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\mode<presentation>

\title{Confirmatory Factor Analysis Part Two\footnote{See last slide for copyright information.}}
\subtitle{STA2053 Fall 2022}
\date{} % To suppress date

\begin{document}

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



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

\begin{frame}
\frametitle{Original and Surrogate Models}
%\framesubtitle{} 

  \begin{itemize}
    \item Original model has expected values, intercepts, and slopes that need not equal one.
    \item Re-parameterization via a change of variables yields a surrogate model.
    \item Centered surrogate model has the same covariance matrix as the original.
%    \item If a model has any factor loadings or variances equal to one, it's a surrogate model.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Why should the variance of the factors equal one?} \pause
%\framesubtitle{} 

  \begin{itemize}
    \item Inherited from exploratory factor analysis, which was mostly a disaster.
    \item The standard answer is something like this:  ``Because it’s arbitrary.  The variance depends upon the scale on which the variable is measured, but we can’t see it to measure it directly.  So set it to one for convenience." \pause
    \item But saying it does not make it so.  If $F$ is a random variable with an unknown variance, then
    \item $Var(F)=\phi$ is an unknown parameter.
  \end{itemize}
\end{frame}

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


\begin{frame}
\frametitle{Centered Model}
\framesubtitle{One factor, four observed variables} 
\begin{columns}  
\column{0.5\textwidth}
\begin{eqnarray*}
d_1 &=& \lambda_1 F + e_1 \\
d_2 &=& \lambda_2 F + e_2 \\
d_3 &=& \lambda_3 F + e_3 \\
d_4 &=& \lambda_4 F + e_4  %  36
\end{eqnarray*} 
\column{0.5\textwidth}
$\begin{array}{l}
e_1,\ldots, e_4, F \mbox{ all independent} \\
Var(e_j) = \omega_j ~~~~ Var(F) = \phi \\
\lambda_1, \lambda_2, \lambda_3 \neq 0
\end{array} $
\end{columns} \pause \vspace{10mm}
% Taken from the old Sage job Fac1 on Big Iron
$\boldsymbol{\Sigma} = 
\left(\begin{array}{rrrr}
\lambda_{1}^{2} \phi + \omega_{1} & \lambda_{1} \lambda_{2} \phi &
\lambda_{1} \lambda_{3} \phi & \lambda_{1} \lambda_{4} \phi \\
\lambda_{1} \lambda_{2} \phi & \lambda_{2}^{2} \phi + \omega_{2} &
\lambda_{2} \lambda_{3} \phi & \lambda_{2} \lambda_{4} \phi \\
\lambda_{1} \lambda_{3} \phi & \lambda_{2} \lambda_{3} \phi &
\lambda_{3}^{2} \phi + \omega_{3} & \lambda_{3} \lambda_{4} \phi \\
\lambda_{1} \lambda_{4} \phi & \lambda_{2} \lambda_{4} \phi &
\lambda_{3} \lambda_{4} \phi & \lambda_{4}^{2} \phi + \omega_{4}
\end{array}\right) $  
\vspace{3mm}

Passes the Counting Rule test with 10 equations in 9 unknowns
\end{frame}

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

\begin{frame}
\frametitle{But for any $c \neq 0$}
%\framesubtitle{} 
\begin{displaymath}
    \begin{array}{c|ccccccccc}
\boldsymbol{\theta}_1 & \phi & \lambda_1 &\lambda_2 &\lambda_3 &\lambda_4  
                      & \omega_1 & \omega_2 & \omega_3 & \omega_4  \\ \hline
\boldsymbol{\theta}_2 & \phi/c^2 & c\lambda_1 &c\lambda_2 &c\lambda_3 & c\lambda_4  
                      & \omega_1 & \omega_2 & \omega_3 & \omega_4 
\end{array} 
\end{displaymath} \pause

Both yield

\begin{displaymath}
    \boldsymbol{\Sigma} = 
\left(\begin{array}{rrrr}
\lambda_{1}^{2} \phi + \omega_{1} & \lambda_{1} \lambda_{2} \phi &
\lambda_{1} \lambda_{3} \phi & \lambda_{1} \lambda_{4} \phi \\
\lambda_{1} \lambda_{2} \phi & \lambda_{2}^{2} \phi + \omega_{2} &
\lambda_{2} \lambda_{3} \phi & \lambda_{2} \lambda_{4} \phi \\
\lambda_{1} \lambda_{3} \phi & \lambda_{2} \lambda_{3} \phi &
\lambda_{3}^{2} \phi + \omega_{3} & \lambda_{3} \lambda_{4} \phi \\
\lambda_{1} \lambda_{4} \phi & \lambda_{2} \lambda_{4} \phi &
\lambda_{3} \lambda_{4} \phi & \lambda_{4}^{2} \phi + \omega_{4}
\end{array}\right) 
\end{displaymath} \pause
The choice $\phi=1$ just sets $c=\sqrt{\phi}$: convenient but seemingly arbitrary.  
 
