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\setbeamertemplate{footline}[frame number] 

\mode<presentation>

\title{Categorical Independent Variables\footnote{See last slide for copyright information.}}
\subtitle{STA 302 Fall 2017}
\date{} % To suppress date

\begin{document}

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

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

% No interactions in 2017

\section{Indicators with Intercept}

\begin{frame}
\frametitle{Independent variables need not be continuous} \pause
%\framesubtitle{} 
Code data so that $x=1$ means Drug, $x=0$ means Placebo. \pause
\vspace{5mm}

  \begin{itemize}
    \item Population mean response is $E(y|x) = \beta_0 + \beta_1 x$. \pause
    \item For patients getting the drug, mean response is \pause $E(y|x=1) = \beta_0 + \beta_1$. \pause
    \item For patients getting the placebo, mean response is \pause $E(y|x=0) = \beta_0$. \pause
    \item Difference (treatment effect) is $\beta_1$.  \pause
    \item Test $H_0:\beta_1=0$.
  \end{itemize}
\end{frame}

\begin{frame}
\frametitle{Scatterplot}
\framesubtitle{Showing the least-squares line}  \pause
\begin{columns}  
\column{0.6\textwidth}
\includegraphics[width=2.5in]{IndicatorScatter} \pause
\column{0.4\textwidth}
Predicted response is $\widehat{y} = b_0 + b_1x $. \pause
\vspace{3mm}
  \begin{itemize}
    \item[] For patients getting the drug, predicted response is \pause $\widehat{y} = b_0 + b_1 \pause = \overline{y}_1$. \pause
    \item[] 
    \item[] For patients getting the placebo, predicted response is \pause $\widehat{y} = b_0 \pause  = \overline{y}_0 $.
  \end{itemize}
  \vspace{10mm}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{More than Two Categories} \pause 
%\framesubtitle{}
Suppose a study has 3 treatment conditions. \pause   For example 
\begin{itemize}
     \item Group 1 gets Drug 1
     \item Group 2 gets Drug 2
     \item Group 3 gets a placebo \pause
     \item So that the Explanatory Variable is Group \pause
     \item Taking values 1,2,3. \pause
     \item The dependent variable $y$ is response to drug. \pause
\end{itemize}

\vspace{10mm}

Why is $E(y|x)  =  \beta_0 + \beta_1x$ (with $x$ = Group) a silly model?
\end{frame}

\begin{frame}
\frametitle{Indicator Dummy Variables}
\framesubtitle{With intercept}  \pause 
\begin{itemize}
    \item $x_1 = 1$ if Drug A, zero otherwise
    \item $x_2 = 1$ if Drug B, zero otherwise \pause 
    \item $E(y|\boldsymbol{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$. \pause 
    \item Fill in the table. \pause 
\end{itemize}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $E(Y|\mathbf{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$ \\ \hline
$A$     &       &       &  $\mu_1$ =                            \\ \hline
$B$     &       &       &  $\mu_2$ =                            \\ \hline
Placebo &       &       &  $\mu_3$ =                            \\ \hline
\end{tabular}
\end{center}}
\end{frame}

\begin{frame}
\frametitle{Answer}
\begin{itemize}
    \item $x_1 = 1$ if Drug A, zero otherwise
    \item $x_2 = 1$ if Drug B, zero otherwise
    \item $E(y|\boldsymbol{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$. \pause 
\end{itemize}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $E(Y|\mathbf{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$ \\ \hline
$A$     &   1   &   0   &  $\mu_1$ =     $\beta_0 + \beta_1$    \\ \hline
$B$     &   0   &   1   &  $\mu_2$ =     $\beta_0 + \beta_2$    \\ \hline
Placebo &   0   &   0   &  $\mu_3$ =     $\beta_0$              \\ \hline
\end{tabular}
\end{center}} \pause 
Regression coefficients are contrasts with the category that has no indicator -- the \emph{reference category}.
\end{frame}

