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  pdftitle = {A Handbook of Statistical Analyses Using R},
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  pdfauthor = {Brian S. Everitt and Torsten Hothorn},
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\begin{document}

%% Title page

\title{A Handbook of Statistical Analyses Using \R}

\author{Brian S. Everitt and Torsten Hothorn}

\maketitle
%%\VignetteIndexEntry{Chapter Simple Inference}
\setcounter{chapter}{1}


\SweaveOpts{prefix.string=figures/HSAUR,eps=FALSE,keep.source=TRUE} 

<<setup, echo = FALSE, results = hide>>=
rm(list = ls())
s <- search()[-1]
s <- s[-match(c("package:base", "package:stats", "package:graphics", "package:grDevices",
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rm(HSAURpkg)
a <- Sys.setlocale("LC_ALL", "C")
book <- TRUE
refs <- cbind(c("AItR", "SI", "CI", "ANOVA", "MLR", "GLM", 
                "DE", "RP", "SA", "ALDI", "ALDII", "MA", "PCA", 
                "MDS", "CA"), 1:15)
ch <- function(x, book = TRUE) {
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@

\pagestyle{headings}

\chapter[Simple Inference]{Simple Inference: Guessing Lengths, 
                           Wave Energy, Water Hardness, Piston Rings, 
                           and Rearrests of Juveniles \label{SI}}

\section{Introduction}

<<SI-setup, echo = FALSE, results = hide>>=
library("vcd")
@



\section{Statistical Tests}


\section{Analysis Using \R{}}

\subsection{Estimating the Width of a Room}

The data shown in Table~\ref{SI-rw-tab} are available as \Robject{roomwidth}
\Rclass{data.frame} from the \Rpackage{HSAUR} package and can be attached by
using
<<SI-roomwidth-data, echo = TRUE>>=
data("roomwidth", package = "HSAUR")
@
If we convert the estimates of the room width in metres into feet by 
multiplying each by $3.28$ then we would like to test the hypothesis that the mean
of the population of `metre' estimates is equal to the mean  %'
of the population of `feet' estimates. We shall do this first %'
by using an independent samples $t$-test, but first it is good
practice to, informally at least, check the normality and equal
variance assumptions. Here we can use a combination of numerical
and graphical approaches. The first step should be to 
convert the metre estimates into feet, i.e., by a factor
<<SI-roomwidth-convert, echo = TRUE>>=
convert <- ifelse(roomwidth$unit == "feet", 1, 3.28)
@
which equals one for all feet measurements and $3.28$ for the measurements
in metres. Now, we get the usual summary statistics and standard deviations 
of each set of estimates using 
<<SI-roomwidth-summary, echo = TRUE>>=
tapply(roomwidth$width * convert, roomwidth$unit, summary)
tapply(roomwidth$width * convert, roomwidth$unit, sd)
@
where \Rcmd{tapply} applies \Rcmd{summary}, or \Rcmd{sd}, to the converted
widths for both groups of measurements given by \Robject{roomwidth\$unit}.
A boxplot of each set of estimates
might be useful and is depicted in Figure~\ref{SI-rw-bxp}. The
\Rcmd{layout} function (line 1 in Figure~\ref{SI-rw-bxp}) divides the
plotting area in three parts. The \Rcmd{boxplot} function produces a boxplot
in the upper part and the two \Rcmd{qqnorm} statements in lines 8 and 11
set up the normal probability plots that can be
used to assess the normality assumption of the $t$-test.

\numberSinput
\begin{figure}
\begin{center}
<<SI-roomwidth-boxplot, echo = TRUE, fig = TRUE, height = 7>>=
layout(matrix(c(1,2,1,3), nrow = 2, ncol = 2, byrow = FALSE))
boxplot(I(width * convert) ~ unit, data = roomwidth,
        ylab = "Estimated width (feet)",
        varwidth = TRUE, names = c("Estimates in feet",
        "Estimates in metres (converted to feet)"))
feet <- roomwidth$unit == "feet"
qqnorm(roomwidth$width[feet], 
       ylab = "Estimated width (feet)")
qqline(roomwidth$width[feet])
qqnorm(roomwidth$width[!feet], 
       ylab = "Estimated width (metres)")
qqline(roomwidth$width[!feet])
@
\caption{Boxplots of estimates of width of room in feet and metres (after
conversion to feet) and normal probability plots of estimates of room width
made in feet and in metres.  \label{SI-rw-bxp}}
\end{center}
\end{figure}
\rawSinput

The boxplots indicate that both sets of estimates contain a number
of outliers and also that the estimates made in metres are skewed and more
variable than those made in feet, a point underlined by the numerical
summary statistics above. Both normal probability plots depart from
linearity, suggesting that the distributions of both sets of estimates
are not normal. The presence of outliers, the apparently different
variances and the evidence of non-normality all suggest caution
in applying the $t$-test, but for the moment we shall apply the 
usual version of the test using the \Rcmd{t.test} function in \R{}.