\end{frame}

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

\begin{frame}
\frametitle{Lack of identifiability}
%\framesubtitle{} 
  \begin{itemize}
    \item For any set of true parameter values, there are infinitely many untrue sets of parameter values that yield the same $\boldsymbol{\Sigma}$ and hence the same probability distribution of the observable data (assuming multivariate normality). 
    \item There is no way to know the full truth based on the data, no matter how large the sample size.
    \item But there is a way to know the partial truth.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Certain \emph{functions} of the parameter vector are identifiable}
%\framesubtitle{} 
At points in the parameter space where $\lambda_1, \lambda_2, \lambda_3 \neq 0$,
\begin{itemize}
     \item $\frac{\sigma_{12}\sigma_{13}}{\sigma_{23}} 
          = \frac{\lambda_1\lambda_2\phi\lambda_1\lambda_3\phi}
                 {\lambda_2\lambda_3\phi} = \lambda_1^2\phi$
     \item And so if $\lambda_1>0$, the function $\lambda_j\phi^{1/2}$ is identifiable \newline for $j = 1, \ldots, 4$. \pause
     \item $\sigma_{11} - \frac{\sigma_{12}\sigma_{13}}{\sigma_{23}} = \omega_1$, and so $\omega_j$ is identifiable for $j = 1, \ldots, 4$. \pause
     \item $\frac{\sigma_{13}}{\sigma_{23}} 
         = \frac{\lambda_1\lambda_3\phi}{\lambda_2\lambda_3\phi}
         = \frac{\lambda_1}{\lambda_2}$, so \emph{ratios} of factor loadings are identifiable.
\end{itemize} 
\end{frame}

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

\begin{frame}
\frametitle{Reliability}
%\framesubtitle{} 

  \begin{itemize}
    \item Reliability is the squared correlation between the observed score and the true score.
    \item The proportion of variance in the observed score that is not error. \pause
    \item For $D_1 = \lambda_1F + e_1$ it's
  \end{itemize}
\begin{eqnarray*}
    \rho^2 & = & \left( \frac{Cov(D_1,F)}{SD(D_1)SD(F)}\right)^2 \\
           & = & \left( \frac{\lambda_1\phi}{\sqrt{\lambda_1^2\phi+\omega_1}\sqrt{\phi}}\right)^2 \\ 
           & = & \frac{\lambda_1^2\phi}{\lambda_1^2\phi+\omega_1}
\end{eqnarray*} \pause
$\lambda_1^2\phi$ and $\omega_1$ are both identifiable, so we've got it.
\end{frame}

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

\begin{frame}
\frametitle{For completeness}
%\framesubtitle{} 
{\small
\begin{displaymath}
    \rho^2 = \frac{\lambda_1^2\phi}{\lambda_1^2\phi+\omega_1} \hspace{5mm}
\boldsymbol{\Sigma} = 
\left(\begin{array}{rrrr}
\lambda_{1}^{2} \phi + \omega_{1} & \lambda_{1} \lambda_{2} \phi &
\lambda_{1} \lambda_{3} \phi & \lambda_{1} \lambda_{4} \phi \\
\lambda_{1} \lambda_{2} \phi & \lambda_{2}^{2} \phi + \omega_{2} &
\lambda_{2} \lambda_{3} \phi & \lambda_{2} \lambda_{4} \phi \\
\lambda_{1} \lambda_{3} \phi & \lambda_{2} \lambda_{3} \phi &
\lambda_{3}^{2} \phi + \omega_{3} & \lambda_{3} \lambda_{4} \phi \\
\lambda_{1} \lambda_{4} \phi & \lambda_{2} \lambda_{4} \phi &
\lambda_{3} \lambda_{4} \phi & \lambda_{4}^{2} \phi + \omega_{4}
\end{array}\right)
\end{displaymath} \pause
} % End size
\begin{eqnarray*}
      \frac{\sigma_{12}\sigma_{13}}{\sigma_{23}\sigma_{11}}  
& = & \frac{\lambda_1\lambda_2\phi\lambda_1\lambda_3\phi}
                 {\lambda_2\lambda_3\phi (\lambda_1^2\phi+\omega_1)} \\ \pause
& = & \frac{\lambda_1^2\phi}{\lambda_1^2\phi+\omega_1} \\
& = & \rho^2
\end{eqnarray*} 
\end{frame}

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

\begin{frame}
\frametitle{What can we successfully estimate?}
%\framesubtitle{} 

  \begin{itemize}
    \item Error variances are knowable.
    \item Factor loadings and variance of the factor are not knowable separately.
    \item But both are knowable up to multiplication by a non-zero constant, so signs of factor loadings are knowable (if one sign is known).
    \item Relative magnitudes (ratios) of factor loadings are knowable.
    \item Reliabilities are knowable.
  \end{itemize}
\end{frame}

% Had a slide here called testing model fit. It claimed that the constraints on the covariances are the same for the original model. That's true for a 1-factor model, but now I'mm not so sure about the general case. See old Powerpoint slides for this one.

