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Categorical Predictors 1 
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Categorical 1
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Paul E.
 Johnson
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1
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and
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Department of Political Science
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and
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Center for Research Methods and Data Analysis, University of Kansas
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K.U.
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2014
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2014
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Basics: Before I get too carried away
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Categorical Coding: Which Dummy is Right for you?
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Differences among approaches are Superficial
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Basics
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Dichotomy
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frametitle{Let's Talk About Sex}
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Sex is coded 
\begin_inset Quotes eld
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M
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 for male or 
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F
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 for female
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manually
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 create two dummy variables, 
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femd
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 and 
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maled
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These are numeric, 0 or 1 (or maybe -1 and 1).
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In SAS (or Stata), one then fits a model using 
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femd
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 or 
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maled
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 as a predictor.
 
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maled
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inline false
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lm (y ~ sex)
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 fits 
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(implicitly) asks for an intercept, plus 
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an 
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intercept shift
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 parameter for a contrast variable for males it calls 
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sexM
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.
 
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R automatically creates a 
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contrast
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 variable, a 0, 1 
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 variable for male
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frametitle{Example: statusquo support in the 1988 Chile Data}
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<<chile20, echo=T, include=F, eval=F>>=
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library(car)
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mod1 <- lm(statusquo ~ sex, data=Chile)
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summary(mod1)
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<<chile21, echo=F, include=F>>=
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library(rockchalk)
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outreg(mod1, tight=FALSE, showAIC=FALSE)
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R's design matrix that looks like this: 
\begin_inset Formula 
\begin{equation}
X=\begin{array}{cc}
constant & sexM\\
1 & 1\\
1 & 0\\
1 & 0\\
 & \vdots
\end{array}
\end{equation}

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Why 
\begin_inset Quotes eld
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M
\begin_inset Quotes erd
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? Female becomes 
\begin_inset Quotes eld
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baseline
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 (in the intercept) because it is alphabetically first (can customize that)
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Same effect as user-created 
\begin_inset Quotes eld
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maled
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 variable.
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fitted intercept represents the effect of 
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being human
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 (or 
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being in the data set
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)
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\begin_layout Itemize
\begin_inset Formula $\hat{b}_{1}sexM$
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; the 
\begin_inset Quotes eld
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difference
\begin_inset Quotes erd
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 effect that distinguishes males from other humans
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\begin_layout Itemize
Model's predicted value is 
\begin_inset Formula $\widehat{statusquo_{i}}=\hat{b}_{0}+\hat{b}_{1}sexM$
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, so for Females predict 
\begin_inset Formula $\hat{b}_{0}$
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 and for males predict 
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.
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frametitle{Regression Equivalent to a "t-test for means"}
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The 
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t test for means
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 calculates the averages within groups and calculates a t value for the
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<<chile10, include=T, echo=T>>=
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by(Chile$statusquo, Chile$sex, mean, na.rm = TRUE)
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t.test(statusquo ~ sex, var.equal=TRUE, data=Chile) 
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@
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Note the Regression intercept and slope re-produce means as predicted values.
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Multichotomy (Polychotomy?)
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As provided, wages has occupation coded as a numeric variable.
 
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<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Management
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Sales
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Clerical
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Service
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Professional
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Other
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages00, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

library(rockchalk)
\end_layout

\begin_layout Plain Layout

dat <- read.table("/home/pauljohn/ps/SVN-guides/stat/DataSets/wages/wages.txt",
 header=T, sep="") 
\end_layout

\begin_layout Plain Layout

colnames(dat) <- tolower(colnames(dat))
\end_layout

\begin_layout Plain Layout

dat$racef <- factor(dat$race, labels=c("Other", "Hispanic","White")) 
\end_layout

\begin_layout Plain Layout

dat$occupationf <- factor(dat$occupation, labels=c("Management", "Sales",
 "Clerical", "Service", "Professional", "Other")) 
\end_layout

\begin_layout Plain Layout

dat$marrf <- factor(dat$marr, levels=c("Unmarried","Married"))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{See Why it is Wrong to treat that as Numeric, Right?}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages10, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

mod1 <- lm(wage ~ occupation, data=dat) 
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages10B, include=F, echo=F, results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(mod1, tight=F, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{Sinput}
\end_layout

\begin_layout Plain Layout

mod1 <- lm(wage ~ occupation, data=dat) 
\end_layout

\begin_layout Plain Layout


\backslash
end{Sinput}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages10B.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Interpret that Termplot}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages10C,fig=T,include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

termplot(mod1, terms=c("occupation"), partial=T, se=T)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=10cm]{plots/t-wages10C}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Recode, Treat Occupation as A Categorical Variable}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Create a new 
\begin_inset Quotes eld
\end_inset

factor
\begin_inset Quotes erd
\end_inset

 variable occupationf, that assigns labels to the categories.
\end_layout

\begin_layout Itemize
When there are 6 occupational categories, the usual approach creates 5 
\begin_inset Quotes eld
\end_inset

dummy variables
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Itemize
In R, those 5 dummy variables are created automatically, called 
\begin_inset Quotes eld
\end_inset

