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\title{Structural Equation Models\footnote{See last slide for copyright information.}}
\subtitle{STA431 Winter/Spring 2017}
\date{} % To suppress date

\begin{document}

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

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

\begin{frame}
\frametitle{Measurement Error}
%\framesubtitle{} 
\begin{itemize}
    \item What you see is not what you really want. \pause
    \item \emph{Latent variable}:  A random variable whose values cannot be directly observed. \pause For example, family income last year. \pause
    \item Contrast with \emph{Observable variable}. \pause For example, reported family income last year. \pause
    \item Usually, interest is in relationships between latent variables. \pause
    \item But all you have in your data set are the observable variables.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Doubly Labeled Water }
\framesubtitle{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}

\begin{frame}
\frametitle{Path diagrams}
\framesubtitle{Example: Exercise and arthritis pain}   \pause
\begin{center}
\includegraphics[width=3.7in]{PainPath}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Comments}
%\framesubtitle{} 
\begin{itemize}
    \item Latent variables are in ovals, observable variables are in boxes.   \pause
    \item Error terms seem to come from nowhere -- often not shown.   \pause
    \item There is real modeling here. Lots of theoretical input is required.   \pause
    \item These are usually interpreted as \emph{causal} models: Models of influence.   \pause
    \item $A \rightarrow B$ means $A$ has an influence on $B$.  \pause
    \item But the data are usually observational.  % \pause
%    \item Omitted variables can cause problems -- more later.
\end{itemize}
\end{frame}

\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.
\end{frame}

\begin{frame}
\frametitle{Regression with observable variables}
%\framesubtitle{} 
\begin{displaymath}
    Y_i = \beta_0 + \beta_1 X_{i,1} + \beta_2 X_{i,2} + \beta_3 X_{i,3} + \epsilon_i
\end{displaymath} \pause

\vspace{4mm}

\begin{center}
\includegraphics[width=3.5in]{RegressionPath}
\end{center}
\end{frame}


\begin{frame}
\frametitle{Tools} \pause
%\framesubtitle{} 
\begin{itemize}
    \item Scalar variance-covariance calculations \pause
    \item Matrices \pause
    \item Random vectors \pause
    \item Multivariate normal \pause
    \item Maximum likelihood \pause
    \item A little large-sample theory \pause
    \item SAS
\end{itemize}
\end{frame}





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\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/431s17} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/431s17}}

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