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

\begin{frame}
\frametitle{Re-parameterization}
%\framesubtitle{} 

  \begin{itemize}
    \item The choice $\phi=1$ is a very smart re-parameterization.
    \item It re-expresses the factor loadings as multiples of the square root of $\phi$.
    \item That is, in standard deviation units. \pause
    \item It preserves what information is accessible about the parameters of the original model.
    \item Much better than exploratory factor analysis, which lost even the signs of the factor loadings.
    \item This is the second major re-parameterization. The first was losing the the means and intercepts.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Re-parameterizations}
%\framesubtitle{} 
%{\large 
\begin{center}
Original model $\rightarrow$ Surrogate model 1 $\rightarrow$ Surrogate model 2 \ldots
\end{center}
%} % End size
\end{frame}

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

\begin{frame}
\frametitle{Add a factor to the centered original model}
%\framesubtitle{}
\begin{center}
\includegraphics[width=3in]{TwoFactors}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Model Equations}
%\framesubtitle{} 
\begin{eqnarray*}
    d_1 &=& \lambda_1 F_1 + e_1 \\
    d_2 &=& \lambda_2 F_1 + e_2 \\
    d_3 &=& \lambda_3 F_1 + e_3 \\
    d_4 &=& \lambda_4 F_2 + e_4 \\
    d_5 &=& \lambda_5 F_2 + e_5 \\
    d_6 &=& \lambda_6 F_2 + e_6  
\end{eqnarray*}
\vspace{5mm}
{\footnotesize
\begin{displaymath}
cov\left(
\begin{array}{c}
F_1 \\ F_2
\end{array}
\right) = 
\left( \begin{array}{c c}
         \phi_{11}  & \phi_{12} \\
         \phi_{12}  & \phi_{22} \\
       \end{array} \right) ~
\begin{array}{l}
e_1,\ldots,e_6 \mbox{ independent of each other and of } F_1, F_2  \\
\lambda_1, \ldots \lambda_6 \neq 0 \\
Var(e_j) = \omega_j  \\
\end{array} 
\end{displaymath}
} % End size
\end{frame}

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

\begin{frame}
\frametitle{Parameters are not identifiable}
%\framesubtitle{} 
$\boldsymbol{\Sigma} = $ \\ \vspace{2mm}
{\scriptsize
$\left(\begin{array}{rrrrrr}
\lambda_{1}^{2} \phi_{11} + \omega_{1} & \lambda_{1} \lambda_{2}
\phi_{11} & \lambda_{1} \lambda_{3} \phi_{11} & \lambda_{1}
\lambda_{4} \phi_{12} & \lambda_{1} \lambda_{5} \phi_{12} &
\lambda_{1} \lambda_{6} \phi_{12} \\
\lambda_{1} \lambda_{2} \phi_{11} & \lambda_{2}^{2} \phi_{11} +
\omega_{2} & \lambda_{2} \lambda_{3} \phi_{11} & \lambda_{2}
\lambda_{4} \phi_{12} & \lambda_{2} \lambda_{5} \phi_{12} &
\lambda_{2} \lambda_{6} \phi_{12} \\
\lambda_{1} \lambda_{3} \phi_{11} & \lambda_{2} \lambda_{3}
\phi_{11} & \lambda_{3}^{2} \phi_{11} + \omega_{3} & \lambda_{3}
\lambda_{4} \phi_{12} & \lambda_{3} \lambda_{5} \phi_{12} &
\lambda_{3} \lambda_{6} \phi_{12} \\
\lambda_{1} \lambda_{4} \phi_{12} & \lambda_{2} \lambda_{4}
\phi_{12} & \lambda_{3} \lambda_{4} \phi_{12} & \lambda_{4}^{2}
\phi_{22} + \omega_{4} & \lambda_{4} \lambda_{5} \phi_{22} &
\lambda_{4} \lambda_{6} \phi_{22} \\
\lambda_{1} \lambda_{5} \phi_{12} & \lambda_{2} \lambda_{5}
\phi_{12} & \lambda_{3} \lambda_{5} \phi_{12} & \lambda_{4}
\lambda_{5} \phi_{22} & \lambda_{5}^{2} \phi_{22} + \omega_{5} &
\lambda_{5} \lambda_{6} \phi_{22} \\
\lambda_{1} \lambda_{6} \phi_{12} & \lambda_{2} \lambda_{6}
\phi_{12} & \lambda_{3} \lambda_{6} \phi_{12} & \lambda_{4}
\lambda_{6} \phi_{22} & \lambda_{5} \lambda_{6} \phi_{22} &
\lambda_{6}^{2} \phi_{22} + \omega_{6}
\end{array}\right)$
} % End size
\begin{eqnarray*}
    \boldsymbol{\theta}_1 &=& 
    (\lambda_1, \ldots, \lambda_6, \phi_{11}, \phi_{12}, \phi_{22},
    \omega_1, \ldots, \omega_6) \\
    \boldsymbol{\theta}_2 &=& 
    (\lambda_1^\prime, \ldots, \lambda_6^\prime, \phi_{11}^\prime, \phi_{12}^\prime, \phi_{22}^\prime,
    \omega_1^\prime, \ldots, \omega_6^\prime) 
\end{eqnarray*}
$\begin{array}{rrrr}
\lambda_1^\prime = c_1\lambda_1 & \lambda_2^\prime = c_1\lambda_2 & \lambda_3^\prime = c_1\lambda_3 & \phi_{11}^\prime = \phi_{11}/c_1^2 \\
\lambda_4^\prime = c_2\lambda_4 & \lambda_5^\prime = c_2\lambda_5 & \lambda_6^\prime = c_2\lambda_6 & \phi_{22}^\prime = \phi_{22}/c_2^2 \\
\phi_{12}^\prime = \frac{\phi_{12}}{c_1c_2}
\end{array} $ \\
~$\omega_j^\prime = \omega_j \mbox{ for } j = 1, \ldots, 6$
\end{frame}
\begin{comment}
---------------------------------------------------------------------
SageMath code