\begin{frame}
\frametitle{Indicator dummy variable coding with intercept} \pause 
%\framesubtitle{} 
\begin{itemize}
    \item With an intercept in the model, need $p-1$ indicators to represent a categorical explanatory variable with $p$ categories. \pause 
    \item If you use $p$ dummy variables and also an intercept, trouble. \pause 
    \item Indicators would add up to the intercept and columns of $X$ would be linearly dependent. \pause
    \item Regression coefficients are contrasts with the category that has no indicator. \pause 
    \item Call this the \emph{reference category}.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{$x_1 = 1$ if Drug A, zero o.w., $x_2 = 1$ if Drug B, zero o.w.} \pause
%\framesubtitle{3-d Scatterplot}
Recall $\sum_{i=1}^n (y_i-m)^2$ is minimized at $m = \overline{y}$ \pause
\begin{center}
\includegraphics[width=3in]{ABCscatter}
\end{center}
\end{frame}

\begin{frame}
\frametitle{What null hypotheses would you test?} \pause
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $E(Y|\mathbf{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$ \\ \hline 
$A$     &   1   &   0   &  $\mu_1$ =     $\beta_0 + \beta_1$    \\ \hline
$B$     &   0   &   1   &  $\mu_2$ =     $\beta_0 + \beta_2$    \\ \hline
Placebo &   0   &   0   &  $\mu_3$ =     $\beta_0$              \\ \hline
\end{tabular}
\end{center}} \pause
\begin{itemize}
    \item Is the effect of Drug $A$ different from the placebo?  \pause $H_0: \beta_1=0$ \pause
    \item Is Drug $A$ better than the placebo? \pause $H_0: \beta_1=0$ \pause
    \item Did Drug $B$ work? \pause $H_0: \beta_2=0$ \pause
    \item Did experimental treatment have an effect?  \pause $H_0: \beta_1=\beta_2=0$ \pause
    \item Is there a difference between the effects of Drug $A$ and Drug $B$? \pause $H_0: \beta_1=\beta_2$ 
\end{itemize}
\end{frame}

% PP Slide 18
\begin{frame}
\frametitle{Now add a quantitative explanatory variable (covariate)}
\framesubtitle{Covariates often come first in the regression equation} \pause 
\begin{itemize}
    
    \item $x_1 = 1$ if Drug A, zero otherwise
    \item $x_2 = 1$ if Drug B, zero otherwise 
    \item $x_3$ = Age \pause 
    \item $E(y|\boldsymbol{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2 + \beta_3 x_3$. \pause 
\end{itemize}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $E(Y|\mathbf{x}) = \beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3$\\ \hline
A       &   1   &   0   &  $\mu_1$ = $(\beta_0+\beta_1)+\beta_3x_3$    \\ \hline
B       &   0   &   1   &  $\mu_2$ = $(\beta_0+\beta_2)+\beta_3x_3$    \\ \hline
Placebo &   0   &   0   &  $\mu_3$ = ~~~~~$\beta_0$~~~~~$+\beta_3x_3$              \\ \hline
\end{tabular}
\end{center}} \pause 
Parallel regression lines.
\end{frame}

\begin{frame}
\frametitle{More comments}
%\framesubtitle{} 
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $E(Y|\mathbf{x}) = \beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3$\\ \hline
A       &   1   &   0   &  $\mu_1$ = $(\beta_0+\beta_1)+\beta_3x_3$    \\ \hline
B       &   0   &   1   &  $\mu_2$ = $(\beta_0+\beta_2)+\beta_3x_3$    \\ \hline
Placebo &   0   &   0   &  $\mu_3$ = ~~~~~$\beta_0$~~~~~$+\beta_3x_3$              \\ \hline
\end{tabular}
\end{center}} \pause 
\begin{itemize}
    \item If more than one covariate, parallel regression planes. \pause 
    \item Non-parallel (interaction) is testable.  \pause 
    \item ``Controlling" interpretation holds. \pause 
    \item In an experimental study, quantitative covariates are usually just observed. \pause 
    \item Could age be related to drug? \pause 
    \item Good covariates reduce $s^2 = \frac{\mathbf{e}^\prime\mathbf{e}}{n-k-1}$, and make tests involving the categorical variables more sensitive.
\end{itemize}
\end{frame}

