%\VignetteEngine{knitr::knitr}
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\cfoot{Statistical Sleuth in R: Chapter 3}
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\newcommand{\cran}{\href{http://www.R-project.org/}{CRAN}}

\title{The Statistical Sleuth in R: \\
Chapter 3}

\author{
Linda Loi \and Ruobing Zhang \and Kate Aloisio \and Nicholas J. Horton\thanks{Department of Mathematics and Statistics, Smith College, nhorton@smith.edu}
} 

\date{\today}

\begin{document}


\maketitle
\tableofcontents

%\parindent=0pt


<<setup, include=FALSE, cache=FALSE>>=
require(knitr)
opts_chunk$set(
  dev="pdf",
  fig.path="figures/",
        fig.height=3,
        fig.width=4,
        out.width=".47\\textwidth",
        fig.keep="high",
        fig.show="hold",
        fig.align="center",
        prompt=TRUE,  # show the prompts; but perhaps we should not do this
        comment=NA    # turn off commenting of ouput (but perhaps we should not do this either
  )
@


<<pvalues, echo=FALSE, message=FALSE>>=
print.pval = function(pval) {
  threshold = 0.0001
    return(ifelse(pval < threshold, paste("p<", sprintf("%.4f", threshold), sep=""),
                ifelse(pval > 0.1, paste("p=",round(pval, 2), sep=""),
                       paste("p=", round(pval, 3), sep=""))))
}
@

<<setup2,echo=FALSE,message=FALSE>>=
require(Sleuth3)
require(mosaic)
trellis.par.set(theme=col.mosaic())  # get a better color scheme 
set.seed(123)
# this allows for code formatting inline.  Use \Sexpr{'function(x,y)'}, for exmaple.
knit_hooks$set(inline = function(x){
if (is.numeric(x)) return(knitr:::format_sci(x, 'latex'))
x = as.character(x)
h = knitr:::hilight_source(x, 'latex', list(prompt=FALSE, size='normalsize'))
h = gsub("([_#$%&])", "\\\\\\1", h)
h = gsub('(["\'])', '\\1{}', h)
gsub('^\\\\begin\\{alltt\\}\\s*|\\\\end\\{alltt\\}\\s*$', '', h)
})
showOriginal=FALSE
showNew=TRUE
@ 

\section{Introduction}

This document is intended to help describe how to undertake analyses introduced as examples in the Third Edition of the \emph{Statistical Sleuth} (2013) by Fred Ramsey and Dan Schafer.
More information about the book can be found at \url{http://www.proaxis.com/~panorama/home.htm}.
This
file as well as the associated \pkg{knitr} reproducible analysis source file can be found at
\url{http://www.math.smith.edu/~nhorton/sleuth3}.


This work leverages initiatives undertaken by Project MOSAIC (\url{http://www.mosaic-web.org}), an NSF-funded effort to improve the teaching of statistics, calculus, science and computing in the undergraduate curriculum. In particular, we utilize the 
\pkg{mosaic} package, which was written to simplify the use of R for introductory statistics courses. A short summary of the R needed to teach introductory statistics can be found in the mosaic package vignette (\url{http://cran.r-project.org/web/packages/mosaic/vignettes/MinimalR.pdf}). 

To use a package within R, it must be installed (one time), and loaded (each session). The package can be installed using the following command:
<<install_mosaic,eval=FALSE>>=
install.packages('mosaic')               # note the quotation marks
@
Once this is installed, it can be loaded by running the command:
<<load_mosaic,eval=FALSE>>=
require(mosaic)
@
This
needs to be done once per session.

In addition the data files for the \emph{Sleuth} case studies can be accessed by installing the \pkg{Sleuth3} package.
<<install_Sleuth3,eval=FALSE>>=
install.packages('Sleuth3')               # note the quotation marks
@
<<load_Sleuth3,eval=FALSE>>=
require(Sleuth3)
@

We also set some options to improve legibility of graphs and output.
<<eval=TRUE>>=
trellis.par.set(theme=col.mosaic())  # get a better color scheme for lattice
options(digits=3, show.signif.stars=FALSE)
@

The specific goal of this document is to demonstrate how to calculate the quantities described in \emph{Sleuth} Chapter 3: A Closer Look at Assumptions using R.

\section{Cloud Seeding to Increase Rainfall}

Does seeding clouds lead to more rainfall? This is the question being addressed by case study 3.1 in the \emph{Sleuth}.

\subsection{Summary statistics and graphical displays (untransformed)}

We begin by reading the data and summarizing the variables.

<<>>=
summary(case0301)
favstats(Rainfall ~ Treatment, data=case0301)
@

A total of \Sexpr{nrow(case0301)} subjects were included in this data: \Sexpr{nrow(subset(case0301, Treatment=="Seeded"))} seeded days  and \Sexpr{nrow(subset(case0301, Treatment=="Unseeded"))} 
unseeded days (Display 3.1, page 59). 