load sem
L = ZeroMatrix(6,2)
L[0,0]= var('lambda1'); L[1,0]= var('lambda2'); L[2,0]= var('lambda3')
L[3,1]= var('lambda4'); L[4,1]= var('lambda5'); L[5,1]= var('lambda6')
P = SymmetricMatrix(2,'phi')
O = DiagonalMatrix(6,symbol='omega')
Sig = FactorAnalysisVar(L,P,O); Sig
Sig=expand(Sig); Sig
print(latex(Sig))

\end{comment}

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

\begin{frame}
\frametitle{Variances and covariances of factors}
%\framesubtitle{} 

  \begin{itemize}
    \item Are knowable only up to multiplication by  positive constants. \pause
    \item Since the parameters of the latent variable model will be recovered from $\boldsymbol{\Phi} = cov(\mathbf{F})$, they also will be knowable only up to multiplication by positive constants – at best. \pause
    \item Luckily, in most applications the interest is in testing (pos-neg-zero) more than estimation.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{$cov(F_1,F_2)$ is un-knowable, but}
%\framesubtitle{} 

  \begin{itemize}
    \item Easy to tell if it’s zero.
    \item Sign is known if one factor loading from each set is known – say $\lambda_1>0$, $\lambda_4>0$. \pause
    \item And,
\begin{eqnarray*}
\frac{\sigma_{14}}
{\sqrt{\frac{\sigma_{12}\sigma_{13}}{\sigma_{23}}}
 \sqrt{\frac{\sigma_{45}\sigma_{46}}{\sigma_{56}}}}
&=& \frac{\lambda_1\lambda_4\phi_{12}}
         {\lambda_1\sqrt{\phi_{11}}\lambda_4\sqrt{\phi_{22}}} \\ \pause
&=& \frac{\phi_{12}}{\sqrt{\phi_{11}}\sqrt{\phi_{22}}} \\ \pause
&=&  Corr(F_1,F_2)
\end{eqnarray*} 
    \item The \emph{correlation} between factors is identifiable!
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{The correlation between factors is identifiable}
%\framesubtitle{} 

  \begin{itemize}
    \item Furthermore, it is the same function of $\boldsymbol{\Sigma}$ that yields $\phi_{12}$ under the surrogate model with $Var(F_1) = Var(F_2) = 1$. \pause
    \item Therefore, $Corr(F_1,F_2) = \phi_{12}$ under the surrogate model is actually  $Corr(F_1,F_2)$ under the original model. \pause
    \item Estimation is very meaningful.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Setting variances of factors to one}
%\framesubtitle{} 

  \begin{itemize}
    \item Is a \emph{very} smart re-parameterization.
    \item Is excellent when the interest is in correlations between factors.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Re-parameterization as a change of variables}
%\framesubtitle{} 
{\LARGE
\begin{eqnarray*}
d_j &=& \lambda_j F_j + e_j \\ \pause
    &=& (\lambda_j \sqrt{\phi_{jj}}) \left(\frac{1}{\sqrt{\phi_{jj}}}F_j\right) + e_j \\ \pause
    &=& \lambda_j^\prime F_j^\prime + e_j
\end{eqnarray*}  
} % End size
\end{frame}

