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\title{Multivariate Linear Model}
\subtitle
{For Between-Within Designs} % (optional)
\date{} % To suppress date


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

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

\begin{frame}{Same study can have both between and within-cases factors} {Example: Grapefruit sales}
\begin{itemize}
     \item Cases are stores
     \item Sales measured at every store with three different price levels (Random order)
     \item Three price levels: Within-stores factor
     \item Incentive program for produce managers (Yes-No): Between-stores factor
\end{itemize}
\end{frame}


\begin{frame}{Multivariate Linear Model}%{Subtitles are optional.}

\begin{displaymath}
{\Large\mathbf{Y} = \mathbf{XB} + \boldsymbol{\epsilon}},
\end{displaymath}
where
  \begin{itemize}
    \item $\mathbf{Y}$ is an $n \times k$ random matrix, with one response variable in each column.
    \item $\mathbf{X}$ is an $n \times p$ matrix of fixed, observable constants. There is one (between-cases) explanatory variable in each column.
    \item $\mathbf{B}$ is a $p \times k$ matrix of unknown parameters (regression coefficients).
    \item $\boldsymbol{\epsilon}$ is an $n \times k$ random matrix. The rows of $\boldsymbol{\epsilon}$ are independent multivariate normals with expected value $\boldsymbol{0}$ and $k \times k$ variance-covariance matrix $\boldsymbol{\Sigma}$.
  \end{itemize}
\end{frame}

\begin{frame}{One Column of $\mathbf{B}$ for Each Response Variable}%{Subtitles are optional.}

    \begin{displaymath}
    \mathbf{B} = 
     \left( \begin{array}{c c c c}
             \beta_{0,1}   & \beta_{0,2}   & \cdots & \beta_{0,k}   \\
             \beta_{1,1}   & \beta_{1,2}   & \cdots & \beta_{1,k}   \\
             \vdots        & \vdots        & \ddots & \vdots        \\
             \beta_{(p-1),1} & \beta_{(p-1),2} & \cdots & \beta_{(p-1),k} \\  
    \end{array} \right) 
    \end{displaymath}
    
\vspace{10mm}

$\widehat{\boldsymbol{B}} = (\mathbf{X}^\prime \mathbf{X})^{-1} \mathbf{X}^\prime \mathbf{Y}$ (a $p \times k$ matrix), so MLEs are what one would get from $k$ univariate regressions.
\end{frame}

\begin{frame}{Null Hypothesis: $\mathbf{LBM=0}$}
\begin{itemize}
     \item $\mathbf{L}$ is $r \times p$ with $r \leq p$.
     \item $\mathbf{B}$ is $p \times k$.
     \item $\mathbf{M}$ is $k \times q$ with $q \leq k$.
     \item With $\mathbf{M=I}$, have
        \begin{itemize}
            \item All the usual linear null hypotheses
            \item Simultaneously for all $k$ response variables
            \item Same null hypothesis for each response variable
        \end{itemize}
     \item The matrix $\mathbf{M}$ specifies linear combinations of the response variables (not obvious).
\end{itemize}
\end{frame}


\begin{frame}{Linear Combinations of the Response Variables}
\begin{displaymath}
\mathbf{Y} = \mathbf{XB} + \boldsymbol{\epsilon} \Rightarrow
\mathbf{YM} = \mathbf{XBM} + \boldsymbol{\epsilon}\mathbf{M}
\end{displaymath}

\begin{itemize}
     \item Each column of $\mathbf{M}$ yields a linear combination of the $k$ response variables.
     \item New ``$\mathbf{Y}$" $=\mathbf{YM}$
     \item New ``$\mathbf{B}$" $=\mathbf{BM}$
     \item New ``$\boldsymbol{\epsilon}$" $=\boldsymbol{\epsilon}\mathbf{M}$
     \item Rows of new ``$\boldsymbol{\epsilon}$" are independent 
     $N_q(\mathbf{0},\mathbf{M}^\prime \boldsymbol{\Sigma} \mathbf{M})$
\end{itemize}
\end{frame}

\begin{frame}{Moral of the Story}
\begin{itemize}
     \item Can easily carry out multivariate tests on collections of linear combinations of the response variables
     \item Multiple response variables could represent measurements at levels of one or more within-cases factors (think 3 Grapefruit Sales numbers)
     \item Linear combinations can correspond to main effects, interactions of within-cases factors
\end{itemize}
\end{frame}



\end{document}


\begin{frame}{Four Exact Likelihood Ratio Tests}

\end{frame}


\begin{frame}{}%{Subtitles are optional.}
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