treatment contrasts
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Quotes eld
\end_inset

first
\begin_inset Quotes erd
\end_inset

 level of factor (or designated level) is excluded, and rest of levels are
 
\begin_inset Quotes eld
\end_inset

dummied up
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{What is R Doing with "occupationf"?}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
R's system of 
\begin_inset Quotes eld
\end_inset

factor
\begin_inset Quotes erd
\end_inset

 variables is intended to make this 
\begin_inset Quotes eld
\end_inset

automatic
\begin_inset Quotes erd
\end_inset

.
 Regression procedures create 
\begin_inset Quotes eld
\end_inset

contrasts
\begin_inset Quotes erd
\end_inset

 
\begin_inset Quotes eld
\end_inset

on the fly
\begin_inset Quotes erd
\end_inset

.
 
\end_layout

\begin_layout Itemize
The factor 
\begin_inset Quotes eld
\end_inset

occupationf
\begin_inset Quotes erd
\end_inset

 is converted thus
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages25A, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

contrasts(dat$occupationf)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout


\backslash
input{plots/t-wages25A}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
So the fitted model for 6 categories is
\begin_inset Formula 
\begin{equation}
\widehat{wages}_{i}=\hat{b}_{0}+\hat{b}_{1}Sales_{i}+\hat{b}_{2}Clerical_{i}+\hat{b}_{3}Service_{i}+\hat{b}_{4}Professional_{_{i}}+\hat{b}_{5}Other_{i}
\end{equation}

\end_inset


\end_layout

\begin_layout Itemize
Maybe I should make this easier to remember
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray*}
\widehat{wages}_{i} & = & \hat{b}_{0}+\hat{b}_{Sales}Sales_{i}+\hat{b}_{Clerical}Clerical_{i}
\end{eqnarray*}

\end_inset


\begin_inset Formula 
\[
+\hat{b}_{Service}Service_{i}+\hat{b}_{Prof}Professional_{_{i}}+\hat{b}_{Other}Other_{i}
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Fitted Regression Model with Categorical Predictor}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages20, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

mod2 <- lm(wage ~ occupationf, data=dat) 
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages20B, include=F, echo=F, results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(mod2, tight=F, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages20B.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Management is the 
\begin_inset Quotes eld
\end_inset

baseline
\begin_inset Quotes erd
\end_inset

.
 Calculate Predicted Values:
\end_layout

\begin_layout Standard
\begin_inset Formula $\hat{y}_{Management}=\hat{b}_{0}=12.704$
\end_inset


\begin_inset space ~
\end_inset


\begin_inset space ~
\end_inset


\begin_inset space ~
\end_inset


\begin_inset space ~
\end_inset


\begin_inset space ~
\end_inset


\begin_inset Formula $\hat{y}_{Sales}=\hat{b}_{0}+\hat{b}_{Sales}=12.704-5.11=7.59$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\hat{y}_{Service}=12.704-6.167=6.537$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Interpret that Termplot}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages20C,fig=T,include=F, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

termplot(mod2, terms=c("occupationf"), partial=T, se=T)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=10cm]{plots/t-wages20C}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim, allowframebreaks]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Contrasts: }
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The default treats the 
\begin_inset Quotes eld
\end_inset

lowest
\begin_inset Quotes erd
\end_inset

 score--the first 
\begin_inset Quotes eld
\end_inset

level
\begin_inset Quotes erd
\end_inset

--as a 
\begin_inset Quotes eld
\end_inset

baseline
\begin_inset Quotes erd
\end_inset

 category.
 
\end_layout

\begin_deeper
\begin_layout Itemize
Meaning: There is no 
\begin_inset Quotes eld
\end_inset

dummy
\begin_inset Quotes erd
\end_inset

 variable for that.
 It is 
\begin_inset Quotes eld
\end_inset

in
\begin_inset Quotes erd
\end_inset

 the intercept.
\end_layout

\end_deeper
\begin_layout Itemize
All other categories are compared against that one.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages25B, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

levels(dat$occupationf)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim, allowframebreaks]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Does the occupationf "Belong" in the Model}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Obviously Yes: 
\begin_inset Quotes eld
\end_inset

occupationf
\begin_inset Quotes erd
\end_inset

 makes a difference--some categories matter
\end_layout

\begin_layout Itemize
Formally test with F test, where null is that none of the differences are
 non-zero.
 
\begin_inset Formula 
\begin{equation}
H_{0}:\hat{b}_{Sales\,}=\hat{b}_{Clerical}=\hat{b}_{Service}=\hat{b}_{Professional}=\hat{b}_{Other}=0
\end{equation}

\end_inset


\end_layout

\begin_layout Itemize
Compare the fitted model against a model that has only the intercept 
\end_layout

\begin_layout Itemize
That's the F test that is reported with most regression models.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages25C, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

summary(mod2)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout


\backslash
input{plots/t-wages25C}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Does the occupationf "Belong" in the Model}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
R's anova function provides a conventional 
\begin_inset Quotes eld
\end_inset

analysis of variance table
\begin_inset Quotes erd
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages25D, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

anova(mod2, test="F")
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout


\backslash
input{plots/t-wages25D}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Simplify the Coding
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{But Do We Really Need All Those Parameters?}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Glance at the estimated slope coefficients.
\end_layout