<<fig.height=8, fig.width=8>>=
bwplot(Rainfall ~ Treatment, data=case0301)
@

<<fig.height=8, fig.width=8>>=
densityplot(~Rainfall, groups=Treatment, auto.key=TRUE, data=case0301)
@

According to the boxplot and the density plot, the rainfall from seeded days seems to be larger than unseeded days. Both density curves are highly skewed to the right.

\subsection{Summary statistics and graphical display (transformed)}

The skewness suggests that there is a need to apply a logarithmic transformation. The transformed data is shown on page 73 (Display 3.9).

<<>>=
case0301 = transform(case0301, lograin=log(Rainfall))
favstats(lograin ~ Treatment, data=case0301)
@

<<fig.height=8, fig.width=8>>=
bwplot(lograin ~ Treatment, data=case0301)
@

<<fig.height=8, fig.width=8>>=
densityplot(~lograin, groups=Treatment, auto.key=TRUE, data=case0301)
@

The log transformation reduces the skewness of these two distributions.

\subsection{Inferential procedures (two-sample t-test)}

 
<<>>=
t.test(Rainfall ~ Treatment, var.equal=FALSE, data=case0301)
t.test(Rainfall ~ Treatment, var.equal=TRUE, data=case0301)
@

The following corresponds to the calculations on page 73.

<<>>=
summary(lm(lograin ~ Treatment, data=case0301))
ttestlog = t.test(lograin ~ Treatment, data=case0301); ttestlog
@

\subsection{Interpretation of log model}

The following code is used to calculate the ``Statistical Conclusion" on page 59. First, we want to calculate the multiplier.

<<>>=
obslogdiff = -diff(mean(lograin ~ Treatment, data=case0301)); obslogdiff
multiplier = exp(obslogdiff); multiplier
@

Next we can calculate the 95\% confidence interval for the multiplier.
<<>>=
ttestlog$conf.int
exp(ttestlog$conf.int)
@

\section{Effects of Agent Orange on Troops in Vietnam}

Is dioxin concentration related to veteran status? This is the question being addressed by case study 3.2 in the \emph{Sleuth}.

\subsection{Summary statistics and graphical display}

We begin by reading the data and summarizing the variables.

<<>>=
summary(case0302)
favstats(Dioxin ~ Veteran, data=case0302)
@

A total of \Sexpr{nrow(case0302)} veterans were included in this data: \Sexpr{nrow(subset(case0302, Veteran=="Vietnam"))} served in Vietnam during 1967 and 1968  and \Sexpr{nrow(subset(case0302, Veteran=="Other"))} served in US or Germany during 1965 and 1971.

<<fig.height=8, fig.width=8>>=
bwplot(Veteran ~ Dioxin, data=case0302)
@

<<fig.height=8, fig.width=8>>=
densityplot(~Dioxin, groups=Veteran, auto.key=TRUE, data=case0302)
@

Both distributions are highly skewed to the right.


\subsection{Inferential procedures (two-sample t-test)}

The following code is used to calculate the ``Statistical Conclusion" on page 62.
<<>>=
t.test(Dioxin ~ Veteran, var.equal=TRUE, alternative="less", data=case0302)
t.test(Dioxin ~ Veteran, var.equal=TRUE, data=case0302)$conf.int
@

So the one-sided $p$-value from a two-sample $t$-test is \Sexpr{pval(t.test(Dioxin ~ Veteran, var.equal=TRUE, alternative="less", data=case0302))}. The 95\% confidence interval is (\Sexpr{round(t.test(Dioxin ~ Veteran, var.equal=TRUE, data=case0302)$conf.int[1], 2)}, \Sexpr{round(t.test(Dioxin ~ Veteran, var.equal=TRUE, data=case0302)$conf.int[2], 2)}). Notice that because of the way we ordered our variables, the confidence interval shown in our analysis is different from that of the book (our confidence intervals are inverse). This is of no consequence, as the difference between the groups is still the same.

\subsection{Removing outliers}

We will remove two extreme observations from the data.  First we remove observation 646 and perform a $t$-test (Display 3.7, page 70).   

<<>>=
case0302.2 = case0302[-c(646), ]
t.test(Dioxin ~ Veteran, alternative="less", data=case0302.2)
@

Next we remove observations 645 and 646 and perform a $t$-test.

<<>>=
dim(case0302)
case0302.3 = case0302[-c(645, 646), ]
dim(case0302.3)
t.test(Dioxin ~ Veteran, alternative="less", data=case0302.3)
@

Notice that after removing these outliers, the $p$-value and the confidence interval have changed but the 
substantive conclusion is unchanged.


\end{document}