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

\begin{frame}
\frametitle{Covariances}
%\framesubtitle{} 
{\large
\begin{eqnarray*}
Cov(F^\prime_j, F^\prime_k) &=& E\left(\frac{1}{\sqrt{\phi_{jj}}}F_j
                                       \frac{1}{\sqrt{\phi_{kk}}}F_k\right) \\ \pause
                            &=& \frac{E(F_jF_k)}{\sqrt{\phi_{jj}}\sqrt{\phi_{kk}}} \\ \pause
                            &=& \frac{\phi_{jk}}{\sqrt{\phi_{jj}}\sqrt{\phi_{kk}}} \\ 
&&\\
                            &=& Corr(F_j,F_k)
\end{eqnarray*} % 24\end{eqnarray* 
} % End size
\end{frame}

% Omitted slides: What happens if there is a latent variable model?

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

\begin{frame}
\frametitle{The other standard trick}
%\framesubtitle{} 

  \begin{itemize}
    \item Setting variances of all the factors to one is an excellent re-parameterization in disguise.
    \item The other standard trick is to set a factor loading equal to one for each factor. \pause
    \item $d = F + e$  is hard to believe if you take it literally.
    \item It's actually a re-parameterization.
    \item Every model you've seen with a factor loading of one is a surrogate model.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Back to a single-factor model with $\lambda_1>0$}
%\framesubtitle{} 
\begin{columns}  
\column{0.5\textwidth}
\begin{eqnarray*}
d_1 &=& \lambda_1 F + e_1 \\
d_2 &=& \lambda_2 F + e_2 \\
d_3 &=& \lambda_3 F + e_3  \\
    & \vdots &
\end{eqnarray*} \pause
\column{0.5\textwidth}
\begin{eqnarray*}
d_j &=& \left( \frac{\lambda_j}{\lambda_1} \right) (\lambda_1 F) + e_j \\
    &=& \lambda_j^\prime F^\prime + e_j % 32
\end{eqnarray*}  \pause
\end{columns} \vspace{5mm}

\begin{eqnarray*}
d_1 &=&  F^\prime + e_1 \\
d_2 &=& \lambda_2^\prime F^\prime + e_2 \\
d_3 &=& \lambda_3^\prime F^\prime + e_3  \\
    & \vdots &
\end{eqnarray*}
\end{frame}

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

\begin{frame}
\frametitle{$\boldsymbol{\Sigma}$ under the surrogate model}
%\framesubtitle{} 
\begin{displaymath}
    \boldsymbol{\Sigma} = 
% Edited
\left(\begin{array}{rrr}
\phi + \omega_{1} & \lambda_{2} \phi & \lambda_{3} \phi \\
\lambda_{2} \phi & \lambda_{2}^{2} \phi + \omega_{2} &
\lambda_{2} \lambda_{3} \phi \\
\lambda_{3} \phi & \lambda_{2} \lambda_{3} \phi &
\lambda_{3}^{2} \phi + \omega_{3}
\end{array}\right) 
\end{displaymath} \pause
\vspace{5mm}

\begin{center}
\begin{tabular}{|c|cc|} \hline
                                  & \multicolumn{2}{c|}{Value under model} \\
Function of $\boldsymbol{\Sigma}$ & Surrogate& Original\\ \hline
$\frac{\sigma_{23}}{\sigma_{13}}$ & $\lambda_2$   & 
$\frac{\lambda_2}{\lambda_1}$   \\ \hline
$\frac{\sigma_{23}}{\sigma_{12}}$ & $\lambda_3$   & 
$\frac{\lambda_3}{\lambda_1}$   \\ \hline
$\frac{\sigma_{12}\sigma_{13}}{\sigma_{23}}$ & $\phi$   & 
$\lambda_1^2\phi$   \\ \hline
\end{tabular} 
\end{center}
\end{frame}

\begin{comment}
% Sage
load sem
L = ZeroMatrix(3,1)
L[0,0]= 1; L[1,0]= var('lambda2'); L[2,0]= var('lambda3')
P = ZeroMatrix(1,1); P[0,0] = var('phi'); P
O = DiagonalMatrix(3,symbol='omega')
Sig = FactorAnalysisVar(L,P,O); Sig
print(latex(Sig))
\end{comment}