\begin_layout Itemize
I suspect the middle 3 categories have 
\begin_inset Quotes eld
\end_inset

about the same
\begin_inset Quotes erd
\end_inset

 effect
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Hypothesis Testing Procedure}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
F test
\end_layout

\begin_layout Itemize
\begin_inset Formula $H_{0}:$
\end_inset

 
\begin_inset Formula $b_{sales}=b_{service}=b_{clerical}$
\end_inset


\end_layout

\begin_layout Itemize
Estimate 
\begin_inset Quotes eld
\end_inset

full
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

unrestricted
\begin_inset Quotes erd
\end_inset

 model with all of the category dummies included
\end_layout

\begin_layout Itemize
Estimate 
\begin_inset Quotes eld
\end_inset

partial
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

restricted
\begin_inset Quotes erd
\end_inset

 model with restriction imposed.
\end_layout

\begin_layout Itemize
Compare the fit, F test indicates whether estimates 
\begin_inset Formula $\hat{b}_{sales},$
\end_inset

 
\begin_inset Formula $\hat{b}_{service}$
\end_inset

, 
\begin_inset Formula $\hat{b}_{clerical},$
\end_inset

 are 
\begin_inset Quotes eld
\end_inset

statistically significantly different
\begin_inset Quotes erd
\end_inset

 from one another.
\end_layout

\begin_layout Itemize
Slang: is 
\begin_inset Quotes eld
\end_inset

predictive power
\begin_inset Quotes erd
\end_inset

 lost by restriction?
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim, allowframebreaks]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Test $
\backslash
hat{b}_{Sales}=
\backslash
hat{b}_{Clerical}=
\backslash
hat{b}_{Service}$}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Testing the restriction that the wage effect for three groups is achieved
 by recoding occupationf variable
\end_layout

\begin_layout Itemize
All 
\begin_inset Quotes eld
\end_inset

Sales
\begin_inset Quotes erd
\end_inset

 
\begin_inset Quotes eld
\end_inset

Clerical
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

Service
\begin_inset Quotes erd
\end_inset

 observations re-coded 1 on new category 
\begin_inset Quotes eld
\end_inset

sales/clerical/service
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages30,include=F,echo=T>>=
\end_layout

\begin_layout Plain Layout

occlev <- levels(dat$occupationf)
\end_layout

\begin_layout Plain Layout

dat$occupationf2 <- dat$occupationf
\end_layout

\begin_layout Plain Layout

dat$occupationf2[dat$occupationf2 %in% occlev[2:4]] <- occlev[2]
\end_layout

\begin_layout Plain Layout

\end_layout

\begin_layout Plain Layout

levels(dat$occupationf2)[2] <- "sales/clerk/serv"
\end_layout

\begin_layout Plain Layout

with(dat, table(occupationf2, occupationf))
\end_layout

\begin_layout Plain Layout

mod3 <- lm(wage ~ occupationf2, data=dat)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages30B, include=F, echo=F, results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(mod3, tight=F, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages30B.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{And the F test result is (drumroll please)}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages30D, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

anova(mod3, mod2, test="F")
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages30D.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{What if I merge "Management" and "Professional"?}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Appears to me 
\begin_inset Formula $\hat{y}_{Professional}$
\end_inset

 and 
\begin_inset Formula $\hat{y}_{Management}$
\end_inset

 are not all that different.
\end_layout

\begin_layout Itemize
Suppose 
\begin_inset Formula $H_{o}:$
\end_inset


\begin_inset Formula $b_{Professional}=0$
\end_inset

 and 
\begin_inset Formula $b_{sales}=b_{service}=b_{clerical}$
\end_inset


\end_layout

\begin_layout Itemize
Then we create an even simpler variable, which leads to 2 
\begin_inset Quotes eld
\end_inset

dummy
\begin_inset Quotes erd
\end_inset

 variables
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages40, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

dat$occupationf2[dat$occupationf2 %in% occlev[5]] <- occlev[1] 
\end_layout

\begin_layout Plain Layout

levels(dat$occupationf2)[1] <- "manag/prof"
\end_layout

\begin_layout Plain Layout

dat$occupationf2 <- dat$occupationf2[, drop=T]
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages40A, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

contrasts(dat$occupationf2)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout


\backslash
input{plots/t-wages40A}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{And the Regression on that Simpler Set of Contrasts is}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages40B, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

mod4 <- lm(wage ~ occupationf2, data=dat)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages40C, include=F, echo=F, results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(mod4, tight=F, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages40C.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{And The F Test says}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Compare the 
\begin_inset Quotes eld
\end_inset

full
\begin_inset Quotes erd
\end_inset

 fitted model with all 5 category differences estimated
\end_layout

\begin_layout Itemize
With the restricted model
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<wages40D, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

anova( mod4, mod2,  test="F")
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-wages40D.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Conclusion: Does not appear the model with 3 categories (intercept + 2 group
 contrasts) has a worse statistical fit.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Coding Schemes
\end_layout

\begin_layout Subsection
G-1 is Over-rated
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{What To Do with a G-Category Nominal Variable?}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
If there are G categories, 
\end_layout

\begin_layout Itemize
Texts usually say 
\begin_inset Quotes eld
\end_inset

regression can provide parameter estimates for G-1 categories
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Itemize
Strinctly Speaking, that's wrong.
 