\section{Cell means coding}

\begin{frame}
\frametitle{Cell means coding: $p$ indicators and no intercept}
%\framesubtitle{} 
Example: Three treatments and no covariate. \pause
{\LARGE
\begin{displaymath}
    E(y|\boldsymbol{x}) = \beta_1x_1 + \beta_2 x_2 + \beta_3 x_3
\end{displaymath}  \pause 
} % End size

\vspace{3mm}

\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
Drug  &$x_1$&$x_2$&$x_3$&$E(y|\mathbf{x}) = \beta_1x_1+\beta_2x_2+\beta_3x_3$ \\ \hline
A       & 1   &   0 &  0  &$\mu_1=\beta_1$                    \\ \hline
B       & 0   &   1 &  0  &$\mu_2=\beta_2$                    \\ \hline
Placebo & 0   &   0 &  1  &$\mu_3=\beta_3$                    \\ \hline
\end{tabular}
\end{center} \pause 

\vspace{3mm}

\begin{itemize}
    \item This model is equivalent to the one with $p-1$ dummy variables and the intercept. \pause 
    \item If you have $p$ dummy variables and the intercept, the model is over-parameterized.
\end{itemize}
\end{frame}


\begin{frame}
\frametitle{Add a covariate: $x_4$}
%\framesubtitle{} 
{\LARGE
\begin{displaymath}
    E(y|\boldsymbol{x}) = \beta_1x_1 + \beta_2 x_2 + \beta_3 x_3  + \beta_4 x_4
\end{displaymath}  \pause 
} % End size

\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
Drug  &$x_1$&$x_2$&$x_3$&$E(Y|\mathbf{x}) = \beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4$ \\ \hline
A       & 1   &   0 &  0  &$\beta_1+\beta_4x_4$   \\ \hline
B       & 0   &   1 &  0  &$\beta_2+\beta_4x_4$   \\ \hline
Placebo & 0   &   0 &  1  &$\beta_3+\beta_4x_4$   \\ \hline
\end{tabular}
\end{center} \pause 
This model is equivalent to the one with the intercept.
\end{frame}

\begin{frame}
\frametitle{Key to the equivalence of dummy variable coding schemes} \pause 
% \framesubtitle{} 
Clearly these $X$ matrices are one-to-one.

\begin{displaymath}
\left(\begin{array}{cccc}
1 & 1 & 0 & x_1 \\
1 & 0 & 1 & x_2 \\
1 & 0 & 0 & x_3 \\
1 & 1 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
1 & 0 & 1 & x_n \\
\end{array}\right) \leftrightarrow
\left(\begin{array}{cccc}
1 & 0 & 0 & x_1 \\
0 & 1 & 0 & x_2 \\
0 & 0 & 1 & x_3 \\
1 & 0 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
0 & 1 & 0 & x_n \\
\end{array}\right)
\end{displaymath} \pause
And it's a linear transformation.
\end{frame}

\begin{frame}
\frametitle{Matrix multiplication} \pause 
% \framesubtitle{} 

\begin{displaymath} 
\left(\begin{array}{cccc}
{\color{red}1} & {\color{red}1} & {\color{red}0} & x_1 \\
{\color{red}1} & {\color{red}0} & {\color{red}1} & x_2 \\
{\color{red}1} & {\color{red}0} & {\color{red}0} & x_3 \\
1 & 1 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
1 & 0 & 1 & x_n \\
\end{array}\right) 
\left(\begin{array}{rrrr}
{\color{red}0} & ~{\color{red}0} & {\color{red}1} & ~0 \\
{\color{red}1} & ~{\color{red}0} & {\color{red}-1} & ~0 \\
{\color{red}0} & ~{\color{red}1} & {\color{red}-1} & ~0 \\
0 & ~0 &  0 & ~1 
\end{array}\right)
=
\left(\begin{array}{cccc}
1 & 0 & 0 & x_1 \\
0 & 1 & 0 & x_2 \\
0 & 0 & 1 & x_3 \\
1 & 0 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
0 & 1 & 0 & x_n \\
\end{array}\right)
\end{displaymath} \pause
{\LARGE
\begin{eqnarray*}
     &  & \mathbf{y} = X \boldsymbol{\beta} + \boldsymbol{\epsilon}  \\  \pause 
     & \Leftrightarrow &  \mathbf{y} = (XA)(A^{-1} \boldsymbol{\beta}) + \boldsymbol{\epsilon} 
\end{eqnarray*} \pause
} % End size
Transformed $X$ requires a transformed $\boldsymbol{\beta}$.
\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/302f17} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f17}}

