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


\begin{document}

\begin{frame}
  \titlepage
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\begin{frame}
\frametitle{Measurement Error}
%\framesubtitle{} 
  \begin{itemize}
    \item Snack food consumption
    \item Exercise
    \item Income
    \item Cause of death
    \item Even amount of drug that reaches animal's blood stream in an experimental study
    \item Is there anything that is \emph{not} measured with error?
  \end{itemize}
\end{frame}

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\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 $H_0:\beta_2=0$.
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\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
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\begin{frame}
\frametitle{True Model versus Naive Model}
\framesubtitle{Independently for $i=1, \ldots,n$} 
%\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},  
\nonumber
\end{eqnarray}

Naive model: $Y_i = \beta_0 + \beta_1 W_{i,1} + \beta_2 W_{i,2} + \epsilon_i$
\vspace{2mm}
\begin{center}
\includegraphics[width=1.9in]{MeReg2Path}
\end{center} 

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\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}}
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% \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