\end_layout

\begin_deeper
\begin_layout Itemize
It is only true if you include an Intercept in your regression
\end_layout

\begin_layout Itemize
Drop the intercept, you can have G category estimates!
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Lets Talk About Sex (again!)}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Recall, the data has a categorical 
\begin_inset Quotes eld
\end_inset

sex
\begin_inset Quotes erd
\end_inset

 (M or F) and we can create 
\begin_inset Quotes eld
\end_inset

dummy
\begin_inset Quotes erd
\end_inset

 variables for females and males.
\end_layout

\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
id
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
constant
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
sex
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
femd
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
maled
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
M
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
F
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
M
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Itemize
You agree, don't you, that:
\end_layout

\begin_deeper
\begin_layout Itemize
We get essentially the same model if we fit a dummy variable for 
\begin_inset Quotes eld
\end_inset

female
\begin_inset Quotes erd
\end_inset

 or for 
\begin_inset Quotes eld
\end_inset

male
\begin_inset Quotes erd
\end_inset

, right?
\end_layout

\begin_layout Itemize
\begin_inset Formula $\hat{y}_{i}=\hat{b}_{0}+\hat{b}_{1}\cdot femd_{i}$
\end_inset

 treats 
\begin_inset Quotes eld
\end_inset

male
\begin_inset Quotes erd
\end_inset

 as baseline and 
\begin_inset Formula $\hat{b}_{1}$
\end_inset

 is the difference for females
\end_layout

\begin_layout Itemize
\begin_inset Formula $\hat{y}_{i}=\hat{b}_{0}+\hat{b}_{1}\cdot maled_{i}$
\end_inset

 treats 
\begin_inset Quotes eld
\end_inset

female
\begin_inset Quotes erd
\end_inset

 as baseline and 
\begin_inset Formula $\hat{b}_{1}$
\end_inset

is the difference for males
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
You Want G Parameters? You Got It!
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Drop the Intercept? Intriguing!}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Drop the intercept? G categories -> G parameter estimates
\end_layout

\begin_layout Itemize
lm(y ~ -1 + sex) : fits no intercept, estimates parameters for both males
 and females
\begin_inset Formula 
\begin{equation}
\begin{array}{cc}
sexF & sexM\\
0 & 1\\
1 & 0
\end{array}
\end{equation}

\end_inset


\end_layout

\begin_layout Itemize
And that is 
\begin_inset Quotes eld
\end_inset

essentially the same model
\begin_inset Quotes erd
\end_inset

 as either of the others.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Problem comes back to Multicollinearity}
\end_layout

\end_inset


\end_layout

\begin_layout ColumnsTopAligned

\end_layout

\begin_deeper
\begin_layout Column
6cm
\end_layout

\begin_layout Itemize
See why you can't estimate this:
\end_layout

\begin_layout Standard
\begin_inset listings
inline false
status open

\begin_layout Plain Layout

lm(y~femd+maled)
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
R automatically inserts an 
\begin_inset Quotes eld
\end_inset

intercept
\begin_inset Quotes erd
\end_inset

 coefficient for you, so this is really
\end_layout

\begin_layout Standard
\begin_inset listings
inline false
status open

\begin_layout Plain Layout

lm(y~1+femd+maled)
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Leading to the design matrix on right: perfect collinearity between constant,
 femd and maled
\end_layout

\begin_layout Column
6cm
\end_layout

\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
constant
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
femd
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
maled
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Your options: 
\end_layout

\begin_deeper
\begin_layout Itemize
include a constant and either femd or maled
\end_layout

\begin_layout Itemize
remove the constant and estimate femd and maled
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Better Check that with the Chile Data}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Traditional model, sexM
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile30M, echo=T, eval=F, include=F>>=
\end_layout

\begin_layout Plain Layout

chile1M <- lm(statusquo ~ sex, data=Chile)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile30M}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Traditional model, sexF
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile30F, echo=T, eval=F, include=F>>=
\end_layout

\begin_layout Plain Layout

Chile$sex <- relevel(Chile$sex, ref="M")
\end_layout

\begin_layout Plain Layout

chile1F <- lm(statusquo ~ sex, data=Chile)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile30F}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
No Intercept Model
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile30NI, echo=T, eval=F, include=F>>=
\end_layout

\begin_layout Plain Layout

chile1NI <- lm(statusquo ~ -1 + sex, data=Chile)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile30NI.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile31, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

<<chile30M>>
\end_layout

\begin_layout Plain Layout

<<chile30F>>
\end_layout

\begin_layout Plain Layout

<<chile30NI>>
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{3 Fits Side By Side}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile32, results=tex, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

outreg(list(chile1M, chile1F, chile1NI), modelLabels=c("M","F","No Int."),
 tight=TRUE, showAIC=FALSE)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile32.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Vital: The Predicted Values Are IDENTICAL!}
\end_layout

