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\cfoot{Introduction to the Practice of Statistics using R: Chapter 10}
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\newcommand{\cran}{\href{http://www.R-project.org/}{CRAN}}

\title{Introduction to the Practice of Statistics using R: \\
Chapter 16}

\author{
Ben Baumer \and Nicholas J. Horton\thanks{Department of Mathematics, Amherst College, nhorton@amherst.edu} 
} 
\date{\today}

\begin{document}


\maketitle
\tableofcontents

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}
@

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showOriginal=FALSE
showNew=TRUE
@ 

\section*{Introduction}

This document is intended to help describe how to undertake analyses introduced as examples in the Sixth Edition of \emph{Introduction to the 
Practice of Statistics} (2002) by David Moore, George McCabe and Bruce
Craig.
More information about the book can be found at \url{http://bcs.whfreeman.com/ips6e/}.  This
file as well as the associated \pkg{knitr} reproducible analysis source file can be found at
\url{http://www.math.smith.edu/~nhorton/ips6e}.

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
@
The {\tt \#} character is a comment in R, and all text after that on the 
current line is ignored.

Once the package is installed (one time only), it can be loaded by running the command:
<<load_mosaic,eval=FALSE>>=
require(mosaic)
@
This
needs to be done once per session.

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)
@

The specific goal of this document is to demonstrate how to replicate the
analysis described in Chapter 10: Inference for Regression.

\section{Simple Linear Regression}

The first example from Chapter 10 is 10.4 (page 566), which assesses fuel economy for 60 cars. 

<<>>=
fuel = read.csv("http://math.smith.edu/ips6eR/ch10/eg10_001.csv")
head(fuel)
@

In this case we are building a model for $MPG$ as a function of $LOGMPG$, which is a precomputed variable. Output similar to that shown in Figure 10.5 can be produced by applying the {\tt summary()} command to an {\tt lm} object.

<<>>=
fm1 = lm(MPG ~ LOGMPH, data=fuel)
summary(fm1)
@

Note that R can compute the same model without using the precomputed variables, by applying the {\tt log()} function to the $MPH$ variables on-the-fly.

<<>>=
fm1a = lm(MPG ~ log(MPH), data=fuel)
summary(fm1a)
@

Like other statistical software packages, R performs a $t$-test for the null hypothesis that $\beta_i = 0$ for all coefficients $\beta_i$ present in the model. The third column of the {\tt summary()} output (labeled {\tt t value}) gives the $t$-statistic, and the fourth column gives the corresponding $p$-value. Confidence intervals can be retrieved using the {\tt confint()} command, which by default returns a 95\% confidence interval.

<<>>=
confint(fm1)
@

Confidence intervals for the mean response, as well as prediction intervals for future observations, can be plotted using the {\tt panel.lmbands} argument to {\tt xyplot()}. The following plot is a mashup of Figure 10.9 (page 573) and Figure 10.10 (page 575). 

<<>>=
xyplot(MPG ~ log(MPH), panel=panel.lmbands, data=fuel)
@

To retrieve the actual values, we can apply the {\tt predict()} command to our regression model object, and specify whether we want confindence intervals or prediction intervals.

<<>>=
# only show the first six rows for clarity
head(predict(fm1, interval="confidence"))
# only show the first six rows for clarity
head(predict(fm1, interval="predict"))
@


\section{More Detail about Simple Linear Regression}

\subsection{The ANOVA $F$-test}

An ANOVA table similar to the one shown in Figure 10.12 (page 583) can be produced by applying the {\tt anova()} command to a regression model object.

<<>>=
anova(fm1)
@

\subsection{Inference for Correlation}

We can test for zero correlation using the {\tt cor.test()} command. In Example 10.22, a $t$-test for non-zero correlation is conducted between the $MPG$ and $LOGMPH$ of 60 cars

<<>>=
with(fuel, cor.test(MPG, LOGMPH))
@


\end{document}