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\title{Structural Equation Models\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{Structural Equation Models}
%\framesubtitle{} 
  \begin{itemize}
    \item An extension of multiple regression. 
    \item Can incorporate latent variables. 
    \item More than one regression-like equation.
%    \item All the variables are random. 
    \item An explanatory variable in one equation can be the response variable in another equation.
  \end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Calories}
\framesubtitle{\textbf{Doubly Labeled Water}: Participants drink water that is enriched with respect to two isotopes, and urine samples allow the measurement of energy expenditure (Graphics used without permission).}  \pause
\begin{center}
\includegraphics[width=3.8in]{CarrollEtAl}
\end{center}
\end{frame}

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\begin{frame}
\frametitle{Path diagrams}
\framesubtitle{Example: Exercise and arthritis pain}   \pause
\begin{center}
\includegraphics[width=3.7in]{PainPath}
\end{center}
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\begin{frame}
\frametitle{Path diagrams correspond to systems of equations}
%\framesubtitle{} 
\begin{columns}  
\column{0.5\textwidth}
\includegraphics[width=2.2in]{PainPath}
\column{0.5\textwidth}
{\scriptsize
\begin{eqnarray*}
    Y_{i,1} & = & \beta_{0,1} + \beta_1 X_i + \epsilon_{i,1} \\
    Y_{i,2} & = & \beta_{0,2} + \beta_2 Y_{i,1} + \epsilon_{i,2} \\ 
    Y_{i,3} & = & \beta_{0,3} + \beta_3 X_i + \beta_4 Y_{i,2} + \epsilon_{i,3} \\ 
    Y_{i,4} & = & \beta_{0,4} + \beta_5 Y_{i,2} + \beta_6 Y_{i,3} + \epsilon_{i,4} \\ 
    D_{i,1} & = & \lambda_{0,1} + \lambda_1 Y_{i,1} + e_{i,1} \\
    D_{i,2} & = & \lambda_{0,2} + \lambda_2 X_i + e_{i,2} \\
    D_{i,3} & = & \lambda_{0,3} + \lambda_3 Y_{i,2} + e_{i,3} \\
    D_{i,4} & = & \lambda_{0,4} + \lambda_4 Y_{i,3} + e_{i,4} \\
    D_{i,5} & = & \lambda_{0,5} + \lambda_2 X_i + e_{i,5} \\
    D_{i,6} & = & \lambda_{0,6} + \lambda_5 Y_{i,4} + e_{i,6} \\
\end{eqnarray*} 
} % End size
\end{columns} \pause

Multivariate normal model is standard.
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\begin{frame}
\frametitle{Strange Vocabulary}
%\framesubtitle{} 
\begin{itemize}
    \item Observed (manifest) versus latent variables.
    \item Endogenous versus exogenous variables.
    \item Exogenous latent variables are sometimes called ``factors" (factor analysis).
\end{itemize}

\begin{columns}  
\column{0.5\textwidth}
\includegraphics[width=2in]{PainPath}
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\includegraphics[width=2in]{efa}
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\begin{frame}
\frametitle{Regression with observable variables}
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    Y_i = \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \beta_3 X_{i,3} + \epsilon_i
\end{displaymath} 

\vspace{4mm}

\begin{center}
\includegraphics[width=3.5in]{RegressionPath}
\end{center}
\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. Except for the picture taken from Carroll et al.'s \emph{Measurement error in non-linear models}, 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:
\vspace{3mm}

\href{http://www.utstat.toronto.edu/brunner/oldclass/2053f22} {\small\texttt{http://www.utstat.toronto.edu/brunner/oldclass/2053f22}}

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