The two-sample test problem is specified by a \Rclass{formula}, here by 
<<SI-roomwidth-formula, echo = FALSE>>=
I(width * convert) ~ unit
@
where the response, \Robject{width}, on
the left hand side needs to be converted first and, because the star has a
special meaning in formulae as will be explained in 
Chapter~4,
the
conversion needs to be embedded by \texttt{I}. The factor \Robject{unit} on
the right hand side specifies the two groups to be compared.

<<SI-roomwidth-tt-T-hide, echo = FALSE, results = hide>>=
tt <- t.test(I(width * convert) ~ unit, data = roomwidth, 
             var.equal = TRUE)
@


\renewcommand{\nextcaption}{\R{} output of the independent samples $t$-test for the
                            \Robject{roomwidth} data. \label{SI-roomwidth-tt-fig}}
\SchunkLabel
<<SI-roomwidth-tt-T, echo = TRUE>>=
t.test(I(width * convert) ~ unit, data = roomwidth, 
       var.equal = TRUE)
@
\SchunkRaw

\renewcommand{\nextcaption}{\R{} output of the independent samples Welch test for the    
                            \Robject{roomwidth} data. \label{SI-roomwidth-welch-fig}}
\SchunkLabel
<<SI-roomwidth-tt-F, echo = TRUE>>=
t.test(I(width * convert) ~ unit, data = roomwidth, 
       var.equal = FALSE)
@
\SchunkRaw


\renewcommand{\nextcaption}{\R{} output of the Wilcoxon rank sum test for the 
                            \Robject{roomwidth} data. \label{SI-roomwidth-wilcox-fig}}
\SchunkLabel
<<SI-roomwidth-wt, echo = TRUE>>=
wilcox.test(I(width * convert) ~ unit, data = roomwidth, 
            conf.int = TRUE)
@
\SchunkRaw
<<SI-roomwidth-wt-hide, echo = FALSE>>=
pwt <- round(wilcox.test(I(width * convert) ~ unit, data =
             roomwidth)$p.value, 3)
@


\subsection{Wave Energy Device Mooring}

The data from Table~\ref{SI-m-tab} are available as \Rclass{data.frame} \Robject{waves} 
<<SI-waves-data, echo = TRUE>>=
data("waves", package = "HSAUR")
@
and requires the use of a matched pairs
$t$-test to answer the question of interest. 
This test assumes that the differences between the matched
observations have a normal distribution so we can begin by checking
this assumption by constructing a boxplot and a normal probability 
plot -- see Figure~\ref{SI-w-bxp}.
\begin{figure}
\begin{center}
<<SI-wavese-boxplot, echo = TRUE, fig = TRUE, height = 4>>=
mooringdiff <- waves$method1 - waves$method2
layout(matrix(1:2, ncol = 2))
boxplot(mooringdiff, ylab = "Differences (Newton metres)", 
        main = "Boxplot")
abline(h = 0, lty = 2)
qqnorm(mooringdiff, ylab = "Differences (Newton metres)")
qqline(mooringdiff)
@
\caption{Boxplot and normal probability plot for differences between the two
mooring methods. \label{SI-w-bxp}}
\end{center}
\end{figure}

\renewcommand{\nextcaption}{\R{} output of the paired $t$-test for the
                            \Robject{waves} data. \label{SI-waves-tt-fig}}
\SchunkLabel
<<SI-waves-tt, echo = TRUE>>=
t.test(mooringdiff)
@
\SchunkRaw

<<SI-waves-wt, echo = FALSE, results = hide>>=
pwt <- round(wilcox.test(mooringdiff)$p.value, 3)
@


\renewcommand{\nextcaption}{\R{} output of the Wilcoxon signed rank test for the
                            \Robject{waves} data. \label{SI-waves-ws-fig}}
\SchunkLabel
<<SI-waves-wt, echo = TRUE>>=
wilcox.test(mooringdiff)
@
\SchunkRaw

\subsection{Mortality and Water Hardness}

There is a wide range of analyses we could apply to the
data in Table~\ref{SI-w-tab} available from
<<SI-water-data, echo = TRUE>>=
data("water", package = "HSAUR")
@
But to begin we will construct a scatterplot
of the data enhanced somewhat by the addition of information
about the marginal distributions of water hardness (calcium concentration)
and mortality, and by adding the estimated linear regression
fit (see 
Chapter~5)
for mortality on hardness. 
The plot and the required \R{} code is given along with 
Figure~\ref{SI-water-sp}. In line 1 of Figure~\ref{SI-water-sp}, we divide
the plotting region into four areas of different size. The scatterplot (line
3) uses a plotting symbol depending on the location of the city (by
the \Rarg{pch} argument), a legend for the location is added in line 6.
We add a least squares fit (see 
Chapter~5)
to the scatterplot and,
finally, depict the marginal distributions by means of a boxplot and a
histogram. The scatterplot
shows that as hardness increases mortality decreases, and the 
histogram for the water hardness shows it has a rather skewed distribution.