\end_inset


\end_layout

\begin_layout ColumnsTopAligned

\end_layout

\begin_deeper
\begin_layout Column
6cm
\end_layout

\end_deeper
\begin_layout ColumnsTopAligned
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile40, fig=T, include=FALSE, height=5, width=4>>=
\end_layout

\begin_layout Plain Layout

termplot(chile1F, partial.resid=TRUE, se=TRUE, ylabs="Statusquo Support")
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\begin_inset listings
inline false
status open

\begin_layout Plain Layout

chile1F <- lm(statusquo ~ sex, data=Chile)
\end_layout

\end_inset


\end_layout

\begin_layout ColumnsTopAligned
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=5cm]{plots/t-chile40}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Column
6cm
\end_layout

\end_deeper
\begin_layout ColumnsTopAligned
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile45, fig=T, include=FALSE, height=5, width=4>>=
\end_layout

\begin_layout Plain Layout

termplot(chile1NI, partial.resid=TRUE, se=TRUE, ylabs=c("Statusquo Support"))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset

 
\begin_inset listings
inline false
status open

\begin_layout Plain Layout

chile1NI <- lm(statusquo ~ -1 + sex, data=Chile)
\end_layout

\end_inset


\end_layout

\begin_layout ColumnsTopAligned
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
includegraphics[width=5cm]{plots/t-chile45}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{I mean Predictions are Completely IDENTICAL! Check the first
 few cases}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile50,echo=T,include=F>>=
\end_layout

\begin_layout Plain Layout

head(predict(chile1F))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile50}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile51,echo=T,include=F>>=
\end_layout

\begin_layout Plain Layout

head(predict(chile1NI))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile51}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Same True With G Categories
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{So, if a Categorical IV has 5 "levels" (as R would call them)}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
We can estimate 4 parameters for levels and 1 for intercept
\end_layout

\begin_layout Itemize
Or we can suppress intercept and estimate 5 parameters for 5 levels
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Treatment Contrasts=="dummy" codes}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Colloquial: Dummy Variable Coding
\end_layout

\begin_layout Itemize
R calls this 
\begin_inset Quotes eld
\end_inset

treatment contrasts
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
id
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Religion
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Cath
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Prot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Musl
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Hindu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Other
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cath
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Prot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Musl
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Hindu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Other
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Regression with Treatment Contrasts}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\hat{y}_{i}\sim\hat{b}_{0}+\hat{b}_{1}Rel.Prot_{i}+\hat{b}_{2}Rel.Musl_{i}+\hat{b}_{3}Rel.Hindu_{i}+\hat{b}_{4}Rel.Other_{i}$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Quotes eld
\end_inset

Catholic
\begin_inset Quotes erd
\end_inset

 is 
\begin_inset Quotes eld
\end_inset

left out?
\begin_inset Quotes erd
\end_inset

 Not really
\end_layout

\begin_layout Itemize
Predicted value for members of 
\end_layout

\begin_deeper
\begin_layout Itemize
Catholic is 
\begin_inset Formula $\hat{b}_{0}$
\end_inset


\end_layout

\begin_layout Itemize
Protestant is 
\begin_inset Formula $\hat{b}_{0}+\hat{b}_{1}$
\end_inset


\end_layout

\begin_layout Itemize
Muslim is 
\begin_inset Formula $\hat{b}_{0}+\hat{b}_{2}$
\end_inset


\end_layout

\begin_layout Itemize
Hindu is 
\begin_inset Formula $\hat{b}_{0}+\hat{b}_{3}$
\end_inset


\end_layout

\begin_layout Itemize
Other is 
\begin_inset Formula $\hat{b}_{0}+\hat{b}_{4}$
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Interpret individual coefficients
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\hat{b}_{1}$
\end_inset

 : difference in predicted value for Protestant (as opposed to Catholic).
\end_layout

\begin_layout Itemize
\begin_inset Formula $\hat{b}_{2}$
\end_inset

 : difference in predicted value for Muslim (as compared against Catholic)
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Any Group Can Serve as the Baseline}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Can make 
\begin_inset Quotes eld
\end_inset

Hindu
\begin_inset Quotes erd
\end_inset

 the baseline group.
\end_layout

\begin_layout Itemize
All estimates treat Hindu as 
\begin_inset Quotes eld
\end_inset

baseline
\begin_inset Quotes erd
\end_inset

 and other estimates are differences in prediction against Hindu category
\end_layout

\begin_layout Itemize
Model predictions and fit indices are still IDENTICAL to other 
\begin_inset Quotes eld
\end_inset

Catholic baseline
\begin_inset Quotes erd
\end_inset

 model.
\end_layout

\begin_layout Itemize
If there are no other predictors in the model, the 
\begin_inset Formula $\hat{b}_{j}'s$
\end_inset

 are simply related to the observed group means (since predicted value is
 
\begin_inset Quotes eld
\end_inset

mean
\begin_inset Quotes erd
\end_inset

 of 
\begin_inset Formula $y$
\end_inset

 for category members).
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Remember $
\backslash
hat{y}$ is the same, no matter how you code these Predictor Contrasts}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Changing 
\begin_inset Quotes eld
\end_inset

dummy codes
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

baseline group
\begin_inset Quotes erd
\end_inset

 alters the 
\begin_inset Formula $\hat{b}$
\end_inset

 estimates
\end_layout

\begin_layout Itemize
It does not alter the essential meaning of the model
\end_layout