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

\begin{frame}
\frametitle{Under the surrogate model}
%\framesubtitle{} 
\begin{itemize}
    \item It looks like $\lambda_j$ is identifiable, but actually it's $\lambda_j/\lambda_1$.
    \item Estimates of $\lambda_j$ for $j\neq 1$ are actually estimates of $\lambda_j/\lambda_1$.
    \item It looks like $\phi$ is identifiable, but actually it's $\lambda_1^2\phi$.
    \item $\phi$ is being expressed as a multiple of $\lambda_1^2$.
    \item Estimates of $\phi$ are actually estimates of $\lambda_1^2\phi$. \pause
    \item[]
    \item Make $d_1$ the clearest representative of the factor.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Add an observable variable to the surrogate model}
%\framesubtitle{} 
  \begin{itemize}
    \item Parameters are all identifiable, even if the factor loading of the new variable equals zero.
    \item Equality restrictions on $\boldsymbol{\Sigma}$ are created, because we are adding more equations than unknowns.
    \item These equality restrictions apply to the original model.
    \item It is straightforward to see what the restrictions are, though the calculations can be time consuming.
  \end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Finding the equality restrictions}
%\framesubtitle{} 
  \begin{itemize}
    \item Calculate $\boldsymbol{\Sigma}(\boldsymbol{\theta})$.
    \item Solve the covariance structure equations explicitly, obtaining $\boldsymbol{\theta}$ as a function of $\boldsymbol{\Sigma}$.
    \item Substitute the solutions back into $\boldsymbol{\Sigma}(\boldsymbol{\theta})$.
    \item Simplify. 
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Example: Add a 4th variable}
%\framesubtitle{} 
%{\LARGE
\begin{eqnarray*}
   D_1 &=& ~~~ F + e_1 \\
D_2 &=& \lambda_2 F + e_2 \\
D_3 &=& \lambda_3 F + e_3 \\
D_4 &=& \lambda_4 F + e_4
\end{eqnarray*} 

\begin{displaymath}
\begin{array}{l}
e_1,\ldots, e_4, F \mbox{ all independent} \\
Var(e_j) = \omega_j ~~~~ Var(F) = \phi \\
\lambda_1, \lambda_2, \lambda_3 \neq 0
\end{array}     
\end{displaymath}
%} % End size
\end{frame}

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



\begin{frame}
\frametitle{Covariance Matrix}
%\framesubtitle{} 
\begin{displaymath}
    \boldsymbol{\Sigma}(\boldsymbol{\theta}) = 
% Edit upper left cell
\left(\begin{array}{rrrr}
\phi+\omega_{1} & \lambda_{2} \phi & \lambda_{3} \phi &
\lambda_{4} \phi \\
\lambda_{2} \phi & \lambda_{2}^{2} \phi + \omega_{2} &
\lambda_{2} \lambda_{3} \phi & \lambda_{2} \lambda_{4} \phi \\
\lambda_{3} \phi & \lambda_{2} \lambda_{3} \phi &
\lambda_{3}^{2} \phi + \omega_{3} & \lambda_{3} \lambda_{4} \phi \\
\lambda_{4} \phi & \lambda_{2} \lambda_{4} \phi & \lambda_{3}
\lambda_{4} \phi & \lambda_{4}^{2} \phi + \omega_{4}
\end{array}\right) 
\end{displaymath} \pause
\begin{columns}  
\column{0.5\textwidth}
Solutions 

\begin{itemize}
     \item[]$\lambda_2 = \frac{\sigma_{23}}{\sigma_{13}}$
     \item[]$\lambda_3 = \frac{\sigma_{23}}{\sigma_{12}}$
     \item[]$\lambda_4 = \frac{\sigma_{24}}{\sigma_{12}}$
     \item[]$\phi = \frac{\sigma_{12}\sigma_{13}}{\sigma_{23}}$
\end{itemize} \pause
\column{0.5\textwidth}
Substitute 
\begin{eqnarray*}
\sigma_{12} &=& \lambda_2 \phi \\ \pause
            &=& \frac{\sigma_{23}}{\sigma_{13}} 
                \frac{\sigma_{12}\sigma_{13}}{\sigma_{23}} \\ \pause
            &=& \sigma_{12}
\end{eqnarray*}
\end{columns}
\end{frame}

% Sagemath
\begin{comment}
load sem
L = ZeroMatrix(4,1)
L[0,0]= 1; L[1,0]= var('lambda2'); L[2,0]= var('lambda3')
L[3,0]= var('lambda4')
P = ZeroMatrix(1,1); P[0,0] = var('phi'); P
O = DiagonalMatrix(4,symbol='omega')
Sig = FactorAnalysisVar(L,P,O); Sig
print(latex(Sig))

