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\title{Measurement Error in the Response Variable\footnote{See last slide for copyright information.}}
\subtitle{STA 2101 Fall 2019}
\date{} % To suppress date

\begin{document}

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

%\section{}


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\begin{frame}
\frametitle{Ignoring measurement error}
%\framesubtitle{} 
\begin{itemize}
    \item We have seen that ignoring measurement error in the explanatory variables can lead to disaster. 
    \item What about measurement error in the response variable?
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Example of Measurement Error in $Y$ only}
\framesubtitle{$X$ could be drug dose, $Y$ could be true anxiety, $V$ could be reported anxiety} \pause
Independently for $i=1, \ldots,n$, let
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    V_i &=&  \nu + Y_i + e_i,  
\end{eqnarray*}
where $Var(X_i)=\sigma^2_x$,  $Var(e_i)=\sigma^2_e$,  $Var(\epsilon_i)=\sigma^2_\epsilon$,
and $X_i, e_i, \epsilon_i$ are all independent. \pause

\begin{picture}(100,100)(75,0)           % 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

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\begin{frame}
\frametitle{Parameters of the true model are not identifiable from the means and covariance matrix}
%\framesubtitle{} 
{\small
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    V_i &=&  \nu + Y_i + e_i,  
\end{eqnarray*}
where $Var(X_i)=\sigma^2_x$,  $Var(e_i)=\sigma^2_e$, and $Var(\epsilon_i)=\sigma^2_\epsilon$. \pause
} % End size
\begin{itemize}
    \item Only the $(X_i,V_i)$ pairs are observable.
    \item There are 5 moments. \pause
    \item $\boldsymbol{\theta} = (\beta_0, \beta_1, \mu_x, \sigma^2_x, \sigma^2_\epsilon, \nu, \sigma^2_e)$: 7 parameters
    \item Fails the test of the parameter count rule.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Ignoring measurement error as usual} \pause
%\framesubtitle{} 
True model:
\begin{eqnarray*}
    Y_i &=& \beta_0 + \beta_1 X_i + \epsilon_i \\
    V_i &=&  \nu + Y_i + e_i,  
\end{eqnarray*}
Naive model: \pause
\begin{displaymath}
    V_i = \beta_0 + \beta_1 X_i + \epsilon_i
\end{displaymath} 
\end{frame}

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\begin{frame}
\frametitle{Fit the Naive Model, using $V_i$ as the response variable}
\framesubtitle{$ V_i = \beta_0 + \beta_1 X_i + \epsilon_i$} \pause
First note that under the \emph{true} model, $Cov(X_i,V_i) = \beta_1 \sigma^2_x$ 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} \nonumber \\ \pause
                  &=& \frac{\widehat{\sigma}_{x,v}}{\widehat{\sigma}^2_x}\nonumber \\ \pause
                  &\stackrel{a.s.}{\rightarrow}&
                       \frac{Cov(X_i,V_i)}{Var(X_i)}  \\ \pause
                  &=& \frac{\beta_1 \sigma^2_x}{\sigma^2_x} \nonumber \\
                  &=& \beta_1 .    \nonumber      
\end{eqnarray*} \pause
So $\widehat{\beta}_1$ is consistent, even though the model is mis-specified.
\end{frame}

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\begin{frame}
\frametitle{Why does the naive model work so well?} \pause
%\framesubtitle{} 
\begin{eqnarray*} 
V_i &=& \nu +  Y_i +  e_i \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  
\end{eqnarray*} \pause

\begin{itemize}
    \item This is a  \emph{re-parameterization}. \pause
    \item Not a one-to-one re-parameterization -- call it a ``collapsing" re-parameterization. \pause
    \item The pair $(\nu,\beta_0)$ is absorbed into $\beta_0^\prime$. \pause
    \item $Var(\epsilon_i +  e_i)= \sigma^2_\epsilon + \sigma^2_e$ is absorbed into a single unknown variance that will probably be called $\sigma^2$. \pause
    \item $\nu$ and $\beta_0$ will never be knowable separately, and also $\sigma^2_\epsilon$ and $\sigma^2_e$ will never be knowable separately. \pause
    \item It's okay. All we care about is $\beta_1$ anyway.
\end{itemize}

\end{frame}

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\begin{frame}
\frametitle{This is very common}
%\framesubtitle{} 
\begin{itemize}
    \item In many models, it will appear that the response variable is being measured without error. \pause 
    \item Of course there really is measurement error in $Y_i$, but it has been absorbed into the error term.\pause
    \item So any model without measurement error in the response variable should be viewed as a re-parameterized version of a more realistic model. \pause
    \item The measurement error should be independent of $X$, or there is real trouble.
\end{itemize}
\end{frame}

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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 Statistical Sciences, 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/2101f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/2101f19}}

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