\begin_layout Itemize
Like saying 
\begin_inset Quotes eld
\end_inset

I am average in height
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

my height is the average plus 0
\begin_inset Quotes erd
\end_inset

 or 
\begin_inset Quotes eld
\end_inset

my height is 36 inches plus one-half of the average
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Effects Coding
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Effects Coding (Unweighted)}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Terminology is 
\begin_inset Quotes eld
\end_inset

new to me
\begin_inset Quotes erd
\end_inset

 in Cohen, et al.
\end_layout

\begin_layout Itemize
Re-code the religion variable like so (for 
\begin_inset Quotes eld
\end_inset

omitted
\begin_inset Quotes erd
\end_inset

 category, put -1 all the way across)
\end_layout

\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="7">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
id
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Religion
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Cath
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Prot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Musl
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Hindu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Rel.Other
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cath
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Prot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Musl
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Hindu
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Other
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Itemize
Called 
\begin_inset Quotes eld
\end_inset

sum-to-zero
\begin_inset Quotes erd
\end_inset

 contrasts in other contexts.
\end_layout

\begin_layout Itemize
We will fit a regression that does not include 
\emph on
Rel.Cath
\end_layout

\begin_deeper
\begin_layout Standard
\begin_inset Formula $\hat{y}_{i}\sim\hat{b}_{0}+\hat{b}_{1}Rel.Prot_{i}+\hat{b}_{2}Rel.Musl_{i}+\hat{b}_{3}Rel.Hindu_{i}+\hat{b}_{4}Rel.Other_{i}$
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Still get 
\begin_inset Formula $\hat{b}$
\end_inset

's as comparisons, but now comparing against a different baseline.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Design Matrix}
\end_layout

\end_inset


\end_layout

\begin_layout ColumnsTopAligned

\end_layout

\begin_deeper
\begin_layout Column
4cm
\end_layout

\end_deeper
\begin_layout ColumnsTopAligned
The 
\begin_inset Quotes eld
\end_inset

design matrix
\begin_inset Quotes erd
\end_inset

:
\end_layout

\begin_layout ColumnsTopAligned
\begin_inset Tabular
<lyxtabular version="3" rows="7" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Const
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Cath
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
P
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
M
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
H
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Oth
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
0
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\vdots$
\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout ColumnsTopAligned
But 
\begin_inset Quotes eld
\end_inset

Cath
\begin_inset Quotes erd
\end_inset

 is omitted from the fitted report
\end_layout

\begin_deeper
\begin_layout Column
6cm
\end_layout

\begin_layout Itemize
Every 
\begin_inset Quotes eld
\end_inset

row
\begin_inset Quotes erd
\end_inset

 gets 
\end_layout

\begin_deeper
\begin_layout Itemize
a 
\begin_inset Formula $1$
\end_inset

 for its 
\begin_inset Quotes eld
\end_inset

own
\begin_inset Quotes erd
\end_inset

 group
\end_layout

\begin_layout Itemize
Except Catholics, who get 
\begin_inset Formula $-1$
\end_inset

 
\end_layout

\end_deeper
\begin_layout Itemize
The 
\begin_inset Formula $-1$
\end_inset

 basically 
\begin_inset Quotes eld
\end_inset

pushes
\begin_inset Quotes erd
\end_inset

 the estimated intercept
\end_layout

\begin_layout Itemize
The other coefficients adjust accordingly to produce same predicted values.
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Where does the Intercept get pushed to?}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Answer: Intercept=mean of group means on 
\begin_inset Formula $y$
\end_inset


\begin_inset Formula 
\begin{equation}
\hat{b}_{0}=\frac{1}{5}\{\bar{Y}_{1}+\bar{Y}_{2}+\bar{Y}_{3}+\bar{Y}_{4}+\bar{Y}_{5}\}
\end{equation}

\end_inset


\end_layout

\begin_layout Itemize
Called 
\begin_inset Quotes eld
\end_inset

unweighted effects coding
\begin_inset Quotes erd
\end_inset

 because the means of the groups are averaged, no matter how many observations
 there are in each group.
\end_layout

\begin_layout Itemize
In order to believe that, I had to run some examples.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Chile Regions: First get the means}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The mean values of 
\begin_inset Quotes eld
\end_inset

statusquo
\begin_inset Quotes erd
\end_inset

 for the regions are
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile110, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

library(car)
\end_layout

\begin_layout Plain Layout

agg1 <- aggregate(Chile$statusquo, by=list(region=Chile$region), mean, na.rm=T)
\end_layout

\begin_layout Plain Layout

agg1
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile110.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Now calculate the 
\begin_inset Quotes eld
\end_inset

mean of the means
\begin_inset Quotes erd
\end_inset

 (no weights)
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile145, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