\end{comment}

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

\begin{frame}
\frametitle{Substitute solutions into expressions for the covariances}
%\framesubtitle{} 


\begin{eqnarray*}
\sigma_{12} &=& \sigma_{12} \\
\sigma_{13} &=& \sigma_{13} \\
\sigma_{14} &=& \frac{\sigma_{24}\sigma_{13}}{\sigma_{23}} \\
\sigma_{23} &=& \sigma_{23} \\
\sigma_{24} &=& \sigma_{24} \\
\sigma_{34} &=& \frac{\sigma_{24}\sigma_{13}}{\sigma_{12}}
\end{eqnarray*}
\end{frame}

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

\begin{frame}
\frametitle{Equality Constraints}
%\framesubtitle{} 
\begin{eqnarray*}
\sigma_{14}\sigma_{23} &=& \sigma_{24}\sigma_{13} \\
\sigma_{12}\sigma_{34} &=& \sigma_{24}\sigma_{13} 
\end{eqnarray*} \pause
These hold regardless of whether factor loadings are zero (1234).
\begin{displaymath}
    \sigma_{12}\sigma_{34} = \sigma_{13}\sigma_{24} = \sigma_{14}\sigma_{23} 
\end{displaymath}
\end{frame}

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

\begin{frame}
\frametitle{Add another 3-variable factor}
%\framesubtitle{} 

  \begin{itemize}
    \item Identifiability is maintained. \pause
    \item The covariance $\phi_{12} = \sigma_{14}$
    \item Actually $\sigma_{14} = \lambda_1 \lambda_4 \phi_{12}$ under the original model.
    \item The covariances of the surrogate model are just those of the surrogate model, multiplied by un-knowable positive constants.
    \item As more variables and more factors are added, all this remains true.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Comparing the surrogate models}
%\framesubtitle{} 

  \begin{itemize}
    \item Either set variances of factors to one, or set one loading per factor to one. \pause
    \item Both arise from a similar change of variables.
    \item $F^\prime_j = \lambda_jF_j$ or $F^\prime_j = \frac{1}{\sqrt{\phi_{jj}}} F_j$. \pause
    \item \emph{Meaning} of surrogate model parameters in terms of the original model is different except for the signs.
    \item Both surrogate models share the same equality constraints, and hence the same goodness of fit results for any given data set. 
    \item Are these constraints also true of the original model?
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{The Equivalence Rule}
\framesubtitle{See text for proof} 
For a centered factor analysis model with at least one reference variable for each factor, suppose that surrogate models are obtained by either standardizing the factors, or by setting the factor loading of a reference variable equal to one for each factor. Then the parameters of one surrogate model are identifiable if and only if the parameters of the other surrogate model are identifiable.
\end{frame}

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

\begin{frame}
\frametitle{Which re-parameterization is better?}
%\framesubtitle{} 
{\small
  \begin{itemize}
    \item Technically, they are equivalent.
    \item Interpretation of the surrogate model parameters is different. \pause
%    \item They both involve setting a single un-knowable parameter to one, for each factor. 
%    \item This seems arbitrary, but actually it results in a very good re-parameterization that preserves what is knowable about the true model.
    \item Standardizing the factors (Surrogate model 2A) is more convenient for estimating correlations between factors.
    \item Setting one loading per factor equal to one (Surrogate model 2B) is more convenient for estimating the relative sizes of factor loadings.
    \item Hand calculations and identifiability proofs with Surrogate model 2B can be easier.
    \item If there is a serious latent variable model, Surrogate model 2B is much easier to specify with lavaan.
    \item Mixing Surrogate model 2B with double measurement is natural. \pause
    \item Don't do both restrictions for the same factor!
  \end{itemize}
  } % End size
\end{frame}

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

\begin{frame}
\frametitle{Why are we doing this? To buy identifiability.}
%\framesubtitle{} 

  \begin{itemize}
    \item The parameters of the original model cannot be estimated directly. For example, maximum likelihood will fail because the maximum is not unique.
    \item The parameters of the surrogate models are identifiable (estimable) functions of the parameters of the true model. \pause
    \item They have the same signs (positive, negative or zero) as the corresponding parameters of the true model.
    \item Hypothesis tests mean what you think they do.
    \item Parameter estimates can be useful if you know what the new parameters mean.  
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{The Crossover Rule}
%\framesubtitle{} 

  \begin{itemize}
    \item It is unfortunate that variables can only be caused by one factor.  In fact, it’s unbelievable most of the time.
    \item A pattern like this would be nicer.
  \end{itemize}
\begin{center}
\includegraphics[width=3in]{RealEstate}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{The Crossover Rule}
\framesubtitle{A weak version of the extra variables rule} 
When you add a set of observable variables to a measurement model whose parameters are \textbf{already identifiable}
  \begin{itemize}
    \item Straight arrows with factor loadings on them may point from each existing factor to each new variable.
    \item  You don't need to include all such arrows.
    \item Error terms for the new set of variables may have non-zero covariances with each other, but not with the error variances or factors of the original model.
    \item Some of the new error terms may have zero covariance with each other.  It’s up to you.
    \item All parameters of the new model are identifiable.
  \end{itemize}
\end{frame}

% The CFA2 Powerpoint slides have a proof of this. One can do better.

