rightarrow}& 
    \frac{\sigma_{11}\sigma_{23} - \sigma_{12}\sigma_{13}}
         {\sigma_{11}\sigma_{22} - \sigma_{12}^2}  \nonumber \\ \pause
        & = & \frac{{\beta_{1} \omega_{1} \phi_{12} 
             + \beta_{2} (\omega_{1}\phi_{22}
             + \phi_{11} \phi_{22}-\phi_{12}^{2}} ) } 
        {(\phi_{1,1} + \omega_1)(\phi_{2,2} + \omega_2) - \phi_{12}^{2}} \\ \pause
        & \neq & \beta_2 
\end{eqnarray*}
} % End size
Inconsistent.
\end{frame}

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

\begin{frame}
\frametitle{When $H_0:\beta_2=0$ is true} 
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
     \widehat{\beta}_2  \stackrel{a.s.}{\rightarrow}
    \frac{\beta_{1} \omega_{1} \phi_{12}}
    {(\phi_{1,1} + \omega_1)(\phi_{2,2} + \omega_2) - \phi_{12}^{2}}
\end{displaymath}  \pause
} % End size

So $ \widehat{\beta}_2$ goes to the wrong target unless  \pause
\begin{itemize}
     \item There is no relationship between $X_1$ and $Y$,  or \pause
     \item There is no measurement error in $W_1$,  or \pause
     \item There is no correlation between $X_1$ and $X_2$.  \pause
\end{itemize}
Also, $t$ statistic goes to plus or minus $\infty$ and the $p$-value $\stackrel{a.s.}{\rightarrow} 0$. \pause
Remember, $H_0$ is true.
\end{frame}



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


\begin{frame}
\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/2053f22} {\footnotesize \texttt{http://www.utstat.toronto.edu/brunner/oldclass/2053f22}}
\end{frame}

\end{document}


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\begin{frame}
\frametitle{}
%\framesubtitle{} 
  \begin{itemize}
    \item 
    \item 
    \item 
  \end{itemize}
\end{frame}

% \stackrel{c}{\mathbf{X}} \stackrel{\top}{\vphantom{r}_i} % Centered X_i Transpose
% \stackrel{c}{X}\stackrel{2}{\vphantom{r}_i}    % Centered X_i^2 
% \stackrel{c}{X}\stackrel{2}{\vphantom{r}}      % Centered X^2