\end{frame}


\end{document}

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

\begin{frame}
\frametitle{}
%\framesubtitle{} 
  \begin{itemize}
    \item 
    \item 
    \item 
  \end{itemize}
\end{frame}

{\LARGE
\begin{displaymath}
    
\end{displaymath} }






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
rm(list=ls()); n=20

# Binary
zero = rnorm(n,4,1); one = rnorm(n,6,1)
x = c(rep(0,n),rep(1,n))
y = c(zero,one)
plot(x,y)
xpts = c(0,1); ypts = c(mean(zero),mean(one))
lines(xpts,ypts)

# 3-d
x1 = c(0,0,1,1); x2 = c(0,1,0,1)
plot(x1,x2,pch=' ',xlab=expression(x[1]),ylab=expression(x[2]))
text(1,0,'A'); text(0,1,'B');  text(0,0,'C')


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


\begin{frame}
\frametitle{Matrix multiplication} \pause 
% \framesubtitle{} 

\begin{displaymath}
\left(\begin{array}{cccc}
{\color{red}1} & 1 & 0 & x_1 \\
1 & 0 & 1 & x_2 \\
1 & 0 & 0 & x_3 \\
1 & 1 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
1 & 0 & 1 & x_n \\
\end{array}\right) 
\left(\begin{array}{rrrr}
0 & ~0 &  1 & ~0 \\
1 & ~0 & -1 & ~0 \\
0 & ~1 & -1 & ~0 \\
0 & ~0 &  0 & ~1 
\end{array}\right)
=
\left(\begin{array}{cccc}
1 & 0 & 0 & x_1 \\
0 & 1 & 0 & x_2 \\
0 & 0 & 1 & x_3 \\
1 & 0 & 0 & x_4 \\
\vdots & \vdots & \vdots & \vdots \\
0 & 1 & 0 & x_n \\
\end{array}\right)
\end{displaymath} \pause
And it's a linear transformation.
\end{frame}


% Not used: 2101 material was better


\begin{frame}
\frametitle{Three categories}
%\framesubtitle{} 
Suppose $x_1=1$ if the subject is in Group 1, and zero otherwise, and $x_2=1$ if the subject is in Group 2, and zero otherwise, and 
$E(y|\boldsymbol{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$. Fill in the table below. \pause

{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug    & $x_1$ & $x_2$ &  $\beta_0 + \beta_1x_1 + \beta_2 x_2$ \\ \hline
$A$     &       &       &  $\mu_1$ =                            \\ \hline
$B$     &       &       &  $\mu_2$ =                            \\ \hline
Placebo &       &       &  $\mu_3$ =                            \\ \hline
\end{tabular}
\end{center}}
\end{frame}


\begin{frame}
\frametitle{Answer}
%\framesubtitle{} 
$x_1=1$ if the subject is in Group 1, and zero otherwise, and $x_2=1$ if the subject is in Group 2, and zero otherwise, and 
$E(y|\boldsymbol{x}) = \beta_0 + \beta_1x_1 + \beta_2 x_2$.

\vspace{4mm}
{\begin{center}
\begin{tabular}{|c|c|c|l|} \hline
Drug   & $x_1$ & $x_2$ &  $\beta_0 + \beta_1x_1 + \beta_2 x_2$ \\ \hline
$A$     &   1   &   0   &  $\mu_1$ =     $\beta_0 + \beta_1$    \\ \hline
$B$     &   0   &   1   &  $\mu_2$ =     $\beta_0 + \beta_2$    \\ \hline
Placebo &   0   &   0   &  $\mu_3$ =     $\beta_0$              \\ \hline
\end{tabular}
\end{center}} \pause
\vspace{3mm}
$x_1$ and $x_2$ are called \emph{dummy variables}.
\end{frame}