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

\begin{frame}
\frametitle{The Crossover Rule}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=4in]{OldCrossover}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{We can do a bit better}
\framesubtitle{Include more covariances between error terms.} 
Call it the extra variables rule.
\end{frame}

% Use hand proof, go back and put in arrows.

% Had "Add an observed variable to the factors," but I did that back in the general model.

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

\begin{frame}
\frametitle{We have some identifiability rules}
%\framesubtitle{} 

  \begin{itemize}
    \item Double Measurement rule.
    \item Scalar three-variable rules.
    \item The equivalence rule.
    \item Combination rule. % Did this one on the board I think; I have it in the hand notes. It's better than the text.
    \item Extra variable rule (enhanced cross-over rule)
    \item Error-free rule. \pause
    \item Need the 2-variable rules.
    \item Need the vector 3-variable rule.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Two-variable Rule}
%\framesubtitle{} 
The parameters of a factor analysis model are identifiable provided
  \begin{itemize}
    \item There are two factors.
    \item There are two reference variables for each factor.
    \item For each factor, either the variance equals one and the sign of one factor loading is known, or the factor loading of at least one reference variable is equal to one.
    \item The two factors have non-zero covariance.
    \item Errors are independent of one another and of the factors. 
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Two-variable Addition Rule}
%\framesubtitle{} 
A factor with just two reference variables may be added to a measurement model whose parameters are identifiable, and the parameters of the combined model will be identifiable provided
    \begin{itemize}
        \item The errors for the two additional reference variables are independent of one another and of the error terms already in the model.
        \item For each factor, either the variance equals one and the sign of one factor loading is known, or the factor loading of at least one reference variable is equal to one.
        \item In the existing model with identifiable parameters, 
                \begin{itemize}
                    \item There is at least one reference variable for each factor, and
                    \item At least one factor has a non-zero covariance with the new factor.
                \end{itemize}
    \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Vector 3-variable rule}
%\framesubtitle{} 
Let
\begin{eqnarray*}
   \mathbf{d}_1 & = & \mathbf{F} + \mathbf{e}_1 \\
   \mathbf{d}_2 & = & \boldsymbol{\Lambda}_2\mathbf{F} + \mathbf{e}_2 \\
   \mathbf{d}_3 & = & \boldsymbol{\Lambda}_3\mathbf{F} + \mathbf{e}_3 \\
\end{eqnarray*} 
where
  \begin{itemize}
    \item $\mathbf{F}$, $\mathbf{d}_1$ and $\mathbf{d}_2$ and $\mathbf{d}_3$ are all $p \times 1$.
    \item $\boldsymbol{\Lambda}_2$ and $\boldsymbol{\Lambda}_3$ have inverses.
    \item $cov(\mathbf{F},\mathbf{e}_j) = cov(\mathbf{e}_i,\mathbf{e}_j) = \mathbf{O}$
  \end{itemize} \pause \vspace{4mm}

This is not quite enough.
\end{frame}

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

\begin{frame}
\frametitle{Scalar 3-variable rule}
\framesubtitle{Put some more arrows} 
\begin{center}
\includegraphics[width=3in]{TwoFactors}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now some pictures from past lectures
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\begin{frame}
\frametitle{Brand Awareness Model 1}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3.5in]{Doughnut1}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Brand Awareness Model 2}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=4.5in]{Doughnut2}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Brand Awareness Model 3}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3in]{Doughnut3}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Brand Awareness Model 4}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=4.5in]{Doughnut4}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Brand Awareness Model 5}
%\framesubtitle{} 
\begin{center}
\includegraphics[width=3.5in]{Doughnut5}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{A big complicated measurement model} % From the text
%\framesubtitle{} 
\begin{center}
\vspace{-5mm}
\includegraphics[width=3.5in]{bigCFA}
\end{center}
\end{frame}

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

\begin{frame}
\frametitle{Bifactor Model} % From Uli
\framesubtitle{From Uli Schimmack's blog} 
\begin{center}
\vspace{-12mm}
\includegraphics[width=3in]{Bifactor}
\end{center}
\end{frame}

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

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

\end{frame}


\end{document}

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

\begin{frame}
\frametitle{Frame Title}
%\framesubtitle{} 

  \begin{itemize}
    \item 
    \item 
    \item 
  \end{itemize}
\end{frame}

{\LARGE
\begin{displaymath}
    
\end{displaymath} 
} % End size


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