(munweighted <- mean(agg1$x))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile145.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
0.076 is a 
\begin_inset Quotes eld
\end_inset

magic number
\begin_inset Quotes erd
\end_inset

.
 Watch out for it later
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Suppress the Intercept: Estimate 5 Params for 5 Regions}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile120, include=F, echo=T, results=tex>>=
\end_layout

\begin_layout Plain Layout

modr1 <- lm( statusquo ~ -1 + region, data=Chile)
\end_layout

\begin_layout Plain Layout

outreg(modr1, tight=FALSE, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile120.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Include the Intercept, Estimate (default) Treatment Contrasts}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile130, include=F, echo=T, results=tex>>=
\end_layout

\begin_layout Plain Layout

modr2 <- lm( statusquo ~ region, data=Chile, x=T, y=T)
\end_layout

\begin_layout Plain Layout

outreg(modr2, tight=FALSE, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile130.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Those Default Contrasts Were }
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout

<<chile135A, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

contrasts(Chile$region)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile135A.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Ask R to use "sum-to-zero" contrasts (aka Unweighted Effects)}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
def
\backslash
Sweavesize{
\backslash
scriptsize}
\end_layout

\begin_layout Plain Layout

<<chile135B, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

options(contrasts=c("contr.sum", "contr.poly"))
\end_layout

\begin_layout Plain Layout

contrasts(Chile$region)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile135B.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Note, the default makes the 
\begin_inset Quotes eld
\end_inset

last
\begin_inset Quotes erd
\end_inset

 category, SA, the reference category.
 Will have to fix that later.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Fitted model with Effects Contrasts}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile140, include=F, echo=T>>=
\end_layout

\begin_layout Plain Layout

modr3 <- lm( statusquo ~ region, data=Chile)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile140B, results=tex, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

outreg(modr3, tight=FALSE, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile140B.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Unfortunately, we lose the region labels here, but they are 1=C, 2=M, 3=N,
 4=S
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{I Had Trouble figuring this Out}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Some patience required :)
\end_layout

\begin_layout Itemize
Note the Effects Coding intercept is 0.076, same as 
\begin_inset Quotes eld
\end_inset

mean of category means
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile142, echo=F, include=F, eval=F>>=
\end_layout

\begin_layout Plain Layout

#agg1 <- aggregate(Chile$statusquo, by=list(region=Chile$region), mean,
 na.rm=T)
\end_layout

\begin_layout Plain Layout

agg1
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile142.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Calculate the difference between the observed means and 0.076 
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile147, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

agg1$diff <- agg1$x - munweighted
\end_layout

\begin_layout Plain Layout

agg1
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile147.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Note those differences exactly reproduce the 
\begin_inset Formula $\hat{b}$
\end_inset

 estimates from the unweighted effects model.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim, allowframebreaks]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{I wish C were the Omitted Category}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Create a new factor 
\begin_inset Quotes eld
\end_inset

region2
\begin_inset Quotes erd
\end_inset

 in which levels are ordered (M, N, S, SA, C)
\end_layout

\begin_layout Itemize
That forces values for cases in C to -1 for all contrasts
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile140C,echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

Chile$region2 <- factor(Chile$region, levels=c("M","N","S","SA","C"))
\end_layout

\begin_layout Plain Layout

modr4 <- lm( statusquo ~ region2, data=Chile)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile140D, echo=F,include=F>>=
\end_layout

\begin_layout Plain Layout

contrasts(Chile$region2)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile140D}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Re-fit with "C" as the reference}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile141D, results=tex, include=F, echo=F>>=
\end_layout

\begin_layout Plain Layout

outreg(modr4, tight=FALSE, showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile141D.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Interpretation benefit to the $
\backslash
hat{b}$'s}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
One can scan down the parameter estimates to see if one category is above
 the unweighted mean
\end_layout

\begin_layout Itemize
Unclear to me why one would want to do that, but one can, if one wants to
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{But they are all Fundamentally the same}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="1" columns="3">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "3.5cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
No Intercept
\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
small
\end_layout

\begin_layout Plain Layout

<<echo=F,results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(modr1, tight=T,showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "3.5cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
Treatment
\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
small
\end_layout

\begin_layout Plain Layout

<<echo=F,results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(modr2, tight=T,showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
inner_pos "t"
use_parbox 0
use_makebox 0
width "3.5cm"
special "none"
height "1in"
height_special "totalheight"
status open

\begin_layout Plain Layout
Effects
\end_layout

\begin_layout Plain Layout
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
small
\end_layout

\begin_layout Plain Layout

<<echo=F,results=tex>>=
\end_layout

\begin_layout Plain Layout

outreg(modr4, tight=T,showAIC=F)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile164, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

pmr1 <- aggregate(predict(modr1), by=list(region=modr1$model$region), mean)
\end_layout

\begin_layout Plain Layout

pmr2 <- aggregate(predict(modr2), by=list(region=modr2$model$region), mean)
\end_layout

