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\title{Random Explanatory Variables\footnote{See last slide for copyright information.}}
\subtitle{STA302 Fall 2020}
\date{} % To suppress date


\begin{document}

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


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preparation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Preparation: Change of Variables Formula}
\framesubtitle{$Y = g(X)$} \pause
Two ways of writing the same thing: \vspace{10mm} \pause
{\LARGE
\begin{eqnarray*}
    E(Y) & = & \int y f_{_Y}(y) \, dy \\
    E(g(X)) & = & \int g(x) f_{_X}(x) \, dx
\end{eqnarray*} 
} % End size
\end{frame}

\begin{comment}
\begin{displaymath}
    E(g(X)) = \int g(x) f_{_X}(x) \, dx
\end{displaymath} 
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{Preparation: Indicator functions}
\framesubtitle{Conditional expectation and the Law of Total Probability} \pause 

$I_A(x)$ is the \emph{indicator function} for the set $A$. It is defined by
\begin{displaymath}
    I_A(x) = 
\left\{ \begin{array}{ll} % ll means left left
   1 & \mbox{for } x \in A  \\
   0     & \mbox{for } x \notin A 
   \end{array}  \right.   % Need that crazy invisible right period!
\end{displaymath}

\pause 

Also sometimes written $I(x \in A)$

\pause

\begin{eqnarray*}
    E(I_A(X)) &=& \sum_x I_A(x) p(x) \pause
               = \sum_{x \in A} p(x) \pause \mbox{, or}\\ \pause
              & & \int_{-\infty}^\infty I_A(x) f(x) \, dx \pause
               = \int_A f(x) \, dx\\ \pause
              &&\\
              &=& P\{ X \in A \}
\end{eqnarray*}
\pause
So the expected value of an indicator is a probability.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}
\frametitle{Applies to conditional probabilities too} \pause 
%\framesubtitle{} 
{\LARGE
\begin{eqnarray*}
    E(I_A(X)|Y) &=& \sum_x I_A(x) p(x|Y)  \mbox{, or}\\ \pause
              & & \int_{-\infty}^\infty I_A(x) f(x|Y) \, dx \\ \pause
              &&\\
              &=& Pr\{ X \in A|Y\}
\end{eqnarray*}   
} % End size
\pause
So the conditional expected value of an indicator is a \emph{conditional} probability.
\end{frame}

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

\begin{frame}
\frametitle{Double expectation}
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
    E\left( X \right) = E\left( E[X|Y]\right) \pause = E(g(Y))
\end{displaymath}
} % End size
\end{frame}

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

\begin{frame}
\frametitle{Showing $E\left( X \right) = E\left( E[X|Y]\right)$}
\framesubtitle{Again note $E\left( E[X|Y]\right)$ is an example of $E(g(Y)$}  \pause
\begin{eqnarray*}
   E\left( E[X|Y]\right) & = & \int  E[X|Y=y] f_y(y) \, dy \\ \pause
   & = & \int  \left(\int x f_{x|y}(x|y) \, dx \right) f_y(y) \, dy \\ \pause
   & = & \int  \left(\int x \frac{f_{x,y}(x,y)}{f_y(y)} \, dx \right) f_y(y) \, dy \\ \pause
   & = & \int  \int x \, f_{x,y}(x,y) \, dx \, dy \\ \pause
   & = & E(h(X,Y)) \\ \pause
   & = & E(X)
\end{eqnarray*} 
\end{frame}

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

\begin{frame}
\frametitle{Double expectation: $E\left( g(X) \right) = E\left( E[g(X)|Y]\right)$}
%\framesubtitle{} 
\pause 
\begin{displaymath}
    E\left(E[I_A(X)|Y]\right) \pause = E[I_A(X)] \pause = Pr\{ X \in A\} \mbox{, so}
\end{displaymath}
\pause 
\begin{eqnarray*}
    Pr\{ X \in A\} &=&  E\left(E[I_A(X)|Y]\right) \pause \\
              &=& E\left(Pr\{ X \in A|Y\}\right) \pause \\
              &=& \int_{-\infty}^\infty Pr\{ X \in A|Y=y\} f_y(y) \, dy \mbox{, or} \pause \\
              & & \sum_y Pr\{ X \in A|Y=y\} p_y(y)
\end{eqnarray*}   

\pause 
This is known as the \emph{Law of Total Probability}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Random Explanatory Variables}