\numberSinput
\begin{figure}
\begin{center}
<<SI-water-plot, echo = TRUE, fig = TRUE>>=
nf <- layout(matrix(c(2, 0, 1, 3), 2, 2, byrow = TRUE),
             c(2, 1), c(1, 2), TRUE)
psymb <- as.numeric(water$location)
plot(mortality ~ hardness, data = water, pch = psymb)
abline(lm(mortality ~ hardness, data = water))
legend("topright", legend = levels(water$location), 
       pch = c(1,2), bty = "n")
hist(water$hardness)
boxplot(water$mortality)
@
\caption{Enhanced scatterplot of water hardness and mortality, showing both
         the joint and the marginal distributions and, in addition, 
         the location of the city by different plotting symbols.
         \label{SI-water-sp}}
\end{center}
\end{figure}
\rawSinput


\renewcommand{\nextcaption}{\R{} output of Pearsons' correlation coefficient %'
                            for the \Robject{water} data. \label{SI-water-c-fig}}
\SchunkLabel
<<SI-water-cor, echo = TRUE>>=
cor.test(~ mortality + hardness, data = water)
@
\SchunkRaw
<<SI-water-cor, echo = FALSE, results = hide>>=
cr <- round(cor.test(~ mortality + hardness, data = water)$estimate, 3)
@


\subsection{Piston-ring Failures}

<<SI-pistonrings-chisq-hide, echo = FALSE, results = hide>>=                                      
chisqt <- chisq.test(pistonrings)
@


\renewcommand{\nextcaption}{\R{} output of the chi-squared test 
                            for the \Robject{pistonrings} data. \label{SI-pr-x2-fig}}
\SchunkLabel
<<SI-pistonrings-chisq, echo = TRUE>>=
data("pistonrings", package = "HSAUR")
chisq.test(pistonrings)
@
\SchunkRaw

Rather than looking  
at the simple differences of observed and expected values for     
each cell which would be unsatisfactory since a difference of     
fixed size is clearly more important for smaller samples, it      
is preferable to consider a \stress{standardised residual}
\index{Standardised residual, for chi-squared tests}
given
by dividing the observed minus expected difference by the square 
root of the appropriate expected value. The $X^2$ statistic 
for assessing independence is simply the sum, over all the cells 
in the table, of the squares of these terms. We can find these   
values extracting the \Robject{residuals} element of the object returned by
the \Rcmd{chisq.test} function
<<SI-pistonrings-resid, echo = TRUE>>=
chisq.test(pistonrings)$residuals
@

A graphical representation of these residuals is called \stress{association
plot}
\index{Association plot}
and is available via the \Rcmd{assoc} function from package
\Rpackage{vcd} \citep{PKG:vcd} applied to the contingency table of the two
categorical variables. Figure \ref{SI-assoc-plot} depicts the residuals for
the piston ring data. The deviations from independence are largest for
C1 and C4 compressors in the centre and south leg.

\begin{figure}  
\begin{center}
<<SI-assoc-plot, echo = TRUE, fig = TRUE>>= 
library("vcd")
assoc(pistonrings)
@
\caption{Association plot of the residuals for the \Robject{pistonrings} data.
         \label{SI-assoc-plot}}
\end{center}
\end{figure}



\subsection{Rearrests of Juveniles}

The data in Table~\ref{SI-r-tab} are available as \Rclass{table} object
via
<<SI-rearrests-data, echo = TRUE>>=
data("rearrests", package = "HSAUR")
rearrests
@
<<SI-rearrests-mcnemar, echo = FALSE, results = hide>>=
mcs <- round(mcnemar.test(rearrests, correct = FALSE)$statistic, 3)
@
and in \Robject{rearrests} the counts in the four cells refer
to the matched pairs of subjects; for example, in $\Sexpr{rearrests[1,1]}$ 
pairs both   
members of the pair were rearrested. Here we need to use McNemar's  %'
test to assess whether rearrest is associated with type of court   
where the juvenile was tried. We can use the \R{} function \Rcmd{mcnemar.test}.
The test statistic shown in Figure~\ref{SI-ra-mc-fig} 
is $\Sexpr{mcs}$ with a single degree of freedom -- the
associated $p$-value is extremely small and there is strong evidence    
that type of court and the probability of rearrest are related.        
It appears that trial at a juvenile court is less likely to result     
in rearrest (see Exercise 2.4).
An exact version of McNemar's test %%'
can be obtained by testing whether $b$ and $c$ are equal using a binomial
test (see Figure \ref{SI-ra-mcbin-fig}).

\renewcommand{\nextcaption}{\R{} output of McNemar's test %'
                            for the \Robject{rearrests} data. \label{SI-ra-mc-fig}}
\SchunkLabel
<<SI-arrests-mcnemar, echo = TRUE>>=
mcnemar.test(rearrests, correct = FALSE)
@
\SchunkRaw


\renewcommand{\nextcaption}{\R{} output of an exact version of McNemar's test %'
                            for the \Robject{rearrests} data computed via a
                            binomal test. \label{SI-ra-mcbin-fig}}
\SchunkLabel
<<SI-arrests-binom, echo = TRUE>>=
binom.test(rearrests[2], n = sum(rearrests[c(2,3)]))
@
\SchunkRaw



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\end{document}