\begin_layout Plain Layout

pmr3 <- aggregate(predict(modr3), by=list(region=modr3$model$region), mean)
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Predicted Values for all Rows are Identical.
 Same, Equivalent, Interchangeable}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile167.5,echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

dat <-data.frame(pmr1,pmr2,pmr3)
\end_layout

\begin_layout Plain Layout

dat <- dat[ ,c(1,2,4,6)]
\end_layout

\begin_layout Plain Layout

colnames(dat) <- c("region", "NoInt", "Treatment", "Effects")
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Note predicted values for all regions are same
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile167.6, echo=F, include=F>>=
\end_layout

\begin_layout Plain Layout

dat
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile167.6.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
R's 
\begin_inset Quotes eld
\end_inset

all.equal
\begin_inset Quotes erd
\end_inset

 verifies that the predictions for each row in data are same.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout

<<chile169, echo=T, include=F>>=
\end_layout

\begin_layout Plain Layout

all.equal(predict(modr1), predict(modr2), predict(modr3))
\end_layout

\begin_layout Plain Layout

@
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
input{plots/t-chile169.tex}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{The Standard Errors of the $
\backslash
hat{b}$ Only Appear to Differ}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
The standard errors are different, but
\end_layout

\begin_layout Itemize
That's only because they are estimating different things!
\end_layout

\begin_layout Itemize
\begin_inset Formula $Std.Err.(\hat{b})$
\end_inset

 varies because each model reports an estimate of a different value
\end_layout

\begin_layout Itemize
The No Intercept model estimates a 
\begin_inset Quotes eld
\end_inset

total effect
\begin_inset Quotes erd
\end_inset

 value for each region
\end_layout

\begin_layout Itemize
The Treatment Contrast model estimates 
\end_layout

\begin_deeper
\begin_layout Itemize
one 
\begin_inset Quotes eld
\end_inset

total effect
\begin_inset Quotes erd
\end_inset

 for baseline 
\end_layout

\begin_layout Itemize
difference for each region against baseline
\end_layout

\end_deeper
\begin_layout Itemize
Effects Contrasts estimate
\end_layout

\begin_deeper
\begin_layout Itemize
one unweighted mean
\end_layout

\begin_layout Itemize
differences for each region against that
\end_layout

\end_deeper
\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{Consider Region S}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
No Intercept model 
\begin_inset Formula $\hat{b}_{S}=0.165$
\end_inset

, 
\begin_inset Formula $Std.Err(\hat{b}_{S})=0.037$
\end_inset


\end_layout

\begin_layout Itemize
Treatment Contrasts, 
\begin_inset Formula $\hat{b}_{S}=0.195$
\end_inset

, 
\begin_inset Formula $Std.Err(\hat{b}_{s})=0.055$
\end_inset


\end_layout

\begin_layout Itemize
Effects Contrasts, 
\begin_inset Formula $\hat{b}_{S}=0.089$
\end_inset

, 
\begin_inset Formula $Std.Err.(\hat{b}_{S})=0.039$
\end_inset


\end_layout

\begin_layout Itemize
From Treatment, can re-construct estimate for 
\begin_inset Quotes eld
\end_inset

total S region effect
\begin_inset Quotes erd
\end_inset


\begin_inset Formula 
\begin{equation}
\hat{b}_{0}+\hat{b}_{S}\, with\, Std.Err.(\sqrt{Var(\hat{b}_{0})+Var(\hat{b}_{S})+2Cov(\hat{b}_{0},\hat{b}_{S})})
\end{equation}

\end_inset


\end_layout

\begin_deeper
\begin_layout Itemize
Inserting values from the Covariance of the 
\begin_inset Formula $\hat{b}$
\end_inset

 from Treatment gives 
\begin_inset Formula $0.037$
\end_inset


\end_layout

\end_deeper
\begin_layout Itemize
Do same with Effects Contrasts, get standard error of 
\begin_inset Formula $0.037$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
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status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{My "Take Away" Message}
\end_layout

\end_inset


\end_layout

\begin_layout Itemize
Regression is a 
\begin_inset Quotes eld
\end_inset

vehicle
\begin_inset Quotes erd
\end_inset

 with which to calculate predicted values
\end_layout

\begin_layout Itemize
Many equivalent 
\begin_inset Quotes eld
\end_inset

design matrices
\begin_inset Quotes erd
\end_inset

 can be used to calculate same predicted values
\end_layout

\begin_layout Itemize
Comfort with one method or its estimates b's drives the selection of one's
 approach.
 There is no 
\begin_inset Quotes eld
\end_inset

real
\begin_inset Quotes erd
\end_inset

 methodological difference between the two.
\end_layout

\begin_layout Itemize
Often choose approach so that 
\begin_inset Quotes eld
\end_inset

free t-tests
\begin_inset Quotes erd
\end_inset

 with regression output are testing the most meaningful questions.
\end_layout

\begin_layout Standard
\begin_inset ERT
status open

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\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
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status open

\begin_layout Plain Layout


\backslash
begin{frame}[containsverbatim]
\end_layout

\begin_layout Plain Layout


\backslash
frametitle{}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
end{frame}
\end_layout

\end_inset


\end_layout

\end_body
\end_document