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

\begin{frame}
\frametitle{Random Explanatory Variables} \pause
%\framesubtitle{} 
Don't you think it’s strange? \pause
\begin{itemize}
    \item In the general linear regression model, the $\mathbf{X}$ matrix is supposed to be full of fixed constants.  \pause
    \item This is convenient mathematically. \pause Think of $E(\widehat{\boldsymbol{\beta}})$. \pause
    \item But in any non-experimental study, if you selected another sample you'd get different $\mathbf{X}$ values\pause, because of random sampling. \pause
    \item So $\mathbf{X}$ should be at least partly random variables, not fixed.  \pause
    \item View the usual model as \emph{conditional} on $\mathcal{X}=\mathbf{X}$.  \pause
    \item All the probabilities and expected values so far in this course are \emph{conditional} probabilities and \emph{conditional} expected values.  \pause
    \item Conditional on $\mathcal{X}=\mathbf{X}$. \pause
    \item We don't want to stop there.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{$\widehat{\boldsymbol{\beta}}$ is (conditionally) unbiased}
%\framesubtitle{} 
{\Large
\begin{displaymath}
    E(\widehat{\boldsymbol{\beta}}|\mathcal{X}=\mathbf{X}) = \boldsymbol{\beta}
    \pause
\end{displaymath} 
} % End size
For \emph{any} fixed $\mathbf{X}$ \pause with linearly independent columns.
\vspace{5mm} \pause

It's \emph{unconditionally} unbiased too. \pause
 
\vspace{5mm}
{\Large
\begin{displaymath}
    E\{\widehat{\boldsymbol{\beta}}\} \pause = E\{E\{\widehat{\boldsymbol{\beta}}|\mathcal{X}\}\} \pause
    = E\{\boldsymbol{\beta}\} \pause = \boldsymbol{\beta}
\end{displaymath} 
} % End size
\end{frame}

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

\begin{frame}
\frametitle{Perhaps Clearer}
%\framesubtitle{} 

\begin{eqnarray*} 
    E\{\widehat{\boldsymbol{\beta}}\}  \pause
    &=&  E\{E\{\widehat{\boldsymbol{\beta}}|\mathcal{X}\}\} 
       \\ \pause
    &=& \int \cdots  \int E\{\widehat{\boldsymbol{\beta}}|\mathcal{X}=\mathbf{X}\} \, 
         f(\mathbf{X}) \, d\mathbf{X} 
        \\ \pause
    &=& \int \cdots  \int \boldsymbol{\beta} \, f(\mathbf{X}) \, d\mathbf{X} 
         \\ \pause
    &=& \boldsymbol{\beta} \int \cdots  \int f(\mathbf{X})\, d\mathbf{X}
        \\ \pause
    &=& \boldsymbol{\beta} \cdot 1 = \boldsymbol{\beta}.
\end{eqnarray*} 
\end{frame}

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

\begin{frame}
\frametitle{Conditional size $\alpha$ test, Critical value $f_\alpha$} \pause
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
    Pr\{F > f_\alpha | \mathcal{X}=\mathbf{X} \} = \alpha
\end{displaymath} \pause
} % End size
% \vspace{3mm}
\begin{eqnarray*}
    Pr\{F > f_\alpha \} &=& \int \cdots  \int Pr\{F > f_\alpha | \mathcal{X}=\mathbf{X} \}
                       f(\mathbf{X})\, d\mathbf{X} \\ \pause
                   &=& \int \cdots  \int \alpha 
                       f(\mathbf{X})\, d\mathbf{X}  \\     \pause
               &=&  \alpha \int \cdots  \int  
                       f(\mathbf{X})\, d\mathbf{X}  \\ \pause
                   &=& \alpha  
\end{eqnarray*} \pause
A similar calculation applies to confidence intervals and prediction intervals.
\end{frame}

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

\begin{frame}
\frametitle{The moral of the story} \pause
%\framesubtitle{} 
\begin{itemize}
    \item Don't worry. \pause
    \item Even though the independent variables are often random, we can apply the usual fixed $\mathbf{X}$ model without fear. \pause
    \item Estimators are still unbiased. \pause
    \item Tests have the right Type I error probability. \pause
    \item Confidence intervals and prediction intervals are still correct. \pause
    \item And it's all distribution-free with respect to $\mathbf{X}$.
\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 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/302f20} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f20}}

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